Essays on Signaling and Social Networks A

ESSAYS ON SIGNALING AND SOCIAL NETWORKS

A DISSERTATION SUBMITTED TO THE DEPARTMENT OF ECONOMICS AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

Tom´asRodr´ıguezBarraquer May 2011

This work is licensed under a Creative Commons Attribution- Noncommercial 3.0 United States License. http://creativecommons.org/licenses/by-nc/3.0/us/

This dissertation is online at: http://purl.stanford.edu/wj208kq6898

ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Matthew Jackson, Primary Adviser

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Kyle Bagwell

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Giacomo DeGiorgi

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Muriel Niederle

Approved for the Stanford University Committee on Graduate Studies. Patricia J. Gumport, Vice Provost Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file in University Archives.

iii Approved for the University Committee on Graduate Studies

iv Preface

Over the last few decades some analytic tools intensely used by economics have produced useful insights in topics formerly in the exclusive reach of other social sciences. In particular game theory, justifiable from either a multi-person decision theoretic perspective or from an evolutionary one, often serves as a generous yet sufficiently tight framework for interdisciplinary dialogue. The three essays in this collection apply game theory to answer questions with some aspects of economic interest. The three of them have in common that they are related to topics to which other social sciences, specially sociology, have made significant contributions. While working within economics I have attempted to use constructively and faithfully some of these ideas. Chapter 1, coauthored with Xu Tan, studies situations in which a set of agents take actions in order to convey private information to an observing third party which then assigns a set of prizes based on its beliefs about the ranking of the agents in terms of the unobservable characteristic. These situations were first studied using game theoretic frameworks by Spence and Akerlof in the early seventies, but some of the key insights date back to the foundational work of Veblen. In our analysis we focus on the competitive aspect of some of these situations and cast signals as random variables whose distributions are determined by the underlying unobservable characteristics. Under this formulation different signals have inherent meanings, preceding any stable conventions that may be established. We use these prior meanings to propose an equilibrium selection criterion, which significantly refines the very large set of sequential equilibria. We apply our framework to examine in detail an argument recently espoused in the philosophy and sociology of science literatures to explain

v what some authors view as the surprising solitude of String Theory as a mainstream contender to a Theory of Everything. The argument is that this preponderance of String Theory may in part be due to the need of scientists in the early stages of their careers to signal their research ability amidst rising competition. While the standard signaling frameworks suffice to account for the existence of multiple stable patterns of behavior of varying efficiency and informativeness, they do not shed light on the possible role of competition in selecting among these equilibria. The framework that we propose contributes to understanding the prevalence of many forms of conspicuous behavior. In Chapter 2, coauthored with Matthew O. Jackson and Xu Tan, we study the structure of social networks that allow individuals to cooperate with one another in settings in which behavior is non-contractible, by supporting schemes of credible ostracism of deviators. There is a significant literature on the subject of cooperation in social networks focusing on the role of the network in transmitting the information necessary for the timely punishment of deviators, and deriving properties of network structures able to sustain cooperation from that perspective. Ours is one of the first efforts to understand the network restrictions that emerge purely from the credibility of ostracism, carefully considering the implications that the dissolution of any given relationship may have over the sustainability of other relations in the community. More concretely we study a model of favor exchange in small communities in which each period an opportunity for a socially efficient exchange of a favor arises among a pair of agents randomly chosen among all those that are linked in the network of relationships that is in place. Agents have complete and perfect information, and we work under the crucial assumption that once a link is deleted it cannot be resuscitated. We define renegotiation-proof networks as networks in which every agent can be compelled not to misbehave along any of his links by the fear of nonnegotiable automatic prosocial retaliation from the offended party plus a credible threat by all or part of his other neighbors to permanently ostracize him. A threat is credible if the link deletions that it entails lead to a network which is itself renegotiation-proof, and importantly, does not expect the community to be able to commit not to forgive itself, and instead implement some other intermediate renegotiation-proof network

vi involving the deletion of a strict subset of the links that the full punishment con- templates. We provide a full characterization of the collection of renegotiation-proof networks assuming homogeneous agents and relationships. One issue with renegotiation proofness is that while the threats upon which the networks stand are credible in the very strong sense just discussed, they may require the deletion of links that do not involve the offending party, and thus be very costly to parties completely unrelated to the behavior being penalized. This observation leads us to consider robust networks, a subclass of renegotiation proof networks that can be supported by punishments that only contemplate the disappearance of relationships involving the offending agent and leading to networks within this same subclass. We fully characterize the collection of robust networks assuming homogeneity and a variety of forms of heterogeneity. In particular we show that a property of all renegotiation proof networks in which no relationship is self-sustainable is that relationships must be supported, that is, all pairs of linked agents (friends) must have at least one friend in common. One implication of this property which illustrates well its reach is that it rules out social structures which include relationship that are not self sustaining, and which bridge otherwise disjoint clans (i.e. structures which can do no better than rely on social norms which depend upon Capulets reproaching Capulets and Montagues reproaching Montagues). The paper ends by exploring the occurrence of support in the networks corresponding to a large variety of relationships in 75 Indian Villages. We simultaneously explore the occurrence of clustering which is the much stronger network structure most commonly associated to cooperation by the prior literature. Our main finding is that while support tends to be very high in networks that can be classified as reflecting favor exchange among a variety of other social networks, clustering tends to be indiscriminately low in all the different networks. In Chapter 3 I study the sets of Pure Strategy Nash equilibria of a variety of binary games of social influence under complete information. In a game of social influence agents simultaneously choose one of two possible strategies(to be inactive or be active), and the optimal choice depends on the strategies of the agents in their social environment. Different social environments and assumptions on the way in which they influence the behavior of the agents lead to different classes of games of

vii varying degrees of tractability. In any such game an equilibrium can be described by the set of agents that are active, and the full set of equilibria can be thus represented as a collection of subsets of the set of agents. I build the analysis of each of the classes of games that I consider around the question: What collections of sets are expressible as the set of equilibria of some game in the class? I am able to provide precise answers to these questions in some of the classes studied, and in other cases just some pointers. The paper begins by studying general games of social influence displaying strategic complements and then focuses on various subclasses that arise from imposing restrictions on the best response behavior of the agents stemming from social structure: Simple games in which each agent is influenced by only one core group of other agents. Nested games in which each agent can only be influenced by agents who are in turn influenced by other agents that influence him. Hierarchical games in which the agents can be embedded in a hierarchy that respects the influence structure. Clan-like games, which are simple games with the additional property that the influence is always mutual. And finally Games of thresholds, in which the way in which agents influence each other can be represented by a network.

viii Acknowledgements

I am forever indebted to Matt Jackson for his guidance and great generosity. It has been extremely pleasant and inspiring to have him as a teacher and a role model. I also greatly beneﬁted from his ability to get people together and spawn very constructive group dynamics. The weekly Jackson blackboard group is a very reliable, fun and fertile idea exchange forum and it allowed me to meet and experience some amazing people. Among them I ought to mention Doru Cojoc, Matthew Elliot, Ben Golub, Carlos Lever and Yair Levine, who provided very useful feedback on my research at various points. I also thank Matt for introducing me to Xu Tan. Xu has been my main peer-at-work and coauthor over the last two years. Working with her has been very enjoyable and her help and teachings, enormous.

I am deeply grateful to Frank Wolak. With extreme patience he gave me invalu- able instruction, advice and countless hours of mentorship. Thanks are also due to Giacomo De Giorgi and Muriel Niederle for their advice and support and to Kyle Bagwell for his generous participation in my reading committee. I also thank Pro- fessors Marcelo Clerici-Arias, Paul David, Klaus Desmet, Peter Hansen, Han Hong, Aprajit Mahajan, Monika Piazzesi, Sudipta Sarangi, Ilya Segal and Faye Steiner for their advice, their example and nourishment.

There are other people whose contributions to this dissertation were essential yet can’t be pinponted. Of these, I ought to mention Andrea Pozzi, Rodrigo Barros, Juan Escobar, Alejandrina Salcedo and Diego Sasson with whom we constructed a wonderful organic work-play friendship-fellowship, Paloma Valencia for all her support, and Alejandro Feged for his omnipresence, innumerable lessons and great patience.

ix I am very grateful to Trudy Haley, Patricia Luna, Susie Madsen, Susan Taylor and Marilyn Watson for their various forms of support during my graduate studies.

Aside from those whom I already mentioned, during these Californian years I had the chance to spend time with some wonderful people. I offer my heartfelt gratitude to Shuaib Ahmed, Andrés Angel,´ Carla Basaldúa,MalúBeltrán,Theodora Bourni, Kazim Buyukboduk, Gabriela Calderón,Juan Cristobal Cerrillo, Cristina Correa, Juana Dávila,Adriana de los Rios, Coni Escobar, Clayton Featherstone, Manuj Garg, Paula Giraldo, Natalia Gómez,Felipe Gómez, David Gutierrez, Zhuo Huang, Elizabeth Husserl, Oliver Kaplan, Katja Kaufmann, Everaldo Lamprea, Tim Larson, Brian Lee, Patricia Macri-Lassus, David Malpica, Jess McNally, Ciro Menotti, Isidora Milin, Pedro Miranda, Andrew Nigrinis, Noemi Pace, Jose Perea, Marcel Priebsch, Nora Raggio, Anat Razon, Jenna Rice, Camilo Rivera, Mariel Saez, Ulrich Schlecht, Hannah Teichmann, Justin Wong, Takuro Yamashita and Jacob Zasada.

Lastly and most importantly I dedicate this thesis to my parents Manuel and Carmen and to my sister Isabel.

x Contents

Preface v

Acknowledgements ix

1 Conspicuous Scholarship: Competitive Unraveling in the Choice of Research Topics 1 1.1 Introduction ...... 1 1.1.1 Related Literature ...... 6 1.2 A Simple Example ...... 9 1.2.1 Productive Efficiency and Screening Efficiency ...... 13 1.3 A More General Model ...... 15 1.3.1 What is it that Explicit Competitions Provide? ...... 16 1.3.2 The Game ...... 18 1.3.3 Competitive Unraveling ...... 21 1.3.4 Some Stylized Facts About Particle Physics in the U.S. . . . . 25 1.4 Uniqueness of Conspicuous Equilibria ...... 26 1.4.1 Fanning Out ...... 27 1.4.2 Increasing Differences ...... 29 1.5 Efficiency ...... 32 1.6 Conclusion ...... 35 1.7 Appendix: Omitted Proofs ...... 37

2 Social Capital and Social Quilts: Network Patterns of Favor Ex- change 41

xi 2.1 Introduction ...... 41 2.1.1 Related Literature ...... 46 2.2 A Model of Favor Exchange ...... 48 2.2.1 Networks, Favors, and Payoffs ...... 48 2.2.2 The Game ...... 50 2.2.3 Equilibrium ...... 53 2.3 Characterizing Renegotiation-Proof Networks ...... 60 2.3.1 Critical Networks and Renegotiation-Proofness ...... 61 2.3.2 Transitively Critical Networks ...... 64 2.4 Robustness ...... 66 2.4.1 Robustness Against Social Contagion ...... 67 2.4.2 The Number of Robust Networks and Subgame Perfect Networks 70 2.5 Asymmetric Payoffs ...... 71 2.5.1 A Special Heterogeneous Case ...... 72 2.6 Support vs. Clustering in Favor Networks in Rural India ...... 73 2.6.1 Description of the Data ...... 74 2.6.2 Measuring Support and Clustering ...... 76 2.6.3 Support in the Data ...... 78 2.6.4 Comparing Support to Clustering ...... 82 2.6.5 Comparing Support in Different Sorts of Relationships . . . . 83 2.6.6 Observed Support and Support in a Random Network . . . . . 85 2.6.7 Support, and Link, Household and Individual Characteristics . 86 2.7 Conclusion ...... 91 2.7.1 Information and Robustness ...... 92 2.8 Appendix: Proofs of Results ...... 92

3 The Structure of the Sets of Pure Strategy Nash Equilibria in Binary Games of Social Influence 101 3.1 Introduction ...... 101 3.2 General Games of Social Influence ...... 104 3.3 Increasing Influence Structures ...... 107

xii 3.4 Simple, Nested, Clan-like and Hierarchical Inﬂuence Structures . . . . 113 3.4.1 Hierarchical Inﬂuence Structures ...... 118 3.5 Structures that Admit a Network Representation ...... 119 3.5.1 A Few Special Cases ...... 122 3.6 Finding All Equilibria and Deciding Expressibility ...... 126

A Social Capital and Social Quilts: Supplementary Results 130 A.1 Critical Networks and Renegotiation-Proofness ...... 130 A.1.1 Critical Networks ...... 130 A.2 Some Renegotiation-Proof Networks ...... 139 A.3 A Special Heterogeneous Case ...... 140 A.4 Weak Robustness ...... 148 A.5 Maximal Equilibria ...... 149 A.5.1 Characterizing Maximal Networks ...... 152 A.5.2 Social Quilts ...... 153 A.5.3 Robustness ...... 154 A.6 Background Statistics on the Indian Village Networks ...... 155 A.6.1 Descriptive Statistics ...... 155 A.6.2 Support in the Data ...... 155 A.6.3 Comparing Support to Clustering ...... 163 A.6.4 Comparing Support in Diﬀerent Sorts of Relationships . . . . 171 A.6.5 Observed Support vs. Support in a Random Network (I) . . . 172 A.6.6 Observed Support vs. Support in a Random Network (II) . . . 174 A.6.7 Bounding Measurement Error ...... 175 A.6.8 Support and Link, Household and Individual Characteristics . 179

Bibliography 185

xiii List of Tables

2.1 The Contents of the Survey ...... 75 2.2 Network Deﬁnitions ...... 77 2.3 Clustering and Support Measures ...... 83 2.4 Comparison of Support Measures ...... 84 2.5 OLS Coeﬃcients for the Village Level Regressions ...... 88 2.6 Support Across Similarity/Dissimilarity Classes: Ratios ...... 89 2.7 Maximum Support Across Similarity/Dissimilarity Classes ...... 90 2.8 Probit Regressions for an Individual’s Support ...... 91

A.1 Non-Isomorphic Renegotiation-Proof and Subgame Perfect Networks 140 A.2 Average Support Measures ...... 171 A.3 Comparison of Support Measures in the All Network ...... 171 A.4 Comparison of Support Measures in the Hed Y F av Network . . . . . 172 A.5 Comparison of Support Measures ...... 172 A.6 Average Support Measures for Observed and Random Networks . . . 173 A.7 Binomial One-Sided Test ...... 175 A.8 Estimated Coeﬃcients of the Ergm ...... 176 A.9 A Closer Look at Figure A.15 ...... 179 A.10 Support by Similarity/Dissimilarity Classes in Education ...... 181 A.11 Support by Similarity/Dissimilarity Classes in Gender ...... 182 A.12 Support by Similarity/Dissimilarity Classes in Age ...... 182 A.13 Support by Similarity/Dissimilarity Classes in Caste ...... 183 A.14 Support by Participation Status in the Micro-Finance Program . . . . 183

xiv A.15 Probit Regression of Support in the All Network of Pairs in the Favors Network (I) ...... 184 A.16 Probit Regression of Support in the All Network of Pairs in the Favors Network (II) ...... 184

xv List of Figures

1.1 A Function p Satisfying pA1q ¡ pA4q ...... 28

2.1 Not an Equilibrium Network ...... 56 2.2 Critical Network with More than m Links per Agent ...... 63 2.3 A Renegotiation Proof, Non-Critical Network ...... 64 2.4 Robust and Non-Robust Renegotiation-Proof Networks ...... 66 2.5 A Union of m-Cliques That is Not an m-Quilt ...... 69 2.6 Support vs. Clustering ...... 72 2.7 The Inverse CDF of Support ...... 79 2.8 The Inverse CDFs of Support of Linked and Unlinked Pairs ...... 81 2.9 The Inverse CDFs of Support and Clustering ...... 82 2.10 Estimated Support Coeﬃcients in the ERGMs ...... 87

3.1 Intended vs. Expressed Lattice when Applying S1) and S2) ...... 110 3.2 A Product Lattice ...... 115 3.3 The Expressive Power Added by Weights...... 125

A.1 Critical Network with Excessive Links ...... 131 A.2 Non Renegotiation-Proof Five Link Network ...... 132 A.3 Non Critical Tree Union of Critical Networks ...... 133 A.4 Non Renegotiation-Proof Tree Union of Critical Networks ...... 134 A.5 A Critical Network with a Bridge ...... 136 A.6 A Union of m-Cliques that is not an m-Quilt ...... 139 A.7 Non Isomorphic Networks for n=1-6, m=2 and m=3 ...... 141

xvi A.8 A Tree Union of Critical Cliques that is Not Robust ...... 145 A.9 A Tree Union of Critical Cliques that is Robust ...... 145 A.10 Centrality Measures ...... 156 A.11 Distribution of the Number of Reported Relationships ...... 162 A.12 Inverse Cumulative Distribution Function of Support in the Villages . 163 A.13 Inverse Cumulative Distribution Function of Support and Clustering . 167 A.14 Support Coeﬃcients in thhe ERGMs for the 75 Villages ...... 178 A.15 Simulated Support as a Function of Measurement Error ...... 180

xvii Chapter 1

Conspicuous Scholarship: Competitive Unraveling in the Choice of Research Topics

1.1 Introduction

Why do scholars spend seemingly disproportionate amounts of time working in projects that systematically yield results that are elegant, technically sound and often beauti- ful, yet in the eyes of others have little use or not many applications? How do scientific research programs persist, often in the midst of significant internal and external questioning of their scientific promise? Why do academic research efforts to answer a given question often focus in a single or a small number of plausible routes instead of simultaneously examining several alternatives? Some sociological overtones in recent rounds of the debate that has surrounded string theory over the last two decades provide an interesting insight that may go a long way in simultaneously answering these three questions. Namely, that the persistence in time of research agendas may have as much to do with signaling and screening within scientific communities as it does with the scientific validity and promise of the theories that they encompass.

String theory arose in the 1960’s and early 1970’s as a theory of a class of subatomic

1 CHAPTER 1. CONSPICUOUS SCHOLARSHIP 2

particles called hadrons. In 1974 Scherk and Schwarz published a paper1 in which they provided a mathematical argument suggesting that the theory could be developed into a theory of gravity. The theory continued to be developed during the 70’s but did not garner much attention beyond the small group of people that were working on it. Four decades later, string theory has come to be highly accepted by the theoretical physics community and is the only theory in progress in mainstream physics widely held to be a candidate Theory of Everything despite the fact that its mathematical support and experimental proof is nowadays fundamentally the same2 as in the 70s (see Zapata (2009) [90]). Moreover, this general embrace of the theory by the physics community is also paralleled by significant enthusiasm outside of academia and extensive media coverage. During he last decade, the process whereby this scientific enterprise rose into the mainstream from the academic backwaters has been the subject of scholarly papers (see for example Hedrich (2007) [51], Schroer (2008) [77], and Zapata (2009) [90]), popular science books3, and several mass media articles4. In their attempts to forge an explanation, a growing number of pieces on the subject have come to address the numerous social mechanisms that may have intervened in the theory’s diffusion.

The current debate surrounding string theory is an example of the way in which no significant contending alternatives may emerge to specific scientific agendas in the midst of substantial external and internal questioning of the social value of their contributions. The current spectrum of skepticism towards string theory is very broad, ranging from the contention that due to its elegance and mathematical beauty it is bound to hold the key elements for a Theory of Everything5, to those who like Roger Penrose (see Penrose (2007) [71]) affirm that it is a case of fashion in science

1J. Scherk and J. H. Schwarz, “Dual Models for Nonhadrons”, Nuclear Physics B, 81 (1974):118- 144. 2This narrative is solely concerned with string theory as a candidate TOE (Theory of Everything). 3The highly controversial books Not Even Wrong: The Failure of String Theory and the Search for Unity in Physical Law and The Trouble with Physics published in 2007 by Peter Woit and Lee Smolin respectively. 4See for example Unstrung an article by Jim Holt published in October 2006 in The New Yorker, or an interview with Roger Penrose published in the September 2009 issue of Discover Magazine. 5“ I can only speak for myself, though I suspect that most others working in this ﬁeld would agree. I believe that we have found the unique mathematical structure that consistently combines quantum mechanics and general relativity. So it must almost certainly be correct.” (Schwarz (1998) [78]) CHAPTER 1. CONSPICUOUS SCHOLARSHIP 3

with little experimental support. Despite the controversy, no significant effort has been devoted to exploring alternative routes, and the majority of graduate students and post-doctoral researchers working in theoretical particle physics continue to train themselves as string theorists. The following quotation, taken from a mass media article authored by a physicist, exemplifies well the growing concern for the lack of alternative theories and suggests that the explanation may lie in the social aspects of the scientific enterprise:

“One reason that only one new theory has blossomed is that graduate students, postdocs and untenured junior faculty interested in speculative areas of mathematical physics beyond the Standard Model are under tremendous pressures. For them, the idea of starting to work on an untested new idea that may very well fail looks a lot like a quick route to professional suicide. So some people who do not believe in string theory work on it anyway. They may be intimidated by the fact that certain leading string theorists are undeniably geniuses. Another motivation is the natural desire to maintain a job, get grants, go to conferences and generally have an intellectual community in which to participate. Hence, few stray very far from the main line of inquiry.” -Peter Woit, American Scientist6 2002, [87]

In this paper we examine in detail the possibility that this phenomenon arises from the need of young scientists to signal their research ability in order to make progress in the academic profession. With that purpose in mind we formulate a signaling game which captures in a stylized fashion scholars’ choice of research projects during the early stages of their academic careers. One of the exogenous parameters of the game is the intensity of competition as given by the number of agents that are competing for saliency. We show that under a plausible set of assumptions and an equilibrium selection criterion, the only patterns of behavior that can be sustained as a sequential equilibria of the game at all possible levels of competition, involve research projects which are highly conspicuous in the sense that they discriminate sharply between diﬀerent research abilities, regardless of the social value that they generate.

6Peter Woit’s book and general posture regarding string theory is highly controversial and has inspired both passionate animosity and support from members of the theoretical physics community. We choose this passage because it illustrates very concisely the major points common to the many scattered observations on the social mechanisms explaining the uncontested prevalence of string theory. CHAPTER 1. CONSPICUOUS SCHOLARSHIP 4

The gist of the argument is that hiring universities and tenure-review committees are concerned with the quality of a candidate’s past and current research output to the extent that it predicts her future productivity. This preoccupation implies that any stable pattern of behavior of individual academics in the early stages of their careers should strengthen the expectation of future productivity that it induces, given the actions of all other candidates and society’s signal interpretive scheme7. We show that if low-yield fields of research make the differences between researchers’ unobservable skills more evident, and provided that society’s beliefs consistently acknowledge this to be the case in and out of the equilibrium path, then the only equilibria that persist amidst rising competition involve researchers working in relatively low yield topics. This implies an irreducible tension between society’s need to screen researchers at different stages of their careers for the sake of future productivity, and their productivity in the periods before and up to those screening instances. The overall significance of the resulting inefficiency depends on the amount of misalignment between highly productive research activities and research-talent discriminating ones, and on the length of the career stages that precede the instances of screening.

More generally the model can also be applied to understand irreducible ineﬃ- ciencies in on-the job screening mechanisms, in contexts in which the actions that discriminate the most between diﬀerent research abilities are not simultaneously the most productive ones.

The fact that in settings of asymmetric information in which actions convey information there can be multiple equilibria of various degrees of informativeness and efficiency has been well established since Spence (1973) [82] and Akerlof (1976) [3]. There is also a large literature which relying on Kreps and Wilson’s concept of sequential equilibrium studies this multiplicity and proposes refinements (See for example Kohlberg and Mertens (1986) [59], Cho and Kreps (1987) [24] , Banks and Sobel (1987) [8] or Mailath et al. (1993) [64]). The argument supporting the beliefs restriction that we propose for the game that we analyze is simply that they are focal. This greater focality is very specific to a narrow class of signaling games in which

7Formally, the equilibrium path and out of equilibrium path beliefs. CHAPTER 1. CONSPICUOUS SCHOLARSHIP 5

agents choose their signaling technologies, and these vary in their ability to express unobservable characteristics in an objective sense. The resulting equilibrium selection criterion is not contemplated by the sequential equilibrium reﬁnements literature.

Our research contributes to the literature in the following ways:

• We provide a simple model of the choice of research topics in the early stages of academic careers, capturing the tension that academic institutions face between providing the appropriate incentives for the production of research of high social value in the present, and screening researchers in order to be able to better allocate them to maximize the value of future research endeavors.

• We contribute to the growing literature examining the role of incentives in the pursuit of knowledge and provide a setting in which these robustly give rise to ineﬃciencies.

• We propose some plausible equilibrium selection criteria which signiﬁcantly re- duce the set of equilibria in a family of signaling games that are appropriate for various applications in which agents must choose from a set of signaling technologies, which vary in their ability to identify some unobservable characteristics. Moreover, the idea behind our equilibrium selection criterion may shed light on other models of conspicuous behavior. The standard renderings of conspicuous behavior that build upon formal signaling frameworks have the property that conspicuousness arises in a small subset of a large variety of equilibria. To our knowledge no general arguments attempt to address whether as a prediction conspicuousness is more or less plausible than its alternatives in a well deﬁned sense.

The paper is structured as follows. In Section 1.2 we introduce our main arguments using a very simple example. In this example agents with private information about their unobservable research abilities choose their research topics from one of two research arenas. We show that the equilibrium research choices display competitive unraveling, that is, they become more conspicuous as competition increases. The driving mechanism behind the result is that due to the inherent properties of CHAPTER 1. CONSPICUOUS SCHOLARSHIP 6

research topics, agents are always able to use more difficult topics in order to screen themselves. In this setting, society does not have much freedom to sustain conventional patterns of behavior because the performance of different types of agents in the more difficult topics is exogenously pinned down. In Section 1.3 we consider a general environment which in contrast to the example discussed in Section 1.2 allows for a broader class of research arenas. In this richer setting there are also exogenous differences between the performance of agents with different research abilities, but the information transmitted by different outcomes can always be overridden by the information conveyed in equilibrium by the choice of research arenas itself through the dependence of strategies on the agents’ types. This feature of the game, as ac- knowledged by the concept of sequential equilibrium gives rise to multiple equilibria. We show that we can recuperate the competitive unraveling result of Section 1.2 by restricting attention to the set of sequential equilibria in which success in more difficult research projects on or off-the equilibrium path is more revealing of having a high research ability than success in less difficult projects, as would be consistent with static maximum likelihood inference. We close Section 1.3 by visiting some stylized facts about the physics job market that are suggestive of very stringent competition, which is the main condition behind the unraveling-towards-conspicuousness result. In Section 1.4 we provide some additional conditions and a further equilibrium selection criterion which imply that in all equilibria, all agent types select research topics of the same level of difficulty and also a stronger version of the result of competitive unraveling towards conspicuousness. We close the paper with some general remarks about efficiency in Section 1.5.

1.1.1 Related Literature

There is a growing philosophy of science and economics literature which uses methods from economic analysis to address various aspects of the pursuit of knowledge and the scientiﬁc enterprise, broadly encompassed by the label Economics of Scientiﬁc Knowledge (see Goldman and Shaked (1990) [47], Kitcher (1993) [58], David (1994) [30], Davis (1997) [33], David (1998a) [31], David (1998b) [32], Goldman (1999) [46], CHAPTER 1. CONSPICUOUS SCHOLARSHIP 7

Brock and Durlauf (1999) [18] and Mandler (2009) [65]). Zamora Bonilla (2005) [89] oﬀers a comprehensive survey and contextualizes the view of the scientiﬁc enterprise as being driven by fallible self-regarding participants, in contrast to the more traditional perspective of the philosophy of science in which the pursuit of knowledge is mainly conducted by truth-seekers.

Goldman and Shaked (1990) [47] analyzes the role of credit-seeking motives in the pursuit of knowledge. The authors formulate a model in which scholars choose among various possible scientific activities in order to maximize the credit that they get from their peers. Credit in turn, is granted to a scholar by the scientific community to the extent that her actions succeed in modifying its beliefs regarding what the true state of the world is. They show that under certain assumptions on the nature of the scientific acts that are available to scholars, the credit-seeking motive fosters socially efficient truth acquisition. The main difference between Goldman and Shaked’s paper and ours is that in our model the key element in society’s impression of a scholar is what it can infer about her unobservable research ability. In their paper what matters is the extent to which her acts are able to shape other scientists’ beliefs about the object of scientific interest. In a similar vein to ours, Brock and Durlauf (1999) [18] try to understand the entrenchment and resilience of long-standing scientific theories and the nonlinear dynamics observed in processes of assessment and adoption of newer ones. They show that these dynamics can be explained by scientists’ willingness to conform to predominant views of the scientific community. Mandler (2009) [65] in turn addresses the underinvestment of scientists in new fields of research. In his framework, new avenues are riskier and tend to yield low returns to pioneers, but have positive externalities as they can give rise to very valuable follow-up work.

A second strand of the literature which addresses issues related to our questions is the one comprised by articles in economics and sociology attempting to understand the institution of tenure and measure its effects over academic output in different stages of scholars’ careers in various disciplines. This literature offers several explanations for the structure of academic careers: The first one is that tenure provides the right amount of job security that scholars need in order to take the risks that CHAPTER 1. CONSPICUOUS SCHOLARSHIP 8

high quality research often entails. The second family of explanations exemplified by Charmichael (1988) [21] focuses on tenure’s role in allowing senior faculty to make optimal hiring decisions, by shielding them from otherwise undesirable competition. Faria and Monteiro (2008) [37] argue that the probationary period in academics’ careers serves the crucial role of building good habits, as seen by the fact that many academics continue to be productive once they obtain tenure. It is also seen as mainly providing an extended observation period that serves to screen agents for a variety of unobservable traits (see for example McPherson and Winston (1983) [68] and McKen- zie (1996) [67]). Finally Carter et al. (2009) [22] contend that tenure is within the class of optimal contracts under the assumption that the value added to a university by a researcher results from her cumulative scientific contributions rather than from her spot contributions. Under this assumption it is optimal for universities to formulate contracts with strong incentives for the accumulation of research achieve- ments in the early stages of scholars’ careers, and which induce less effort close to retirement, once the future period over which the benefits from research successes can be reaped is short enough. While as a whole, the literature acknowledges the effects of the lengthy probationary periods in academic careers over the actions taken by researchers, to our knowledge ours is the first paper to specifically focus on the incentives that these create regarding the choice of research topics. The question of whether these incentives should be seen as a side-product of a mechanism mainly concerned with other objectives, or as pertaining the institution’s main role is beyond the scope of this paper.

Finally, our paper is related to the broad literature on social status and conspicuous behavior. This literature explains a variety of behaviors as resulting from the human drive to gain better standing in hierarchies. Beginning with Veblen (1899) [85] this theme has surfaced over the past century by way of different concepts and applications: Veblen effects, snob effects, positional goods and status goods to name a few. More recently Frank (1987, 1995, 2005) [41] [43] [42] and Basu (1989) [9] provide general frameworks which allow them to offer explanations for a large variety of socioeconomic phenomena including wage dispersion, initiation rituals, wastefulness and individual patterns of consumption in time, in terms of conspicuousness. While CHAPTER 1. CONSPICUOUS SCHOLARSHIP 9

most of the recent theoretical and empirical papers in this literature are related to consumption (see for example Bagwell and Bernheim (1996), [6], Corneo and Jeanne (1997) [28], Ravina (2007) [76] and Charles et al. (2009) [23]) there are also a number of articles motivated by other applications. Glazer and Konrad (1996) [45] presents a model philanthropy and Bloch et al. [15] apply the idea to explain wedding celebrations in India. In general all this literature implicitly builds upon the arguments of Spence [82] and Akerlof (1976) [3] that show how information can be conveyed in equilibrium and make explicit the necessary correspondence between the underlying costs of actions and the meanings that these can convey. One issue however is that in signaling models conspicuousness emerges as one among many possible stable patterns of behavior. Our paper provides a simple argument showing that in many of these settings the conspicuous equilibria may be the most plausible.

1.2 A Simple Example

Two agents, indexed by i P t1, 2u are solely concerned by their status as given by the belief about their private types θ1 and θ2 that society forms upon observing their actions, and the outcomes of these actions. The types of the agents are independently drawn from tθL, θH u and represent their unobservable research abilities. The type of an agent is θL with probability qL and θH with probability qH 1 ¡ qL. After each agent observes his own type, they simultaneously choose a topic within one of two possible research arenas t1 and t2 to work on. The outcomes from the research eﬀorts are random variables whose distributions depend on the agent’s type and the research arena chosen as follows. An agent of type θH has success with probability p1,H if he chooses to work on arena t1 and p2,H if he chooses to work on arena t2. An agent of type θL has success with probability p1,L if he chooses to work on arena t1 and 0 if he chooses to work on arena t2. We assume that 0 p1,L p1,H . Society values the agents of type θH more than the agents of type θL and therefore after observing the agents’ arena choices and the outcomes of the research eﬀorts, it praises the agent which it deems more likely to have a higher research ability. Agents do not value any quality of their research output directly and only care about social recognition. CHAPTER 1. CONSPICUOUS SCHOLARSHIP 10

We normalize the utility that an agent derives from being praised to 1, and assume that an agent gets no utility in case he is not recognized. Finally, we assume that the agents’ labels contain no additional information allowing society to make a decision, and that as a result it is forced to choose the recipient of its praise uniformly at random in case that it believes that either agent is equally likely the one with higher research ability. In what follows, we characterize the set of pure strategy symmetric sequential equilibria of the game8.

Claim 1 There are no separating equilibria.

Proof of Claim 1: There are no separating equilibria since rather than revealing himself, the low type θL could do strictly better by imitating the high type θH . This results from the fact that the outcomes are probabilistic and that society treats the agents equally when it cannot distinguish them. ¡ © ¡ ¡ 1 p1,H p1,L Claim 2 If p2,H 2 qL 2 then in all pure strategy pooling equilibria of the game described above the agents select t2 with probability 1.

Proof of Claim 2:

Suppose there existed a sequential equilibrium in which agents pooled on t1. In such an equilibrium, the expected payoﬀ of an agent of type θH would be given by:

¢ ¡ © q p p p1 ¡ p qp1 ¡ p q 1 p ¡ p H q 1,L 1,H 1,L 1,H p1 ¡ p qp q 1,H 1,L 2 L 2 2 1,L 1,H 2 L 2

Since agents of type θL have no probability of succeeding in research arena t2, upon observing a success in t2 society must conclude that the agent is a high type for sure. As a result, the minimum expected payoﬀ that¡ the agent© gets from deviating ¡ ¡ 1 p1,H p1,L to t2 is p2,H . Therefore, provided that p2,H 2 qL 2 there does not exist a sequential equilibrium in which the agents pool on t1.

8That is, sequential equilibria in pure strategies in which the players’ strategies are independent of their labels. CHAPTER 1. CONSPICUOUS SCHOLARSHIP 11

To finish the proof we just note that there exist sequential equilibria in which both agent types select arena t2 with probability 1. Specifically, the strategy profile according to which both types of agents select t2 for sure can be sustained as a sequential equilibrium of the game under all belief structures which deem it more likely that an agent deviating to t1 is of type θL than of type θH . We therefore conclude that under the parametric restrictions set forth in Claim 2, all sequential pure strategy equilibria of the game entail both players choosing t2 with probability 1. This example allows us to closely examine the following arguments: A pattern of behavior in which researchers with different abilities behave evidently differently (in pure strategies) can’t be stable because the low types would be revealing themselves. Instead they could imitate the high types, case in which the worst that could happen is that society detected them, and more so, with positive probability they would get misclassified to their advantage. On the other hand if society’s expectation is that both types are to work on t1, by deviating to t2 and in case of success, the high type could force on society the correct conclusion that she is indeed a high type. Note that in this game and in sustaining a pooling equilibrium in t1, the concept of sequential equilibrium allows society to rely on any posterior beliefs at the information sets in which it observes an agent choose t2 and fail. In particular it could then automatically brand the player in question as a low type. On the other hand the posterior beliefs after observing some agent succeed in t2 cannot be defined conventionally since due to the laws of nature (p2,L 0) it is impossible that the agent in question has low research ability. In this sense we speak of t2 as being the more conspicuous research arena. If in addition, the high type’s success probability in t2 is sufficiently high, then she finds it strictly profitable to deviate. It is worth noting that her success probability can nevertheless be much lower than in t1, since when both types select the same arena, their ability to distinguish themselves gets trebled down by competition.

In order to further highlight this last point we now examine an environment like the one just analyzed, but where we have an arbitrary number of agents, n ¥ 2, competing to be selected by society. As above, society only selects one agent, and we CHAPTER 1. CONSPICUOUS SCHOLARSHIP 12

impose the restriction that it chooses uniformly at random in case its beliefs make it indiﬀerent among a subset of the agents. We ﬁrst note that with the same arguments as those already espoused in the case of n 2 it can be seen that there are no separating equilibria. ¡ © n¡1 1 p1,H ¡p1,L 1¡p1¡p1,Lq Claim 3 If p2,H ¡ then all the pooling equilibria involve n n p1,L all agents choosing topics in arena t2.

Proof of Claim 3: Suppose that there existed a sequential equilibrium in which both types of agents chose research arena t1. The expected utility to a an agent of type θH in such an equilibrium would be bounded above by her expected utility conditional on the remaining n ¡ 1 agents being of type θL. We can compute this upper bound as follows. For notational ease denote p1,H by pH and p1,L by pL:

¡ ¢ n¸1 n ¡ 1 pk p1 ¡ p qn¡1¡k p1 ¡ p qp1 ¡ p qn¡1 p L L H L H k k 1 n k0 ¡ ¢ p n¸1 n ¡ 1 pk 1p1 ¡ p qn¡pk 1q p1 ¡ p qp1 ¡ p qn¡1 H L L H L p k k 1 n L k0 ¢ 1 p ¸n n p1 ¡ p qp1 ¡ p qn¡1 H pz p1 ¡ p qn¡z H L n p z L L n L z1 p ¡ qp ¡ qn¡1 1 pH n 1 pH 1 pL p1 ¡ p1 ¡ pLq q n pL ¢ n ¡ ¡ © ¡ p ¡ qn¡1 1 pH pL 1 1 pL n n pL

Since in case of success in t2 a high type player gets immediately identified by society as such, his expected payoff from deviating from the equilibrium strategy profile would be bounded from below by p2,H . The existence of a pooling sequential equilibrium in which both players choose t1 therefore requires that p2,H does not exceed the upper bound that we have just computed.

In a similar fashion it can be seen that in a pooling equilibrium in t1 the probability that a low type player is selected is bounded from above by 1 and bounded from below ¨ ¡ © n ¡ ¡p ¡ qn¡1 by 1 ¡ pH pL 1 1 pH . The excess probability that the high type has of being n n pH selected by society due to her greater research ability vanishes as n grows, just as CHAPTER 1. CONSPICUOUS SCHOLARSHIP 13

1 the mean probability, n , that a player has of being selected before learning her type.

Therefore as long as p2,H ¡ 0 and regardless of the parameter values governing the success probabilities in t1, there always exists a level of competitiveness, as indexed by the number of players in the game, above which the unique pure strategy sequential equilibrium involves all players selecting t2.

1.2.1 Productive Eﬃciency and Screening Eﬃciency

There are two key considerations in assessing the social eﬃciency of behavior in our context:

1. Productive eﬃciency: The direct social value of the successful research projects.

2. Screening eﬃciency: The future social value of the information about the unobservable research abilities that society extracts from observing the agents’ choices and their performances.

The problem is interesting to the extent that there is a tension between the screening and productive efficiencies of behavior. The crucial aspect of the example is that the nonexistence of a sequential equilibrium in which both agents choose t1 for sure does not depend on the relative productive efficiency of research efforts in t1 and t2. In order to fix ideas, suppose that the direct value of a successful research endeavor 9 in either arena is v. One plausible interpretation for t1 and t2 is the following:

• t1 represents relatively new topics in which not too much research has been carried out. Reaching fertile grounds at the forefront of the literature in one of these topics may therefore require a moderate amount of time, and the probability of producing a signiﬁcant contribution is high.

• t2 encompasses well established research topics that have been addressed by scholars for a long time, and which have thereby acquired technical maturity.

9Since a research arena may include a variety of research topics with potentially diﬀerent values, this can be seen as a distributional assumption on the values of these diﬀerent research topics, and the preferences of researchers over research topics (which lie outside the scope of this project) within each arena. CHAPTER 1. CONSPICUOUS SCHOLARSHIP 14

Reaching the forefront of the literature in these topics in general takes more

time than for topics in t1, and the probability of making a valuable contribution

is lower. Research in t2 is more diﬃcult than research in t1 in the sense that success is less likely for agents of any research ability. The parametric restriction

p2,L 0 p1,L already guarantees this to be the case for the low type agents,

and p1,L p1,H already captures the greater research ability of θH . In order to

complete the picture we just add p2,H p1,H .

• The crucial assumption in the example is that t2 is conspicuous, in the sense

that p2,L 0 p2,H . Interpreting it as meaning that more diﬃcult research arenas are more discriminating of research talent, this assumption ﬁts nicely 10 into the picture of t1 and t2 that we have just described .

In this interpretation of t1 and t2, the most conspicuous research topics are also the ones with the lowest yields. Specifically, the expected value of the contribution of an agent working in t1 is vpqLp1,L qH p1,H q, whereas the expected contribution of an agent working in a topic in t2 is vpqH p2,H q. The pooling equilibrium in t2 is therefore the worst possible allocation of scholars to research topics in terms of productive efficiency. As discussed in Claim 1, there are no separating equilibria, and the most efficient in terms of information revelation among the pooling equilibria is the one in which both agents choose t2. The problem is that in this setting there is a tradeoff between the screening efficiency and productive efficiency of behaviors, and society is unable to choose the optimal level of screening in order to balance both considerations. One of the main ideas that we want to set forth in this paper is that a broad class of status mechanisms unavoidably reach the screening efficiency upper bound, in detriment of productive efficiency.

10More diﬃcult tasks (in the sense of success probabilities) need not be more discriminating of research abilities than easy tasks. It seems to be a plausible assumption regarding research, and the proposed properties of t1 and t2. CHAPTER 1. CONSPICUOUS SCHOLARSHIP 15

1.3 A More General Model

The simple example developed in detail above sought to showcase what in our view are the crucial elements in obtaining the unraveling that characterizes certain competitive signaling processes: strong incentives for agents to differentiate themselves from the group when everyone is acting in a similar fashion, and strong incentives to follow agents that attempt to differentiate themselves, thus effectively precluding differentiation in equilibrium. While lending a lot of clarity, the environment that we have used to explore the possibility of these kind of instabilities fails to capture the early stages of academic careers eventually leading to hiring contests in two important respects:

1) The “status assignment” by society does not deﬁne a winner takes-all situation

but rather sorts individuals into categories (J1,J2, ...., Jr) which they all rank in a similar fashion.

2) Standard signaling models are very good for representing instances in which society relies on individuals’ behaviors in order to classify them, but upon a closer examination society seldom faces a pure classiﬁcation problem regarding the competition between academics in the early stages of their careers.

The second problem is the more urgent one since it demands the greatest mod- iﬁcations to the framework of the example, so we begin by discussing it in greater detail.

A classification problem requires well defined exogenous type definitions to begin with. The “classifying” social institution’s main goal is then to produce some beliefs as to the correspondence of individuals to the collection of unobservable types, based upon some self-sustaining conventional patterns of behavior associated to them. Therefore, a mechanism whose objective is to define who are the best in a group, in general cannot be thought of as a classification mechanism to the extent that being the best is not an individual property, but rather depends on the joint characteristics of all the agents involved in the competition. When screening groups of young academics in order to allocate them efficiently to the different positions available, society CHAPTER 1. CONSPICUOUS SCHOLARSHIP 16

is usually simply ranking them, allowing for the possibility of significantly different cohorts associated to different instances of the problem. In general the problems of ranking and classifying only become the same in the rare cases where occupying a specific position in a ranking comes in itself with some crucial defining characteris- tics11.

1.3.1 What is it that Explicit Competitions Provide?

Ranking a set of agents can be achieved by first classifying them, but classification is in general a more complex problem than ranking, with the exception of situations in which it is known ex-ante that the number of agents in the group corresponds exactly to the number of different types and that there is precisely one agent representing each of the possible types.

In particular, if the potential number of types is much larger than the set of agents that society wishes to rank, the approach of classifying in order to rank seems horribly wasteful from an algorithmic point of view. Consider as an example the task of deciding which among two individuals is physically fitter. One possibility is to have each of them go through a detailed battery of tests in order to carefully identify their locations in the huge spectrum of adult human fitness, and then rank them based upon this detailed classification. An intuitive alternative to the cumbersome classification process is to have them run a race; intuitively a well chosen trial or a small series of trials would provide a ranking at least as accurate as the “classificatory” one, while being a lot less complex.

Although one could interpret a race between two individuals as two simultaneous, independent, ﬁtness classiﬁcatory tests, the striking changes in behavior elicited by the introduction into a trial of any kind of explicit competitive elements suggests otherwise. Most of the game theory literature to date has represented the process of status attainment by individuals using elaborations of the Spence’s signaling model.

11One example of such a situation is when the ﬁeld of participants is large enough to guarantee the existence of at least one representative of each exogenous type. CHAPTER 1. CONSPICUOUS SCHOLARSHIP 17

While that kind of representation can accurately render certain aspects of such processes, one major disadvantage of the approach is the multiplicity of equilibria that arise. So for example, while the possibility of conspicuous consumption equilibria can be well established and understood with the aid of a Spencian signaling model, it arises as just one possibility among many, and this fact may in part be responsible for the relative small amount of attention that the phenomenon has received. From the perspective of classiﬁcation problems it is natural to have multiple equilibria, and for the number of equilibria to grow very rapidly with the number of potential different types; in a way this property becomes an accurate rendering of how arbitrary human conventions can be. However, when the setting in question has an explicitly competitive ﬂavor, some natural restrictions to the concept of sequential equilibrium arise which produce much sharper predictions.

The view that we take in what follows is that when formulating an explicit competition, society effectively commits to uphold a simple global interpretation of relative outcomes. In the case of the simple example of the race used to determine fitness this would amount to concluding that whomever is fastest is the fittest among the two agents. Note that modeling the race in this example as a signaling game implies uncountably many uninformative pooling equilibria, sustained purely conventionally, unless it were the case that certain levels of performance were ex-ante unattainable to certain types. Competitions however seem to be widely applied as part of ranking mechanisms12, under the robust impression that they often lead to patterns of collective behavior (conspicuous actions) which effectively amplify the relation between unobservable and visible differences, facilitating the discriminating efforts. This suggests that the “classification” approach often taken to model conventions in environments that are inherently competitive13, may not be the most appropriate.

In our subsequent analysis we explore the extent to which restricting attention to

12Competitions are often used to rank the players directly in terms of the object of the race itself. e.g. Running the 100m dash in order determine the ranking of players in terms of how fast they run the 100m dash. In this paper we are just concerned with instances in which there are unavoidable discrepancies between the intended and literal meanings of the outcomes of competitions. 13e.g. The original application to job market signaling through educational attainment in Spence’s 1973 paper. CHAPTER 1. CONSPICUOUS SCHOLARSHIP 18

sequential equilibria that uphold the “direction” of the discriminating qualities inherent to the diﬀerent research topics, suﬃces to obtain the kind of univocal competitive unraveling seen in the example of Section 1.2.

1.3.2 The Game

Consider an environment in which n agents are competing for r ¤ n jobs, regarding which they have identical preferences, represented by some utility func- t u Ñ tion u : J1,J2, ..., Jr R. Their common strict ranking of the jobs is given by

J1 ¡ J2, ..., ¡ Jr. They also prefer job Jr to remaining unemployed which gives them a payoﬀ of 0. Each agent’s type, representing his research ability is drawn independently from r0, 1s according to a continuous distribution described by a cdf F .

After privately observing their types, the players simultaneously choose among 14 t1, t2, ..., tm the research arena in which they each want to work . The performance of an agent in a given arena is a binary random variable which behaves according to the function p : r0, 1s ¢ t1, 2, 3, ..., mu Ñ p0, 1q, representing for each type θ P r0, 1s the probability of success in each of the m possible arenas. Usually we denote the probability of success of an agent of type θ in arena tj by pθ,j, but sometimes use ppθ, jq when the context demands that p be treated as a function. We assume that for each j P t1, 2, ..., mu, the function pj : r0, 1s Ñ p0, 1q deﬁned by pjpθq ppθ, jq is measurable and diﬀerentiable15

The essential characteristics of the environment that our model seeks to represent are captured by the following three assumptions on p. We explicitly state which of these underly each of the individual results as we present them.

(A1) Diﬃculty: Success is less likely in arenas with higher indexes 1 For all θ, k ¡ k ñ pθ,k1 ¡ pθ,k

14We think of a research arena as a collection of research topics all entailing the same relationship between research abilities and the quality of research outcomes (performance). The precise meaning of the term is made clear below as we deﬁne the properties of research arenas. 15The diﬀerentiability assumption is just for expository convenience. CHAPTER 1. CONSPICUOUS SCHOLARSHIP 19

(A2) Meaning of θ: Higher types are more likely to succeed in all research arenas: ¡ ñ ¡ For all k, θ2 θ1 pθ2,k pθ1,k

(A3) Increasing differences of success probabilities: The difference in success probabilities between types strictly increases with the difficulty of the research arenas. 1 ¡ ¡ ñ 1 ¡ 1 ¡ ¡ Formally, k k, θ2 θ1 pθ2,k pθ1,k pθ2,k pθ1,k

Each of the r principals associated to th r jobs is seeking to hire someone from the pool of n agents in order to maximize the expected value Ergrpθqs where gr is a strictly increasing function of θ. After observing the arena chosen by each of the n agents and whether each of them is successful or unsuccessful in their endeavor, the principals update their beliefs about the type of each of the n players, and based upon these posterior distributions each of them ranks the agents based on his expected utility function Ergrpθqs and makes job offers. Since the players rank the different jobs in the same way we can think of this stage as a serial process whereby the principal of J1 offers the job to his most preferred candidate, who automatically accepts and leaves the pool of applicants, then the principal of J2 hires his most preferred option from the remaining candidates, and so forth. We assume that a principal cannot discriminate in any way between the agents by relying on the labels, so if he is going to assign some job Jq to a player in a set within which he is unable to differentiate by research talents, then he must assign the job uniformly at random among the agents in that set. For the purpose of clarity, we speak of society as the subject when we refer to judgment calls that must be held in common by all principals16. In what follows we analyze this game using the concept of sequential equilibrium. We focus on symmetric equilibria, that is, equilibria in which the strategies of the players are label-independent17.

16Since the principals are allowed to have different utility functions they may rank the players differently, however in any sequential equilibrium their posterior beliefs about the players’ types must coincide on or off the equilibrium path, since they rely on exactly the same information when constructing them. 17Throughout we use the labels of the agents only for analytic purposes. The underlying assumption is that all the aspects of a player that are relevant for the game, are fully captured by his type. CHAPTER 1. CONSPICUOUS SCHOLARSHIP 20

Before continuing we introduce some more notation. Denote by ai P tt1, t2, ..., tmu 1 the research arena chosen by agent i, and by si P t0, 1u i s performance, where si 1 is meant to represent success and si 0 failure. For any events x and y, let πspx|yq denote the probability that society attaches to event x at the information set18 that corresponds to having observed event y. Throughout σ : r0, 1s Ñ r0, 1sm th denotes a possibly mixed strategy, where the k entry σpθqk evaluated at θ represents the probability with which an agent of type θ selects research arena tk. Based on the above, a symmetric sequential equilibrium will be denoted by xσ, πsy where σ represents a label-independent strategy profile and πs the complete system of posterior beliefs. Given some equilibrium xσ, πsy we will denote by Sk an agent’s expected payoff from being successful in arena tk, and Uk her expected payoff conditional on failure in arena tk. Throughout the rest of this chapter we say that research arena k1 makes research p 1 p ñ θ2,k ¡ θ2,k abilities more conspicuous than research arena k, if θ1 θ2 p 1 p , that is θ1,k θ1,k if success of a fixed agent is more indicative of research ability in k1 than in k, under maximum likelihood inference of types. We say that a collection of research arenas has property pCq if it is the case that whenever an arena k1 is more difficult for all types than arena k, then k1 is more conspicuous than k. In light of pA1q, we can formally write pCq in our context as:

p 1 p p q ¡ 1 ¡ ñ θ2,k ¡ θ2,k C The success likelihood ratio is increasing in k: θ2 θ1 and k k p 1 p θ1,k θ1,k We end this section by noting that pA1q pA2q and pA3q imply pCq.

Lemma 1 If pA1q, pA2q and pA3q hold then so does pCq.

The proof of this lemma is relegated to the appendix. The main point is that given that all the terms involved are positive, condition pCq is equivalent to increasing differences in the logarithms of the success probabilities which is weaker than increasing diﬀerences when success probabilities are increasing in θ pA2q and decreasing in k, pA1q19.

18Formally, an information set or the union of a number of information sets. 19Special thanks Guofu Tan for pointing this out. CHAPTER 1. CONSPICUOUS SCHOLARSHIP 21

1.3.3 Competitive Unraveling

In this section, we show that the conspicuousness of the sequential equilibria is increasing in the amount of competition of the environment if we restrict attention to the subclass of equilibria sustained by beliefs which recognize successes in research arenas with higher indexes to be more indicative of an agent having a high research talent than successes in research arenas with lower indexes. The precise sense in which this is the case is captured by Proposition 1. Formally the equilibrium selection criterion that we work with is the following:

(R1) If society observes an agent i succeed in research arena tk1 and j succeed in 1 research arena tk where k ¡ k then, in terms of its posterior beliefs, θi must

ﬁrst order stochastically dominate θj. That is

@u P r0, 1s, πspθi ¥ u|ai tk ^ si 1q ¥ πspθj ¥ u|aj tk ^ sj 1q

and the inequality must be strict for some u P r0, 1s.

Note that in the current setting, the agents only know their own types, which implies that in a sequential equilibrium society’s beliefs about a given player’s type cannot be inﬂuenced in any way by what it observes other players do, or how other players perform. This implies that under restriction pR1q if player i undertakes a research topic in an arena with a higher index than the one chosen by player j and they are both successful, then regardless of what other players do and how they perform, θi will ﬁrst-order stochastically dominate θj at the job assignment stage of the game.

Given some function p : r0, 1s ¢ t1, 2, 3, ..., mu Ñ p0, 1q, and a utility function u over tJ1,J2, ..., Jru, such that J1 ¡ J2, ..., ¡ Jr, let Γp,u,n denote a game with n players assembled from p and u as described in Section 1.3.2. CHAPTER 1. CONSPICUOUS SCHOLARSHIP 22

Proposition 1 Suppose that p satisﬁes pA2q and let σ be a symmetric strategy pro- 20 21 ﬁle which places positive probability on some research arena tk with k m. There exists nσ such that if n ¡ nσ then there are no beliefs πs such that xσ, πsy is a sequential equilibrium of Γn satisfying pR1q.

The proof of Proposition 1 expresses the main intuition behind the result and for that reason we present it below. It relies on the following lemma, whose proof we leave for the Appendix.

Lemma 2 If pA2q holds, then in any sequential equilibrium, Sk ¥ UK @k (in or out of the equilibrium path).

Proof of Proposition 1: Consider some strategy profile such that for some θ P r0, 1s, σpθq places positive ³1 probability on tk for some k m, and for notational convenience let α 1tσpθq ¡ 0 tkudF pσq. When an agent is called on to select tk by his strategy, then under any system of beliefs πs satisfying pR1q he may only get a job in the event that at most r¡1 of the agents that select some research arena with a higher index than k succeed. For 22 pαn¡r¡1q large n we can bound the probability of this event from above by p1 ¡ p0,mq , pαn¡r¡1q and the agent’s payoff by UpJ1qp1 ¡ p0,mq . If instead he chooses tm and succeeds, then by pR1q, pA2q and Lemma 2 he has at least an equal chance among all other successful agents in research arenas with indexes higher than k of getting one of the r jobs. Specifically pR1q guarantees that society’s posterior beliefs upon observing his success first order stochastically dominate its beliefs regarding any agent succeeding in research arenas with lower indexes. Moreover, pA2q and Lemma 2 along with pR1q imply that they also first order stochastically dominate society’s beliefs upon observing a failure at any arena. We can therefore bound his expected payoff

20Symmetric strategy profiles σ are by definition label independent, and therefore apply to the game defined for any number of players n. Recall that throughout, we only consider symmetric sequential equilibria. 21m is the highest possible index. 22This follows from the fact that we can choose an n large enough in order to approximate Ep|ti : σpθiq ¡ tku|q by αn as well as we wish, and by applying Jensen’s inequality to the strictly x convex function gpxq p1 ¡ p0,mq . CHAPTER 1. CONSPICUOUS SCHOLARSHIP 23

p ¡ q p q r 1 p0,m from deviating to tm from below by U Jr n¡1 , which is strictly greater than the upper bound on his expected payoﬀ from choosing arena tk for large enough n.

Proposition 2 Let σ¦ denote the strategy proﬁle prescribing that all agent types ¦ choose tm for sure. For every n there exists a belief system πs such that xσ , πsy is a sequential equilibrium of Γp,u,n satisfying pR1q.

The proof of Proposition 2 just entails showing that for any distribution of types F we can construct a system of beliefs consistent with every agent type choosing tm with probability 1 satisfying pR1q, and that given those beliefs it is sequentially rational for every type to choose tm with probability 1. In the appendix we provide the details of one such construction.

Together, Propositions 1 and 2 imply that the only equilibria satisfying pR1q that survive the intensification of competition for the r jobs, involve all agents selecting tm for sure. This result only depends on assumption pA2q and the equilibrium selection criterion pR1q. The main idea is the one captured by the example in Section 1.2: As competition intensifies, the ability of agents to single themselves out in any given equilibrium diminishes, and forces them towards topics in arenas where success is most indicative of unobservable research ability, as defined by the equilibrium selection criterion pR1q. PropertypCq, which formally is not required by Propositions 1 and 2 is the key justification for the equilibrium selection criterion pR1q and it follows from assumptions pA1q through pA3q as shown in Lemma 1. Under pCq a maximum likelihood test of the null hypothesis that the ability of a given agent is below some given threshold versus the alternative hypothesis that his research ability exceeds that threshold would have a lower size in arenas with higher indexes. Claim 4 examines the precise implication of pCq. We relegate the proof to the appendix.

Claim 4 Assume that pCq holds and suppose that k1 ¡ k. The type of a given agent conditional on success in tk1 ﬁrst order stochastically dominates the type of that same agent conditional on success in tk.

In this sense success in research arenas with higher indexes is naturally more indicative of research ability than success in research arenas with lower indexes, so given CHAPTER 1. CONSPICUOUS SCHOLARSHIP 24

pCq, belief systems that satisfy pR1q are more focal. Like pCq, pA1q plays no direct role in either of the two previous propositions, but it provides an important characteristic of the environment: it captures the idea that the research arenas with higher indexes encompass globally more difficult topics, in the sense that the expected yield of projects is higher in research arenas with lower indexes regardless of the underlying research ability of the agent. Moreover, as shown in Lemma 1, when assumptions pA1q and pA2q hold, the increasing likelihood ratio of success probabilities pCq which justifies pR1q, is implied by increasing differences pA3q.

Pooling equilibria, separating equilibria and mixed equilibria satisfying belief restriction pR1q involving research arenas other than tm are plausible only to the extent that the level of competition for the r jobs available is sufficiently low. The equilibrium selection criterion pR1q is the essential element behind this result. For example, pooling on the least conspicuous research arena, t1, can be sustained as part of a sequential equilibrium in which after observing an agent choose any arena other than t1 and independently of performance, society automatically concludes that the agent in question has the lowest type among all the competitors. While this pattern of behavior constitutes a self-sustaining institution in the sense captured by the concept of sequential equilibrium, any system of beliefs serving to support it would need to depart from those implied by property pCq upon acting on randomly chosen agents. An argument that makes the focality of belief systems satisfying pR1q more palpable is that it amounts to a property of the different research arenas that can be inferred by society through simple experimentation: by having randomly selected agents try out the different research arenas. It is plausible that along with this inference process, society also learns or develops23 the belief system which supports the whole screening institution. pR1q differs from other restrictions studied by the literature on equilibrium selection criteria in that rather than appealing to rationality it rests on the idea that belief systems which are consistent with the simple inference rule of thumb that is optimal for decisions in the absence of strategic complications, are more plausible than all others. Studying in detail the epistemic conditions and learning processes

23And to its disadvantage, under the appropriate conditions on the relative importance of productive versus screening eﬃciency as discussed in section 1.2.1. CHAPTER 1. CONSPICUOUS SCHOLARSHIP 25

under which these beliefs are indeed more likely or under which this fails to be the case is left for future work.

1.3.4 Some Stylized Facts About Particle Physics in the U.S.

String theorists belong to the broader community of particle physicists. The following are some stylized facts obtained from the census of particle physicists in the U.S. carried out by the Particle Data Group24 in 2007 and 2008 suggesting that the competition in the initial stages of careers in particle physics and more specifically among particle theorists is stringent, as noted by some of the sources that we refer to in the introduction. In 2008 there were 1363 tenured faculty, 936 retired and non-tenured open term faculty and research staff, and 1008 post-docs and other fixed-term research staff in the post-doctoral particle physics community. In the same year there were 1409 graduate students. The figures corresponding to the preceding 4 years are very similar. To place these numbers in perspective it is worth noting that in a 2006 survey carried out by the American Institute of Physics25 60% of the physics Phds minted that year reported aspiring to an academic job. This figure takes into account the relatively large number of doctorates in applied fields who tend to gravitate towards government and private sector jobs more frequently than particle physicists. The data pertaining the sub-community of particle theorists suggests that young scholars in that subfield face an even greater scarcity of future academic jobs. There were a total of 1483 researchers in the community in 2008, of which 469 were tenured faculty, 209 were in the category of open-term non-tenured faculty and staff, 266 were post-docs or had other fixed-term research positions, and 536 were graduate students.

Based on historical data the Particle Data Group estimates that less than 20% of the particle physics entering graduate students in 2008 can expect to obtain a research position in the US upon completing their Phds, and less than 8% can expect

24The Particle Data Group is a international collaborative venture which compiles, summarizes and publishes particle physics results obtained throughout the world. The census attempts to include the 153 institutions carrying out particle physics research in the US and it enjoys a response rate close to 100%. 25This is a survey conducted annually by the AIP among all the degree-granting physics departments in the U.S. and Puerto Rico (252 departments in 2006). The response rate is around 95%. CHAPTER 1. CONSPICUOUS SCHOLARSHIP 26

to eventually achieve a tenured position.

1.4 Uniqueness of Conspicuous Equilibria

The results presented in Section 1.3.3 show that if agent types with higher research abilities succeed with greater probability (assumption pA2q) the game displays competitive unraveling within the subset of equilibria that rely on beliefs that are consistent with the inherent research-ability-discrimination properties of the research arenas captured by property pCq. Another property of the simple example in Section 1.2, is that it has no pure strategy separating sequential equilibria. The intuitive reason for this to be the case in that setting is that under any pattern of behavior in which the strategies themselves convey too much information about the agents’ types, lower ability agents would avoid self revelation by imitating higher ability agents. In the example, the full revelation follows from the fact that as there are only two types, any pure strategy equilibrium other than the pooling ones must be fully revealing of the ranking of the players. This property still holds in the general game, but only in a very restricted sense: There do not exist any equilibria in which at least two distinct arenas tk and tk1 get chosen such that the lowest type choosing arena k is weakly larger than all the types choosing k1 and strictly larger than at least one of them. If this were the case then one of the types choosing arena tk1 would ﬁnd it strictly beneﬁcial to choose tk. With this kind of argument, even under the equilibrium selection criterion pR1q not much more can be said about the existence of other separating equilibria. The reason is that with many types, separating equilibria can be almost as uninformative about research abilities as pooling equilibria.

But there are other reasons why some types may be driven to “imitate” other types in settings such as the ones motivating this paper. In particular the fact that a player with a given research ability prefers the lottery offered by the topics in some research arena may imply restrictions over the arenas that different types may prefer. In this section we investigate the traction that this method of reasoning has in our setting. Specifically we show that separating equilibria vanish in our game provided some additional assumptions on the connection between research abilities and performance CHAPTER 1. CONSPICUOUS SCHOLARSHIP 27

in topics of varying difficulties. We present two different sets of additional restrictions to the model inducing a form of increasing differences in the choice of research arenas yielding similar consequences:

1. A strong form of increasing diﬀerences in success probabilities which we call fanning out.

2. Increasing diﬀerences in success probabilities and an additional equilibrium selection criterion which further restricts attention to equilibria in which higher index arenas are riskier.

The next two sections make use of the following assumption on the function p:

1 (A4) Fanning out of success probabilities: For all k k , θ1 θ2 ¤ θ3 θ4 we have p 1 ¡p 1 p 1 ¡p 1 k ,θ4 k ,θ3 k ,θ2 k ,θ1 that ¡ ¡ . pk,θ4 pk,θ3 pk,θ2 pk,θ1 Assumption pA3q establishes that research endeavors in arenas with higher indexes are more discriminating between different research abilities. pA4q in turn implies that the relative advantage of higher types in arenas with higher indexes, is greater for pairs of types lower in the type spectrum. Together pA3q and pA4q imply that the success probabilities of different types fan down with the index of the research arenas. Figure 1.1 sketches a function p satisfying assumptions pA1q through pA4q: Research arenas with higher indexes encompass more difficult topics in the sense that all types have a lower probability of succeeding in them than in topics in arenas with lower indexes pA1q. An agent is more likely to succeed in any arena than agents with lower research ability pA2q, that is, the curves representing the success probabilities of different types never cross. The more difficult topics also make research abilities more conspicuous, in the sense that they are more discriminating between different types pA3q . And finally, the extent to which the more difficult arenas are more discriminating is greater for lower types pA4q.

1.4.1 Fanning Out

In this section we show that when the success probability proﬁle p satisﬁes pA4q, in all pure strategy sequential equilibria consistent with pR1q, all agent types select the CHAPTER 1. CONSPICUOUS SCHOLARSHIP 28

0 1 2 . . k . . m Research Arena (Subindex)

Figure 1.1: A sketch of a function p satisfying pA1q ¡ pA4q, for 6 diﬀerent types. The upper curves represent the success proﬁles of higher types. same research arena.

Proposition 3 If assumptions pA2q and pA4q hold then there are no separating sequential equilibria in pure strategies satisfying pR1q.

Proof of Proposition 3: Consider some sequential equilibrium in pure strategies 1 1 xσ, πsy satisfying pR1q. Suppose that there exist types θ, θ P r0, 1s, such that θ θ , 1 1 σpθqk 1 and σpθ qk1 1 where k ¡ k. Then there must exist θ1, θ2 P r0, 1s with ³1 θ2 ¡ θ1 such that σpθ2qk 1, σpθ1qk1 1 and σpθqkdF pθq ¡ 0 since otherwise a θ2 player of type θ would be strictly better oﬀ by deviating from tk to tk1 . By pR1q this in turn implies that there exists θ3 ¡ θ2 such that σpθ3qk1 1. We therefore have:

p ¡ qp ¡ q p p ¡ q q ¡ p p ¡ q q pk,θ2 pk,θ1 Sk Uk pk,θ2 Sk 1 pk,θ2 Uk pk,θ2 Sk 1 pk,θ2 Uk

¥ p 1 p ¡ 1 q 1 q ¡ p 1 1 p ¡ 1 q 1 q pk ,θ2 Sk 1 pk ,θ2 Uk pk ,θ1 Sk 1 pk ,θ1 Uk

p 1 ¡ 1 qp 1 ¡ 1 q pk ,θ2 pk ,θ1 Sk Uk CHAPTER 1. CONSPICUOUS SCHOLARSHIP 29

p ¡ qp ¡ q p p ¡ q q ¡ p p ¡ q q pk,θ3 pk,θ2 Sk Uk pk,θ3 Sk 1 pk,θ3 Uk pk,θ2 Sk 1 pk,θ2 Uk

¤ p 1 1 p ¡ 1 q 1 q ¡ p 1 1 p ¡ 1 q 1 q pk ,θ3 Sk 1 pk ,θ3 Uk pk ,θ2 Sk 1 pk ,θ2 Uk

p 1 ¡ 1 qp 1 ¡ 1 q pk ,θ3 pk ,θ2 Sk Uk

p 1 ¡ q p 2 ¡ 1 q ñ pk,θ pk,θ ¤ pk,θ pk,θ pph,θ1 ¡ ph,θq pph,θ2 ¡ ph,θ1 q The last inequality follows from pA2q and contradicts assumption pA4q.

In this context, Propositions 1 and 2 can be applied directly. To sum up we have that if assumptions pA2q and pA4q hold then all pure strategy sequential equilibria satisfying selection criterion pR1q involve all agent types choosing research topics in the same arena. Furthermore, for each research arena tk with k m, there exists a critical threshold nk such that for all n ¡ nk there are no pure strategy sequential equilibria of Γp,u,n satisfying pR1q in which players work on research topics in arenas with indexes lower than k.

1.4.2 Increasing Diﬀerences

Proposition 3 depends crucially on the the fact that it restricts attention to pure strategies. pA4q coupled with pA2q induces a weak single crossing property of the preferences of different types over research arenas. As we show, this single crossing property is necessarily violated when the equilibrium beliefs are congruent with pR1q. Obtaining stronger results requires further restrictions on the set of equilibria, as in general, differences in success probabilities only allow us to make statements relating the preferences of different agent types over more difficult research arenas, to the extent that greater difficulty implies something about the stakes in success.

In this section we introduce pR2q, an additional equilibrium selection criterion. We show that if the success probability proﬁle p satisﬁes pA2q and pA3q then in all pure strategy sequential equilibria consistent with pR1q and pR2q all player types choose CHAPTER 1. CONSPICUOUS SCHOLARSHIP 30

research topics in the same arena, and there do not exist any sequential equilibria in mixed strategies.

1 (R2) The equilibrium beliefs πs must guarantee that if k ¡ k then Sk ¡Uk ¤ Sk1 ¡Uk1 .

The payoff to an agent of type θ from working in research arena tk is given by pθ,kSk p1 ¡ pθ,kqUk Uk pθ,kpSk ¡ Ukq. Criterion pR2q restricts attention to equilibria in which working in research arenas with higher indexes is riskier in the sense that the part of the payoff that depends on success, pSk1 ¡ Uk1 q, is at least as large as in lower index research arenas. That is, the stakes in success over failure are greater in arenas that encompass more difficult topics. One reason why the system of beliefs may satisfy restriction pR2q is that society could seek to raise the thresholds in the level of competition at which research choices unravel away from some socially desirable mapping of types to arenas. One simple way of doing it within the class of systems of beliefs that satisfy pR1q is by severely punishing failures in the more difficult arenas by uniformly raising the stakes in this fashion. As we show below, while raising the stakes may be effective in raising the thresholds for unraveling, it has a stark consequence: it precludes the existence of separating or mixed strategy equilibria. To the extent that the optimal variety of research cannot be fulfilled through topical differences within a fixed arena, this implication might be very costly.

Proposition 4 If pA2q and pA3q hold then all sequential equilibria satisfying pR1q and pR2q are weakly monotonic in the sense that θ2 ¡ θ1 implies that if σpθ1qk ¡ 0 1 and σpθ2qk1 ¡ 0 then k ¥ k.

Proof of Proposition 4: Let θ2 ¡ θ1 and suppose that σpθ1qk ¡ 0 and σpθ2qk1 ¡ 0 where k1 k. This implies that: p ¡ q p ¡ q p ¡ qp ¡ q pθ2,kSk 1 pθ2,k Uk pθ1,kSk 1 pθ1,k Uk pθ2,k pθ1,k Sk Uk

¥ 1 1 p ¡ 1 q 1 p ¡ qp ¡ q pθ1,k Sk 1 pθ1,k Uk pθ2,k pθ1,k Sk Uk

1 1 p ¡ 1 q 1 p 1 ¡ 1 qp 1 ¡ 1 q p ¡ qp ¡ q pθ2,k Sk 1 pθ2,k Uk pθ1,k pθ2,k Sk Uk pθ2,k pθ1,k Sk Uk

¡ 1 1 p ¡ 1 q 1 p 1 ¡ 1 qpp ¡ q ¡ p 1 ¡ 1 qq pθ2,k Sk 1 pθ2,k Uk pθ2,k pθ1,k Sk Uk Sk Uk

¥ 1 1 p ¡ 1 q 1 pθ2,k Sk 1 pθ2,k Uk CHAPTER 1. CONSPICUOUS SCHOLARSHIP 31

The ﬁrst inequality follows from the fact that σpθ1qk ¡ 0 and the third inequality follows from restriction pR2q and assumption pA2q. pR2q and the assumption that 1 1 ¡ 1 ¤ ¡ p q p 1 ¡ 1 q ¡ k k, together imply that Sk Uk Sk Uk, while A2 implies that pθ2,k pθ1,k 0. The second inequality follows from pA2q and pA3q, and Lemma 2 which together

p ¡ qp ¡ q ¥ p 1 ¡ 1 qp ¡ q guarantee that pθ2,k pθ1,k Sk Uk pθ2,k pθ1,k Sk Uk . This last inequality must actually be strict because if it were the case that Sk Uk, then by pR1q and 1 k ¡ k we would have Uk ¡ Sk1 contradicting σpθ2qk1 ¡ 0. We therefore have that

p ¡ q ¡ 1 1 p ¡ 1 q 1 pθ2,kSk 1 pθ2,k Uk pθ2,k Sk 1 pθ2,k Uk and that as a result a player of type

θ2 strictly prefers tk to tk1 , which contradicts σpθ2qk1 ¡ 0.

Proposition 5 There are no pure strategy separating equilibria.

Proof of Proposition 5: Suppose there existed a pure strategy separating equilibrium and let tk and tk1 be two distinct research topics chosen by some types in 1 equilibrium such that k ¡ k. By Proposition 4 all the types selecting tk1 must be higher than the types selecting tk, but this means that some type choosing tk could do strictly better by deviating to tk1 regardless of her research ability, which is a contradiction.

Proposition 6 There are no mixed strategy equilibria.

Proof of Proposition 6: Suppose that in equilibrium σpθq prescribed choosing two 1 distinct arenas tk and tk1 with positive probability, where k k . This implies that the player is indifferent between the two arenas. By Proposition 4 this means that all higher types than θ only choose arenas th where h ¥ k and all types lower than θ only choose arenas th with h ¤ k. As a result the only way in which θ can be indifferent is if no other types’ equilibrium strategy profiles prescribe choosing arenas tk and tk1 . But this means that, unless θ 1, a player of type θ would be strictly better off by deviating to some higher arena, only chosen in equilibrium by strictly higher types.

If θ 1, then some lower type could do strictly better by deviating to tk or tk1 . CHAPTER 1. CONSPICUOUS SCHOLARSHIP 32

Propositions 5 and 6 imply that under assumptions pA2q and pA3q, the game can only have pooling sequential equilibria satisfying selection criteria pR1q and pR2q. Moreover the argument used in the proof of Proposition 2 to construct the equilibrium beliefs satisfying pR1q only needs to be modiﬁed slightly, by specifying Gk to be an atom for each k. Together with Proposition 1 this set of results means that if pA2q and pA3q hold, then in all equilibria satisfying pR1q and pR2q all agent types choose topics in the same research arena, and furthermore, there exists n¦ such that for all ¦ n ¡ n all equilibria xσ, πsy of Γp,u,n require σpθqm 1 for all types θ P r0, 1s.

1.5 Eﬃciency

In order to examine the implications for efficiency of the results discussed in the previous sections we first need to make some assumptions about the social value of the information revealed by the game about the agents’ underlying abilities. We represent the value of the screening mechanism using the functions Zj : r0, 1s Ñ r0,Bs, for j P t1, 2, ..., Bu where Zjpθq denotes the value created by an agent of type θ when assigned to job j. An assumption already implicit in this notation is that different jobs generate social value independently from each other.

We make the following assumptions on the social value generated by the academics:

(E1) The social value generated by an agent that is not awarded one of the r jobs is independent of his research ability26, and therefore the expected social value created by the agents that are excluded (denoted by B) only depends on the fact that r agents are being selected, and not on the informativeness of the particular equilibrium of the game.

(E2) Any given research arena contains a wide range of topics, and conditional on the research arena the agents select their topic randomly. The value of successful

26An alternative plausible assumption is that there is also a positive relationship between the agent’s underlying research ability and the social value of his outside option. This assumption would further decrease the global value of highly discriminating equilibria, with respect to our expressions. CHAPTER 1. CONSPICUOUS SCHOLARSHIP 33

contributions may vary across diﬀerent topics, but the expected social value of a successful research project is the same across diﬀerent research arenas. We denote this value by h. Unsuccessful research endeavors have no social value.

For large enough n it is almost certain that the r agents are selected among the successful ones, so we can compute the social value generated by a pooling equilibria on research arena k as:

° r ErZjpθqppθ, kqs V nhErppθ, kqs j 1 pn¡rqB p1q k Erppθ, kqs

The diﬀerence in value between pooling equilibria in research arenas tk1 and tk with k1 ¡ k is therefore:

° ° r r p qp p qs r r p qp p 1qs 1 j1 E Zj θ p θ, k j1 E Zj θ p θ, k V ¡V 1 nhpErppθ, kqs¡Erppθ, k qsq ¡ p2q k k Erppθ, kqs Erppθ, k1qs

So in our framework, we can decompose the efficiency implications of greater competitiveness into the effects over the expected value of current research and the effects over the value of future research as influenced by the allocation of young scholars to jobs. As captured by the Propositions of Sections 1.3.3, 1.4.1 and 1.4.2 higher competitiveness pushes the players towards the more conspicuous research arenas, improving the quality of the inference that society makes about their unobservable research abilities. This increases the value of the industry’s future output for two reasons. First of all just purely mechanically, if there are more candidates for the future jobs it is more likely that some of these have very high research abilities. Secondly, to the extent that it implies more discriminating equilibria, the allocation of agents to the jobs available is better, given any specific pool of candidates. The first effect is seen by the fact that as n increases, the probability that a job gets assigned to a candidate that is not among those having succeeded after selecting the most conspicuous research arena that is part of a given equilibrium, decreases. Moreover provided that the ranking of the jobs according to the social value that they can generate coincides CHAPTER 1. CONSPICUOUS SCHOLARSHIP 34

with the preference ranking27, then this probability drops down faster in the case of the most preferred jobs. However for sufficiently large n this probability is very small, and this effect becomes insignificant. Intuitively, with a sufficiently high number of candidates, the screening ability of any given equilibrium gets fully exploited, and no significant further gains from increasing the amount of competition are possible. The second effect is bounded by ° ° r ErZ pθqpppθ, mqs r ErZ pθqpppθ, 1qs j1 j ¡ j1 j Erppθ, mqs Erppθ, 1q which is the asymptotic difference between screening social value generated by the most discriminating arena and the least discriminating arena. This means that for large enough n the efficiency gains due to the finer screening generated by the greater competition are dominated by the losses in the expected value of current research. In particular from expression p2q above it is clear that for sufficiently large n, having all agent types work on topics in research arena t1 dominates all other configurations of behavior.

The main message of the paper is that the conspicuousness in behavior induced by competition is independent of the social value generated by the activities that can be selected by the agents. In this section we have explicitly analyzed the case in which the value process is such, that expectations are well deﬁned and by virtue of assumption pA1q the more conspicuous research arenas are also the ones that generate the least social value. These assumptions seem to be consistent with the motivating landscape of theoretical particle physics as described by the skeptics of this research program. Note however that pA1q is the one assumption among pA1q ¡ pA4q with respect to which proponents and detractors of string theory would most patently disagree.

Regarding theoretical particle physics the main point of the paper is that the persistence of this research agenda, the lack of viable alternatives, and the pride of of the discipline in the depth and beauty of its mathematical propositions, may be

27This is a natural assumption if the institutions being analyzed have existed for a long time and there are no further frictions distorting the future alignment of incentives and eﬃciency. CHAPTER 1. CONSPICUOUS SCHOLARSHIP 35

explained by competitive unraveling. Whether this persistence proves to be a good thing or not depends on the highly contentious assessment of whether this is the correct scientific path or not. In the history of science there are abundant examples of research agendas which ultimately proved to yield revolutionary breakthroughs after very long periods of no apparent progress and strong28 opposition from within the scholarly/scientific communities and from the broader society. From this perspective, the resilience that the drive towards conspicuousness affords may be fundamental for scientific progress.

1.6 Conclusion

We study in detail a signaling model that captures in a stylized fashion scholars’ choice of research topics as they seek to make progress in the early stages of their academic careers. The key feature of the setting29 is that the more difficult topics are more conspicuous in the sense that they express more clearly the difference between different underlying research abilities. As it is often the case in settings with asymmetric information, in our model there are multiple sequential equilibria, reflecting a variety of conventions which society can uphold by relying on appropriate off-the-equilibrium path systems of beliefs. Different equilibria can be distinguished by two features which are essential in our context: their direct productive efficiency and their screening efficiency. We focus on the subset of sequential equilibria in which success in a given research arena induces beliefs about an agent’s underlying research ability which first order stochastically dominate the beliefs induced by success in all lower index (higher yield, less difficult) research arenas. We show that with the exception of the equilibrium in the lowest yield, most conspicuous research arena, all equilibria in this subset vanish as competition increases; that is, the game displays competitive unraveling. Furthermore we show that under some additional assumptions which effectively allow us to link the preferences over research arenas of agents with different underlying research abilities, we can rule out separating and mixed strategy equilibria

28And frequently violent. 29Formally represented by assumptions pA2q and pA3q. CHAPTER 1. CONSPICUOUS SCHOLARSHIP 36

altogether. The main conclusion from our investigation is that the behavior which predominates in the theoretical physics community in the eyes of some observers is not simply one among many plausible patterns of institutionalized behavior, but the only one that is likely to emerge under sufficiently high levels of competition. The essential underlying assumption in this perspective, is that conveying information about their research abilities is one fundamental motivation in the choice of research topics in the early stages of academics’ careers. Moreover, this kind of competitive unraveling is likely in other academic disciplines, and in general in industries which rely on significant amounts of on-the-job screening and in which the information revealed by agents’ performance varies across the different tasks to which they can choose to devote their time.

The model offers a number of empirical implications which could be tested using data on publication and citation patterns, regarding not only theoretical physics but other academic disciplines and sub-disciplines devoted to research with no immediate applications30, during periods of reputed high competitiveness. Some of these are: 1) We should observe young researchers working on well established fields, with an abundance of long standing questions with a strong reputation of being difficult. 2) Research in any given topic should persist well beyond the last observed significant contribution, and to the extent that the topic has had fewer recent breakthroughs it should be relatively more attractive for young scholars. 3) Older, well established scientists should be the only ones opening new research avenues.

While this tendency towards conspicuousness may be seen as detrimental in certain contexts, it can also serve the crucial role of providing the kind of continuity and persistence required for certain scientiﬁc breakthroughs. In fact, as can be seen throughout the history of science, the apparent fertility or barrenness of a scientiﬁc agenda at a given moment in time is often a bad predictor of its future.

30In applied ﬁelds, it may be the case that the research topics are directly determined by funding agencies, and external market forces. CHAPTER 1. CONSPICUOUS SCHOLARSHIP 37

1.7 Appendix: Omitted Proofs

Proof of Lemma 1: Note that we can always construct a twice diﬀerentiable extension of p in r0, 1s, r1, ms. The cross derivative of logpppθ, kqq is

B p p qq p q p q ¡ p q p q log p θ, k p1,2 θ, k p θ, k p1 θ, k p2 θ, k BθBk ppθ, kq2 which is strictly positive as the ﬁrst term in the numerator is strictly positive due to pA3q and the second term is strictly negative due to pA1q and pA2q31.

Proof of Lemma 2: We will show that society’s beliefs about an agent’s type after observing her a succeed in any given arena must first order stochastically dominate its beliefs after observing her fail. This implies that conditional on the behavior and performance of all other players, i’s position in any employer’s ranking after succeeding in a research arena must always be weakly higher than her ranking after a failure in that same research arena, which in turn means that the global expected payoff of a player conditional on success in a research arena Sk must be weakly greater than her expected payoff conditional on failure in that same research arena

Uk. Formally we will show that

@k, u P r0, 1s πspθi ¤ u|ai tk, si 0q ¥ πspθi ¤ u|ai tk, si 1q

1 In a sequential equilibrium, society’s beliefs about i s research ability θi at any information set following i’s choice ai of research arena, and her subsequent performance must be consistent with its beliefs after seeing the agent’s choice but not her performance32. That is:

31That under pA1q and pA2q increasing differences of the logarithm of the success probabilities is weaker than increasing differences can be shown to be true without appealing to the existence of the second partial derivatives and differentiability, but the proof is not any more enlightening than the above. 32Formally, a sequential equilibrium only specifies the beliefs of agents at information sets at which they are called on to move, which is what is required in order to verify sequential rationality. This means that in our game, society’s beliefs are only specified for when it has already seen an agent’s performance. However, on the equilibrium path, its beliefs at any other node of the extensive form are fully implied by the equilibrium strategies. Moreover, any perturbation of the players’ strategies that reaches all information sets with positive probability implies specific beliefs at all nodes. By the CHAPTER 1. CONSPICUOUS SCHOLARSHIP 38

Proof of Proposition 2: Consider the strategy proﬁle given by σpθq tm. The following beliefs are consistent, and they satisfy restriction pR1q: deﬁnition of sequential equilibrium, society’s beliefs at any of its information sets must be consistent, in the sense of being the limits of some sequences of beliefs obtainable from sequences of perturbations converging to the equilibrium strategies. Given any such sequence, the post-performance beliefs can be derived using Bayes rule from the pre-performance, post-choice implicit beliefs. It follows that a necessary condition for consistency is the existence of some pre-performance, post-choice beliefs that can simultaneously give rise to society’s beliefs at all subsequent post-performance information sets. CHAPTER 1. CONSPICUOUS SCHOLARSHIP 39

• On the equilibrium path beliefs. These are defined by Baye’s rule. Specifically, given u P r0, 1s let: ³u ppθ, mqdF pθq π pθ ¤ u|a t , s 1q 0 s i i m i ³1 ppθ, mqdF pθq 0 ³u p1 ¡ ppθ, mqqdF pθq π pθ ¤ u|a t , s 0q 0 s i i m i ³1 p1 ¡ ppθ, mqqdF pθq 0 • Off-the equilibrium path beliefs. Consider some ε P p0, 1q and for k m and u P r0, 1s let:

k πspθi ¤ u|ai tk, si 1q πspθi ¤ u|ai tm, si 0q ε p1 ¡ πspθi ¤ u|ai

tm, si 0qq,

Let Gk be a CDF with domain on r0, 1s satisfying: ³u ppθ, mqdGkpθq π pθ ¤ u|a t , s 1q 0 s i i m i ³1 ppθ, mqdGkpθq 0

Given Gk we can then deﬁne: ³u p1 ¡ ppθ, mqqdGkpθq π pθ ¤ u|a t , s 0q 0 s i i m i ³1 p1 ¡ ppθ, mqqdGkpθq 0

Since the beliefs that an agent induces by failing at tm ﬁrst order stochastically dominate the beliefs that he would induce by succeeding in any research arena other than tm no type would deviate. Moreover, these beliefs satisfy restriction pR1q by construction.

Proof of Claim 4: Let k1 ¡ k, and suppose that assumption pA3q holds. Let G be the CDF representing society’s prior beliefs about agent i’s type. Then we have that for all u P r0, 1s: CHAPTER 1. CONSPICUOUS SCHOLARSHIP 40

πspθi ¤ u|ai tk1 , si 1q ¤ πspθi ¤ u|ai tm, si 0q ³u ³u ppθ, k1qdGpθq ppθ, kqdGpθq ô 0 ¤ 0 ³1 ³1 ppθ, k1qdGpθq ppθ, kqdGpθq 0 0 ³1 ³1 ppθ,kq p q ppθ,k1q p q ppu,kq dG θ ppu,k1q dG θ ô u ¤ u ³u ³u ppθ,kq p q ppθ,k1q p q ppu,kq dG θ ppu,k1q dG θ 0 0 ppθ,kq P p s which is true given that ppu,kq is increasing in k for all θ u, 1 and decreasing in k for all θ P r0, uq. This establishes the weak inequality Moreover for all u such that G places a strictly positive probability on θ being smaller than u, the inequality is strict. Chapter 2

Social Capital and Social Quilts: Network Patterns of Favor Exchange

2.1 Introduction

Human beings rely on cooperation with others for their survival and growth. Although some forms of cooperation and behavior are enforced by social, religious, legal, and political institutions that have emerged throughout history, much of development, growth, and basic day-to-day functioning relies on a society’s ability to “informally” encourage cooperative behavior. This sort of informal enforcement of cooperation ranges from basic forms of quid-pro-quo (or tit-for-tat in the game theory parlance) to more elaborate forms of social norms and culture, all of which must function without enforceable contracts or laws.1 Indeed, contracting costs are prohibitive for many day-to-day favors that people exchange, ranging from oﬀering advice to a colleague, a small loan to a friend, or emergency help to an acquaintance. Such informal favor exchange and cooperative behaviors, in one sort or another, underly much of the

1In fact, the term “ostracism” (which has Greek origins based on a practice of banishments that originated in the Athenian democracy) has come to embody the idea of individuals cutting ties with members of society who do not perform properly.

41 CHAPTER 2. SOCIAL QUILTS 42

literature on social capital. Although there is a large literature on social capital, there is a paucity of work that provides careful foundations for how social structure relates to such favor exchange and cooperative behavior. Moreover, as we show here, favor networks do not exhibit the suggested patterns predicted by the previous literature that has considered network architecture. These points are related to each other since some standard network measures have emerged only loosely from the literature discussing the role of networks in fostering cooperation. In particular, the importance of social pressures on fostering cooperation has deep roots in the sociology literature including seminal work by Simmel (1950) [79], Coleman (1988) [25] and more recently by Krackhardt (1996) [60], among others (see the literature discussion below). Standard measures of network clustering and transitivity have grown in part out of those works. Clustering measures examine the extent to which two friends of a given agent are friends of each other. In the data on favor exchange networks in Rural India that we examine here, clustering is on the order of ten to thirty percent. A puzzle emerges as to why one sees that level of clustering, and not some other higher level, and even whether clustering is really the appropriate measure for capturing social pressures. In contrast, the new concept of “support” that emerges from our theoretical analysis measures the number of pairs of friends that have some other friend in common. As we shall see in the data, support is several times higher than clustering, and indeed this distinction is consistent with the theory presented here. To be specific, in this paper we provide a game theoretic foundation for social enforcement of informal favor exchange, and also examine network patterns of favor exchange from 75 rural villages. In particular, we consider settings where simple bilateral quid-pro-quo enforcement is insufficient to sustain favor exchange. Some bilateral interactions may be infrequent enough that they fail to allow natural self- enforcement of cooperation or favor exchange. However, when such interactions are embedded in a network of interactions whose functioning can be tied to each other, then individuals can find it in their interest to cooperate given (credible) threats of ostracism or loss of multiple relationships for failure to behave well in any given relationship. We provide complete characterizations of the network patterns of favor CHAPTER 2. SOCIAL QUILTS 43

exchange that are sustainable by a form of equilibrium satisfying two robustness criteria. The setting that we examine is such that opportunities for one agent to do a favor for another agent arrive randomly over time. Providing a favor is costly, but the benefit outweighs the cost, so that it is efficient for agents to provide favors over time. However, it could be that the cost of providing a favor today is sufficiently high that it is not in an agent’s selfish interest to provide the favor even if that means that he or she will not receive favors from that person again. Thus, networks of relationships are needed to provide sufficient incentives for favor exchange, and it may be that an agent risks losing several relationships by failing to provide a favor. We characterize the network structures that correspond to robust equilibria of favor exchanges. The criteria that we examine are twofold: first, the threats of which relationships will be terminated in response to an agent’s failure to deliver a favor must be credible. Credibility is captured by the game theoretic concept of “renegotiation- proofness”.2 After an agent has failed to deliver a favor, that relationship is lost, but which additional relationships are lost in the continuation equilibrium, must be such that there is not another equilibrium continuation that all agents prefer to the given continuation. This sort of renegotiation-proofness rules out unreasonable equilibria such as the “grim-trigger” sort of equilibrium where once anyone fails to provide a single favor the whole society grinds to a halt and nobody provides any favors in the future. At that point, it would be in the society’s interest to return to some equilibrium where at least some favors are provided. Renegotiation-proof equilibria can be complex, but have some nice intuitions underlying their structure as we explain in detail in the paper. The second criterion that we impose is a robustness condition that we call “robustness against social contagion.” It is clear that to sustain favor exchange, an agent must expect to lose some relationships if the agent fails to deliver a favor. Those lost relationships can in turn cause other agents to lose some of their relationships since the incentives to provide favors change with the network structure. This can lead to some fragility of a society, as one agent’s bad behavior can ripple

2Although there are several definitions in the literature for infinitely repeated games, our games have a structure such that there is a natural definition which has an inductive structure reminiscent of that of Benoit and Krishna (1993) [10]. CHAPTER 2. SOCIAL QUILTS 44

through the society. The robustness against social contagion requires that the ripple effects of some agent’s bad behavior be confined to that agent’s neighbors and not propagate throughout the network. The combination of renegotiation-proofness and robustness tie down a unique type of network configuration of favor exchanges that are possible. We call those configurations “social quilts.” A social quilt is a union of small cliques (completely connected subnetworks), where each clique is just large enough to sustain cooperation by all of its members and where the cliques are laced together in a tree-like pattern. One of our main theoretical results shows that configurations of favor exchange that are sustained in robust equilibria are precisely the social quilts. We then extend the model to allow heterogeneity in the cost and value of favors to various individuals. Under that extension, we prove that all robust equilibrium networks must exhibit a specific trait: each of its links must be “supported.” That is, if some agent i is linked to an agent j, then there must be some agent k linked to both of them. This is related to, but turns out to be quite different from, various clustering measures that are common in the social network literature. With the theoretical underpinnings in hand, we then examine social networks in 75 villages in southern rural India.3 These data are particularly well-suited for our study as they provide network structure for various favor relationships, and moreover have this for many separate villages. We are not aware of any other data set having these attributes. In particular, in these data we have information about who borrows rice and kerosene from whom, who borrows small sums of money from whom, who gets advice from whom, who seeks emergency medical aid from whom, and a variety of other sorts of relationships, as well as gps data. Using these data we can examine the networks of various forms of social interaction including specific sorts of favor exchange. In line with the theoretical predictions, we find that the number of favor links that have this sort of social support is in the range of eighty percent in these

3Although we apply some of our ﬁndings to favor relationships in Indian Villages, such informal favor exchange is clearly not limited to developing countries. For example, a recent New York Times/CBS News poll (reported in the New York Times, December 15 2009) found that 53 percent of surveyed unemployed workers in the U.S. had borrowed money from friends or family as a result of being unemployed. CHAPTER 2. SOCIAL QUILTS 45

villages. Moreover, the level of support is significantly higher than what would arise if links were formed at random (even with some geographic bias to formation), and significantly higher than levels of clustering. We analyze various aspects of the levels of support and also find that it is significantly higher for favor relationships than other sorts of relationships. Our research contributes to the understanding of informal favor exchange as well as social networks in several ways:

• We provide an analysis of repeated interactions where individual’s decisions are inﬂuenced by the network pattern of behavior in the community, and this provides new insights into repeated games on networks.

• Our model includes dynamic choices of both favor provision and relationship choices and provides new insights into the co-evolution of networks and behavior, and in particular into the phenomenon of ostracism.4

• Our analysis suggests a new source of ineﬃciency in informal risk and favor sharing, showing why individuals may have to limit the number of relationships in which they take part.

• A by product of our analysis is an operational definition of social capital that is more specific and tighter than many existing definitions, and it makes tight predictions about how relationships in a society must organized.

• We provide a new property of networks that we call “support” and show how this is distinguished from standard clustering measures.

• We examine data that include many sorts of interactions and cover 75 different villages, and find that the networks exhibit substantial and significant distinc- tions between our measure of support and standard measures of clustering.

4As we shall see, ostracism has further consequences in terms of lost relationships, beyond those directly involving the individual being punished. CHAPTER 2. SOCIAL QUILTS 46

2.1.1 Related Literature

As mentioned above, there is a large literature on social capital that studies the ability of a society to foster trust and cooperation among its members.5 Although that literature is extensive and contains important empirical studies and many intuitive ideas, it has struggled in providing firm theoretical foundations and the term “social capital” has at times been used very loosely and as a result has lost some of its bite.6 Part of the contribution of our paper is to provide an explicit modeling of how societies can enforce cooperative favor exchange and how this is linked to the social network structure within a society. In this way, our paper provides a very concrete definition of social capital that is embedded in three components: a notion of equilibrium that embodies notions of ostracism and social expectations of individual behaviors, implications of this for resulting social network structure, and individual payoffs from the resulting behaviors. Coleman (1988) [25] discusses closure in social networks, emphasizing the ability of small groups to monitor and pressure each other to behave. Here we provide a new argument for, and a very specific variety of, closure. Here a specific form of minimal clique structures emerge because of a combination of renegotiation-proofness and a local robustness condition, rather than for informational, monitoring, or pressuring reasons. Minimal sized cliques offer both credible threats of dissolving in the face of bad behavior, and in terms of minimal contagion for a society. Our analysis also formalizes this in terms of support and contrasts it with clustering. The most closely related previous literature in terms of the theoretical analysis of a repeated game on a network is a series of papers that study prisoners’ dilemmas in network settings, including Raub and Weesie(1990) [75], Ali and Miller (2009) [4], Lip- pert and Spagnolo(2011) [62], and Mihm, Toth, Lang (2009) [69].7 In particular, Raub

5For example, see Homans (1958) [52], Loury (1977) [63], Bourdieu (1986) [16], Coleman (1988, 1994) [25] [26] , Woolcock (1998) [88], Dasgupta (2001) [29], Putnam (1993, 1995, 2001) [72] [73] [74], Glaeser, Laibson, and Sacerdote (2002) [44], Guiso, Sapienza, and Zingales (2004) [49], Tabellini (2010) [83], among others. 6See Sobel (2002) [81] for an illuminating overview and critique of the literature. 7Other studies of network structure and cooperative or various forms of risk-sharing behavior and the relationship to social network structures include Fafchamps and Lund (2003) [36], De Weerdt and Dercon (2006) [86], Bramoull´eand Kranton (2007) [17], Bloch, Genicot, and Ray (2007, 2008) CHAPTER 2. SOCIAL QUILTS 47

and Weesie(1990) [75] and Ali and Miller (2009) [4] show how completely connected networks shorten the travel time of contagion of bad behavior which can quicken punishment for deviations. Although cliques also play a prominent role in some of those papers, it is for very different reasons. In those settings, individuals do not have information about others’ behaviors except through what they observe in terms of their own interactions. Thus, punishments only travel through the network through contagious behavior (or word-of-mouth), and the main hurdle to enforce individual cooperation is how long it takes for someone’s bad behavior to come to reach their neighbors through chains of contagion. This builds on earlier work by Greif (1989) [48], Kandori(1992) [55], Ellison (1994) [35], Okuno-Fujiwara and Postlewaite(1995) [70] among others, who studied the ability of a society to sustain cooperation via threats of contagions of bad behavior. Our analysis is in a very different setting, where individuals have complete information. The quilts in our setting emerge because they do not lead to large contagions but instead compartmentalize the damage from an individual’s defection. Moreover, the quilts consist of minimal sized cliques because only those sorts of implicit punishments are immune to renegotiation. Haag and Lagunoff (2004) [50] provide another reason favoring small cliques: heterogeneity. In their analysis large differences in preferences can preclude cooperative behavior, and so partitioning a group into more homogeneous subgroups can enable cooperative behavior which might not be feasible otherwise. Although our reasoning behind cliques comes from very different reasons, when we examine heterogeneous societies we do find assortativity in who exchanges favors with whom. Here, the reasoning is not because of direct reciprocity considerations, but because robustness requires balanced cliques and so agents need to have similar valuations of favors in order for their cliques to be critical. In this way, we provide new insights into ho- mophily, where relationships of agents are biased towards others who have similar characteristics in terms of their values and arrival rates of favors. Finally, our analysis of the data not only provides support for the support measure, but also uncovers significant differences between different sorts of networks. Differ- ences between the network structure of various sorts of relationships is something

[13] [14], Karlan, Mobius, Rosenblat and Szeidl (2009) [56], and Balmaceda and Escobar (2011) [7]. CHAPTER 2. SOCIAL QUILTS 48

that might be expected based on the diﬀerent ways in which links might form across applications (e.g., see Jackson (2008) [53]) and here we add a new angle to this understanding, ﬁnding statistically distinct patterns of support in various sorts of favor and social networks. These suggest some interesting questions for future research.

2.2 A Model of Favor Exchange

2.2.1 Networks, Favors, and Payoﬀs

A ﬁnite set N t1, . . . , nu of agents are connected in a social network described by an undirected8 graph. Given that the set of agents or nodes N is ﬁxed throughout the analysis, we represent a network, generically denoted g, simply by the set of its links or edges. Let gN be the set of all links (so the set of all subsets of N of size 2), and let G tg | g gN u be the set of all possible networks. For simplicity, we write ij to represent the link ti, ju, and so ij P g indicates that i and j are linked under the network g. We write g ¡ ij to denote the network obtained from g by deleting a link ij. For an integer k, 0 ¤ k ¤ npn ¡ 1q{2, let Gk be the set of all networks that have exactly k links, so that Gk t g P G : |g| ku. The neighbors of agent i are denoted

Nipgq tj | ij P gu.

We follow a convention that rules out self-links, and so all agents in Nipgq are distinct from i. The degree of agent i in the network g is the number of his or her neighbors denoted by dipgq |Nipgq|. Time proceeds in discrete periods indexed by t P t0, 1,...u and in any given period, there is a chance that an agent will need a favor from a friend or will be called upon to grant a favor to a friend. In particular, an agent i who is connected to an agent j (so that ij P g) anticipates a probability p ¡ 0 that j will need a favor from i in period t and a probability p that i will need a favor from j. It is assumed that at

8This is not necessary for the analysis, and we comment later on possible extensions to directed networks. CHAPTER 2. SOCIAL QUILTS 49

most one favor will be needed across all agents in any given period, and so we require that npn ¡ 1qp ¤ 1. we allow the sum to be less than one to admit the possibility that no favor is needed in a given period. This is a proxy for a Poisson arrival process, where the chance that two favors are needed precisely at the same moment is 0. By letting the time between periods be small, the chance of more than one favor being called upon in the same period goes to 0. Thus, when applying the model it is important to keep in mind that periods are relatively small compared to the arrival rate of favors. A restriction of this formulation is that p does not depend on the network structure. More generally, the chance that i needs a favor from j will depend on many things including how many other friends i has. We characterize the equilibrium networks for the general case in Section 2.5. We begin with the current case since it more clearly provides the basic intuitions, but the results have very intuitive analogs for the general case that are easy to describe once we have presented the simpler case. Doing a favor costs an agent an amount c ¡ 0 and the value of the favor to an agent is an amount v ¡ c. “Favors” can embody many things including asking for advice, to borrow some good, to borrow money, or to perform some service. The important aspect is that the value of a favor to one agent exceeds the cost, so that it is ex ante Pareto eﬃcient for agents to exchange favors over time. However, we examine settings where it is impossible (or too costly) for agents to write binding contracts to perform favors whenever called upon to do so. This applies in many developing countries, and also in developed countries where it is prohibitively costly and complex to write complete contracts covering the everyday sort of favors that one might need from friends. Thus, we examine self-enforcing favor exchange. Agents discount over time according to a factor 0 δ 1. Thus, if there were just two agents who always performed favors for each other, then they would each CHAPTER 2. SOCIAL QUILTS 50

expect a discounted stream of utility of

p pv ¡ cq . 1 ¡ δ

The more interesting case from a network perspective is the one that we examine, where δp pv ¡ cq c ¡ . 1 ¡ δ In this case, favor exchange between two agents in isolation is not sustainable. When called upon to perform a favor, the agent sees a cost that exceeds the future value of potential favor exchange (in isolation) and so favor exchange cannot be sustained between two people alone, but must be embedded in a larger context in order to be sustained. Sustaining favor exchange between two individuals requires a high enough frequency of arrival coupled with a high enough marginal benefit from a favor and sufficient patience. In a marriage, there are generally sufficiently many opportunities for each spouse to help the other out with some task or need that bilateral favor exchange can be sustained. However, in other contexts, where such needs are rarer - such as a need to borrow cash due to an emergency, or a need for medical advice, etc., one might need a multilateral setting to sustain favor exchange. A society is described by the profile pN, p, v, c, δq.

2.2.2 The Game

The favor exchange game is described as follows.

• The game begins with some initial network in place, denoted g0.

• Period t begins with a network gt¡1 in place.

• Agents (simultaneously)9 announce the links that they are willing to retain: p q 1 t | P P u Li Ni gt¡1 . The resulting network is gt ij j Li and i Lj . 9Given the equilibrium reﬁnements that we use, whether or not the link choices are simultaneous is eﬀectively irrelevant. CHAPTER 2. SOCIAL QUILTS 51

1 • Let kt be the number of links in gt. With probability 2pkt need for a (single)

favor arises and with probability 1 ¡ 2pkt there is no need for a favor in the 1 period. If a favor is needed, then it could apply to any link in gt with equal

likelihood and then go either direction. If a favor is needed, then let it denote

the agent called upon to do the favor and jt the agent who needs the favor, P 1 where itjt gt.

• Agent it chooses whether or not to perform the favor. If the favor is performed

then it incurs the cost c and agent jt enjoys the beneﬁt v. Otherwise no cost or beneﬁt is incurred.

1 ¡ • The ending network, denoted gt, is gt itjt if the need for a favor arose and it 1 was not performed, and is gt otherwise.

People make two sorts of choices: they can choose with whom they associate and they can choose to do favors or not to do favors. Opportunities for favor exchange arise randomly, as in a Poisson game, and people must choose whether to act on favors as the need arises. Choices of which relationships to maintain, however, can be made essentially at any time. In the model this is captured by subdividing the period into link choices and favor choices, so that agents have a chance to adjust the network after any favor choice, as well as before any potential favor arises. This structure embodies several things. First, favor relationships can either be sustained or not. Once a favor is denied, that relationship cannot be resuscitated. Thus, at any point in time an agent’s decision is which relationships to maintain. This simplifies the analysis in that it eliminates complicated forms of punishments where various agents withhold favors from an agent over time, in order to punish an agent, but then eventually revert to providing favors. It can be motivated on various behavioral (e.g., emotional) or pro-social grounds and effectively it acts as a sort of refinement of the set of all possible punishments that might occur, as it requires that one of the ostracizing agents be the one who failed to get the favor. Eventually, one would like to extend the analysis to situations where after some period of time forgiveness is possible, but this simplification allows us to gain a handle on sustainable network structures as the problem is already very complex (as CHAPTER 2. SOCIAL QUILTS 52

will become clear shortly), and it appears that much of the intuition carries over to the more ﬂexible case, but that is a subject for further research. As will be clear, this approach generates quite a rich, natural, and interesting set of conclusions. Second, we do not consider the formation of new links, but only the dissolution of links. This embodies the idea that the formation of new relationships is a longer-term process and that decisions to provide favors and/or ostracize an agent can be taken more quickly and are shorter term actions. It is important to note that we cover the case where society starts with the complete network, so we do not a priori restrict the links that might be formed, and so our results do make predictions about which networks can be formed/sustained in a society. The important wedge that we impose is that an agent who has lost a relationship cannot (quickly) replace it with a newly formed one. One other aspect of the model is important to mention. Agents do not exchange money for favors although at least hypothetically, all favor exchange could simply be monetarized. Of course we do not see this in reality, as when a colleague asks to borrow a book we would not charge her or him a rental fee; but that observation does not explain why we do not charge our friends and acquaintances for every little favor that we perform. One explanation is a behavioral one: that monetarizing favors would fundamentally change the way in which people perceive the relationship, and this explanation is consistent with people no longer viewing a monetarized relationship as a long run relationship. For more discussion of this point see Kreps (1997) [61]. The speciﬁcs of why all favors are not monetarized is outside of our scope, and our starting point is one of simple favor exchange. For now, we consider a complete information version of the game, in which all agents observe all moves in the game at every node. We discuss more limited information variations in section 2.7.1. An agent i’s expected utility from being in a network g that he or she expects to exist forever10 is d pgqp pv ¡ cq u pgq i i 1 ¡ δ 10This applies at any point within the period other than at the a time at which the agent is called to receive or produce a favor. CHAPTER 2. SOCIAL QUILTS 53

If an agent i is called to do a favor to j and chooses to perform the favor and expects a network g to be played in perpetuity thereafter, then he or she expects a utility of

¡c δuipgq.

Similarly if agent i is called to receive a favor from agent j and expects to receive the favor and then anticipates a network g to be played in perpetuity thereafter then his or her expected discounted stream of utility is given by

v δuipgq.

For ease of expression, we assume that the discounting parameter δ enters the agents’ calculations between the announcement stage and the favor stage in any given pe- riod11.

We note that although we work with favor exchange as a building block in our model, the analysis here directly extends to supporting cooperation more generally, and the same results apply to the play of a prisoners dilemma between agents, or other forms of trust and cooperation games with free-rider or short-term deviation challenges.

2.2.3 Equilibrium

In this setting, any network of favor exchange g can be sustained in perpetuity as a pure strategy subgame perfect equilibrium as long as

δp pv ¡ cq c d pgq i 1 ¡ δ 11This is purely for expository ease as it slightly simplifies the expressions for the critical utility levels for different behaviors, but does not alter the basic structure of the arguments or conclusions. Effectively, it ensures that whenever an agent is either making a decision of which links to announce or whether to follow through on a favor, any potential future favors that might be influenced by the decision are discounted. If discounting happens after the favor period, then when making link choices agents would not discount one round of future favors, but when making favor decisions they would. This simply makes sure that all future favors are discounted in the same way. It is also convenient to begin with an announcement phase, but again this is not essential to the conclusions. CHAPTER 2. SOCIAL QUILTS 54

for every agent i. One way in which this is sustained is by a sort of “grim-trigger” strategy where all relationships are sustained and favors are provided as long as no agent refuses a favor, and once any favor is denied then all agents delete all their links and never expect to receive any favors again in the future. Thus, if each agent has enough relationships that he or she might lose, then favor exchange can be sustained. So, for instance, if δp pv ¡ cq c pn ¡ 1q 1 ¡ δ then the complete network with the most eﬃcient favor exchange could be sustained.

Renegotiation-Proofness

While the above conclusion offers some optimism regarding a society’s ability to efficiently sustain favor exchange, it rests upon drastic threats that are not always credible. If for some reason a favor was not performed and some link disappears, a society might wish to reconsider its complete dissolution. Indeed, the idea that if some person in a society acts selfishly and fails to provide a favor, the whole society collapses and no favors are ever exchanged again is drastic and unrealistic. This sort of observation is not unique to this setting, but has been an issue in repeated games for some time (e.g., see the discussion in Bernheim, Peleg and Whinston (1987) [11]). The basic problem is that if agents have some chance to communicate with each other (and perhaps even if they cannot), then when beginning some phase of equilibrium which involves low payoffs, if there is some other equilibrium continuation, in which all agents are better off, then they have a strong incentive to change to the play that gives them all better payoffs. Even though this sort of “renegotiation” problem with many sorts of equilibria is well known, it is rare for researchers to do more than to acknowledge it and forge ahead. The reason for this is that properly accounting for renegotiation becomes quite complicated, especially in infinite settings where it is not even clear how to define equilibrium in the face of renegotiation (e.g., see Farrell and Maskin (1989) [38], Bernheim and Ray (1989) [12], and Abreu, Pearce and Stacchetti (1993) [2]). Thus, there are few analyses of renegotiation-proofness outside of some of the original papers working out the definitions, although some instances of it appear CHAPTER 2. SOCIAL QUILTS 55

in other forms (e.g., repentance strategies in Lippert and Spagnolo (2011) [62]). Our setting has a nice structure that makes it relatively easy to provide a natural deﬁnition of renegotiation-proofness and to characterize such equilibria. Before moving to the formal deﬁnitions, we present an example that illustrates the ideas.

Example 1 The Logic of Renegotiation-Proofness

Let there be 4 agents and consider a case such that

δppv ¡ cq δppv ¡ cq 2 ¡ c ¡ 1 ¡ δ 1 ¡ δ

Here, no link is sustainable in isolation, since the value of providing a favor c is greater pδpv¡cq than the value of the future expected stream of giving and receiving favors: 1¡δ . However, if an agent risks losing two links by not performing a favor, then links could pδpv¡cq be sustainable depending on the configuration of the network, since c 2 1¡δ . In this case, note that networks where each agent has exactly two links, for example, g t12, 23, 34, 41u, can be sustained as a subgame perfect equilibrium. If any agent ever fails to perform a favor, then a link will be lost. For example, suppose that 1 fails to deliver a favor to 2, and so the link 12 is lost. At this point, agent 1 only has one relationship left: 14. It is now clear that agent 1 will never perform future favors for 4 and so the link 14 is effectively useless as well. The same is true of 23. Iterating on this logic, there is no subnetwork that could be sustained as a subgame perfect equilibrium. As such, an agent realizes that if he or she fails to provide a favor, then that will lead to a collapse of all favors and so the network of favor trading is sustained in equilibrium, as failing to provide one favor induces a loss of two relationships. So, starting with such a minimal network there is no difficulty with renegotiation, as following any deviation from the prescribed favor exchange the equilibrium continuation is unique. So, favor exchange sustaining this network is renegotiation-proof as an equilibrium (to be defined shortly). The problematic subgame perfect equilibria come with k 5 or more links. Consider the network g1 t12, 23, 34, 41, 13u as pictured in Figure A.2. Agents 1 and 3 each have three links and agents 2 and 4 have two links. There is a subgame CHAPTER 2. SOCIAL QUILTS 56

Figure 2.1: A ﬁve link network that is not sustained as a renegotiation-proof equilibrium perfect equilibrium sustaining this network: if any link is ever cut, then all agents cut every link in the future. However, there is no renegotiation-proof equilibrium sustaining this network. To see this, suppose that agent 1 is called upon to do a favor for agent 3. If agent 1 does not do the favor, then the resulting network is g1 t12, 23, 34, 41u. Note that g is sustainable as a subgame perfect equilibrium as argued above (and in fact is sustainable as part of a renegotiation-proof equilibrium). The logic is now that if g is reached, then it will be sustained rather than having agents delete all links, since it is not credible for agents to destroy these links as they are all strictly better oﬀ sustaining g than going to autarchy. Thus, when reaching g, in the absence of some exogenous commitment device, the agents’ previous threat to delete all links lacks credibility. As a result of this, agent 1 can cut the link 13 and still expect the network g to endure, and so this is the unique best response for agent 1 and so g1 is not sustainable as an equilibrium if we require that continuations not be Pareto dominated by another (renegotiation-proof) equilibrium continuation.

We define renegotiation-proof networks to be networks that are sustainable via pure strategy renegotiation-proof equilibria. It is easiest to define the networks directly, in a way that simultaneously implicitly defines renegotiation-proof equilibrium and explicitly tracks the networks that are sustainable via those equilibria. We define the set of pure strategy renegotiation-proof equilibria inductively.12,13

12Our analysis concentrates on pure strategy equilibria. As will become clear, considering mixed strategy equilibria would not add much to the insights regarding sustainable network structures. 13Our deﬁnition is in the spirit of Pareto perfection of Bernheim, Peleg, and Whinston (1987) [11], CHAPTER 2. SOCIAL QUILTS 57

The induction operates via the number of links in the starting network. In terms of notation, it will be useful to keep track of the set of all networks that have exactly k links and can be sustained in perpetuity as part of a pure strategy renegotiation-proof equilibrium if we start at that network.

We let RPNk denote the set of renegotiation-proof networks with k links.

• Let RPN0 tHu

• Let RPNk denote the subset of Gk such that g P RPNk if and only if beginning 14 with g0 g implies there exists a pure strategy subgame perfect equilibrium such that

– on the equilibrium path g is always sustained (all favors are performed and all links are maintained), and

15 1 1 2 – in any subgame starting with some network g P Gk1 with k k if g is played in perpetuity with some probability16 in the continuation then 2 2 3 1 g P RPNk2 for some k and there does not exist any g g such that 3 3 2 g P RPNk3 and uipg q ¥ uipg q for all i with strict inequality for some i.17

The concept is fairly straightforward even though the definition is inductive. It requires that any continuation equilibrium (e.g., a threat of what will happen if some agent deviates) not be Pareto dominated by some other continuation equilibrium. Thus, if society gets to some point where they are supposed to follow through with some equilibrium continuation, they should not all find it better to follow some other (feasible) continuation. The self-referential nature of the definition comes from the but instead of induction with respect to the stages of the game, we work with induction on the size of the networks as our game is infinitely repeated. 14That the equilibrium is in pure strategies requires that agents use pure strategies at all nodes on and off the equilibrium path. 15This includes subgames starting at any node, not just beginning of period nodes. 16Even though agents use pure strategies, nature randomly recognizes favors, and so there can be some randomness in a continuation path. 17 1 1 Note that this condition implies that in any subgame starting with a network g P RPNk1 , g is played in the continuation. CHAPTER 2. SOCIAL QUILTS 58

fact that we want society only to view some continuation as a feasible option if it is renegotiation-proof itself - as otherwise the society should not expect that to be stable either. Thus, the definition is inductive, since the logic of renegotiation-proofness requires that a network sustained in some continuation not be Pareto dominated by any other continuation that itself is renegotiation-proof. The self-referential logic is what generally provides difficulties in identifying an unambiguously “correct” definition in an infinite setting. Here, despite our infinite horizon, we can define renegotiation- proofness cleanly since relationships can be severed but not resuscitated and so there is a natural induction on the number of links. We say that a network g is renegotiation-proof or a renegotiation-proof network if there exists some k such that g P RPNk. As a further illustration of the definition, let us return to Example 1 and characterize all of the renegotiation-proof networks.

Example 2 Renegotiation-Proof Networks

Let there be 4 agents and consider a case as in Example 1 such that

δppv ¡ cq δppv ¡ cq 2 ¡ c ¡ . 1 ¡ δ 1 ¡ δ

Here, RPN1 H since no isolated links are sustainable.

Similarly, RPN2 H since any agent who only has one link will never perform a favor.

RPN3 tg tij, jh, ihu : for some distinct h, i, ju. Thus, triads are sustainable, since if any agent severs a link, then that will lead to a two-link network which is not sustainable, and so becomes an empty network. Thus, not performing a favor leads to an empty network, and so it is a best response to perform a favor, anticipating favors by other agents.

RPN4 tg tij, jh, h`, ìu : for distinct h, i, j, ù. This is an obvious extension of the logic from three-link networks. Following the argument in Example 1 it is easy to check that there are no five-link renegotiation-proof equilibria. Thus RPN5 H. CHAPTER 2. SOCIAL QUILTS 59

Next, note that RPN6 H as well. To see this, note that if some agent i deletes a link ij, then a continuation must result in a pure strategy renegotiation- proof equilibrium, which would be either a triad, four link network (with a cycle), or the empty network. The remaining four link network that has a cycle Pareto dominates any other continuation. Thus, if an agent i severs a link ij, then the agent is sure that he or she will still have two links in the continuation and so only loses one link.18

Given this example, let us re-emphasize an aspect of our model that simplifies the analysis. Renegotiation-proofness serves as as a natural and plausible way to limit potential punishments, which tightens predictions of which networks of favor exchange are sustainable. However, its tractability relies in part on our assumption that when an agent fails to do a favor for another agent the relationship in question is deleted. This allows the analysis to move forward, as the renegotiation-proofness then applies to the subnetwork that remains and not the original network. By assuming that the link in question must be part of the punishment that ensues, renegotiation can be anchored via an induction argument and we avoid the substantial difficulties associated with non-inductive, self-referential definitions of renegotiation-proofness. In terms of justifying this approach, it is important to note that clearly some punishment must occur following a failure to provide a favor if favors are to be sustained. Thus, the assumption may be viewed as focusing on equilibria where the relationship in question is necessarily involved in that punishment. Let us offer two possible justifications for this focus, without explicitly expanding the model. One justification is that there is a behavioral or emotional response that induces an agent whose favor is denied to place a sufficiently negative weight on the deviator’s utility so as not to want to do any favors for the deviator in the future, even if that means losing some other relationships in the continuation. Justifications for such other-regarding preferences generally rely on psychological evidence or evolutionary-style arguments, and

18This conclusion depends on the way in which renegotiation is deﬁned. Here we have deﬁned it taking all agents into account. Another possibility is to only take the non-deviating agents into account, in which case the complete network can be sustained by having the three remaining agents sever all ties to any deviating agent. We explore this variation in the supplementary appendix. CHAPTER 2. SOCIAL QUILTS 60

are well-debated in the behavioral literature and so we do not discuss them here.19 Effectively, this redirects the justification of the assumption to a well-trodden discussion in behavioral economics. Another avenue for rationalizing this behavior is that there is a relationship specific type variable for each agent (some sort of “compatibility”) that agents learn about over time by observing others’ behaviors. If one agent fails to deliver a favor for another agent it could signal that the agent has a bad “type” for the compatibility of that relationship, and so will also fail to be able to perform favors in the future in that particular relationship, leading that relationship to no longer function. Having this be relationship-specific is important as then subsequent deletion of other relationships need not be tied to subsequent inferences of a player’s type in further relationships (just because i and j are incompatible, it does not necessarily imply that j and k will also be incompatible). Independence of types across relationships allows further deletions to be based on the viability of the remaining network structure, and not further inference about types.20

2.3 Characterizing Renegotiation-Proof Networks

In this section we provide a complete characterization of renegotiation-proof networks. Before providing the complete characterization, however, we provide some intuitive sufficient conditions that give insight into the structure of renegotiation-proof networks. 19See Camerer (2003) [20], Fehr and Schmidt (2006) [39], and Cooper and Kagel (2009) [27] for background discussion and evidence on other-regarding preferences. 20Note that expanding the model in this direction would allow for an analysis of favor exchange along our lines for some period of time; but after agents’ had sufficiently learned about each other’s types, then we would face an issue that agents would have an incentive to deviate and ask for forgiveness. Thus, analyzing the full long-term dynamics of such a model, and not just the short to medium run, would involve enriching the model to allow for some occasional deviations and potential forgiveness. That again complicates the analysis, so that the results presented here would hold for some period of time, but we do not have firm conjectures for what would happen in the very long run when all types are learned, and forgiveness is allowed. In that regard, the behavioral justification provides a longer-run justification for our results than the incomplete information story. CHAPTER 2. SOCIAL QUILTS 61

Let m be the whole number deﬁned by

δppv ¡ cq δppv ¡ cq m ¡ c ¡ pm ¡ 1q . (2.1) 1 ¡ δ 1 ¡ δ

It is clear that there is at most one such m and that m ¥ 1. We work with the generic case, ignoring exact equality on either side above. The δppv¡cq parameter m captures how many relationships, each with a future value of 1¡δ , an agent has to risk losing in the future in order to have incentives to perform a favor today at the cost of c. Throughout the remainder of the paper, the definitions will all be relative to m, and so we take it to be fixed and defined by (A.1) and omit explicit mention of it in some of the definitions. We begin with a formal statement of the proposition on subgame perfect equilibria that motivates our analysis of renegotiation-proof equilibria.

Proposition 7 A network is sustainable in perpetuity on the equilibrium path of a subgame perfect equilibrium of the favor exchange game if and only if each agent has either 0 links or at least m links.

The proof of Proposition 7 is obvious and thus omitted.

Now we examine renegotiation-proof networks.

2.3.1 Critical Networks and Renegotiation-Proofness

Before proceeding to the complete characterization of renegotiation-proof networks, we examine a natural class of renegotiation-proof equilibria.21 These suﬃcient conditions provide an intuitive look at equilibrium structure and help motivate the complete characterization. Let

Gpmq tg | @i, dipgq ¥ m or dipgq 0u

21Additional classes of renegotiation-proof equilibria are discussed in the supplementary appendix. CHAPTER 2. SOCIAL QUILTS 62

denote the set of networks in which each agent has either at least m links or 0 links. So Gpmq is the set of networks sustainable as subgame perfect equilibria. Note that any renegotiation-proof network must be in Gpmq and since any network sustained in any off-equilibrium path continuation must also be a renegotiation-proof network it must also be contained in Gpmq. Following the argument above, one way to build a sustainable network is to offer proper incentives for sustaining favors: if an agent deletes a link in a network or fails to provide a favor, it is sufficient that the agent expect to lose at least m links in the sequel. That is captured by the following definition. We say that a network g is m-critical, if

• g P Gpmq

1 1 • for any i and ij P g, there is no subnetwork g g ¡ ij such that dipg q ¡ 1 dipgq ¡ m and g P Gpmq.

As m ¥ 2 will generally be a given in the analysis (although we consider diﬀerent levels in some examples), we simply omit its reference and use the term “critical” in what follows. An easy way to build a critical network is to have each agent have exactly m links. However, we remark that criticality does not require all agents to have exactly m links. It only requires that any possible continuation equilibrium after some agent fails to provide a favor be such that the agent lose at least m links. To see how this can allow agents to have more than m links, consider Figure 2.2, which pictures such a critical network for a case with m 3 where agent 1 has four links. There is no proper nonempty subnetwork in which all agents who still have links, have at least 3 links each. Thus, if agent 1 (or any other agent) severs any link, the entire network would have to collapse since any proper subnetwork that is nonempty will have some agent with fewer than 3 links, and thus that agent will not have incentives to provide favors. Therefore, if agent 1 fails to provide a favor on some link, then agent 1 would lose all four links.

Critical networks provide an important and nonempty class of networks that are CHAPTER 2. SOCIAL QUILTS 63

Figure 2.2: A critical network for m 3, but where agent 1 has 4 links. renegotiation-proof. In the sense of Proposition 8, they are a foundational class of networks.

Proposition 8 Any nonempty network g P Gpmq contains a nonempty critical network, and any critical network is renegotiation-proof.

The last part of Proposition 8 follows from Theorem 1, but we mention the idea behind how this can be proven directly. In order to prove this, we need to show that there exists a pure strategy renegotiation-proof equilibrium such that for any i and ij P g, if i is called to grant a favor to j and refuses the favor, i must lose at least m links in any continuation network g1. Renegotiation-proofness requires that the continuation g1 be a subnetwork g1 g ¡ ij and be sustainable as a pure strategy renegotiation-proof equilibrium and so it must be in Gpmq. By the deﬁnition 1 1 of criticality, it then follows that any such g P Gpmq be such that dipg q ¤ dipgq ¡ m (and there always exists at least one such g1 since the empty network is in Gpmq), and so provides the necessary incentives. To see that criticality is not necessary for renegotiation-proofness, consider the network in Figure A.3: let m 2 and g t12, 23, 13, 14, 15, 45, 26, 27, 67u is a tree union of three triads. This is not critical since if 1 cuts the link 12, then all agents in the sub-network still have at least 2 links. Nonetheless, we shall see that it is renegotiation-proof below. CHAPTER 2. SOCIAL QUILTS 64

Figure 2.3: A non-critical network, but still renegotiation-proof.

2.3.2 A Complete Characterization of Renegotiation-Proof Networks: Transitively Critical Networks

We now turn to the complete characterization of renegotiation-proof networks. Let Dpgq denote the proﬁle of degrees associated with a network g:

Dpgq pd1pgq, . . . , dnpgqq.

Write Dpgq ¡ Dpg1q if Dpgq ¥ Dpg1q and Dpgq Dpg1q. We deﬁne transitively critical networks as follows.

Given a whole number m satisfying (A.1), let TCkpmq Gk denote the set of transitively critical networks with k links.

• Let TC0pmq tHu.

• Inductively on k, TCkpmq Gk is such that g P TCkpmq if and only if for any i 1 1 1 and ij P g, there exists g g ¡ ij such that g P TCk1 pmq for some k ¤ k ¡ m, 1 2 2 dipg q ¤ dipgq ¡ m, and there is no g P TCk2 pmq such that g g ¡ ij and Dpg2q ¡ Dpg1q.

A network g is transitively critical if when some link ij of i’s is deleted, then the next largest transitively critical network that is a subset of the remaining network involves a loss to i of at least m links. Consider, for instance, starting with a critical network where each agent has exactly m links. Now, if we start to add links for some CHAPTER 2. SOCIAL QUILTS 65

agent to such a network, we will have to add at least m links for that agent. This follows from renegotiation-proofness, as otherwise the agent could delete a link and expect to lose fewer than m links in landing at a critical network in the continuation. There are complicated interrelationships in the ways that links must be added, as adding links must keep the incentives for all agents in line. The definition of transitive criticality is inductive in a way that accounts for all of those incentive constraints at the same time. Basically, starting with some network that is transitively critical, the next superset of that network that will also be transitively critical, will have to add minimal numbers of links for each agent that gains links. Even though this is also an inductive definition (not surprisingly, given that renegotiation-proof equilibria are so defined), it does not involve any incentive descriptions and is effectively an algorithm that can identify equilibria directly from m. In fact, we use the equivalence set forth by Theorem 1 to build a computer program that calculates renegotiation-proof networks. In the supplementary appendix we present a table with the number of non-isomorphic transitively critical networks for small values of n and m, along with renderings of some of these networks.

Theorem 1 A network is renegotiation-proof if and only if it is transitively critical.

Although one might expect the proof to be straightforward given that both definitions are inductive, there are some subtleties and challenges in proving Theorem 1. The main one is that there are many strategy profiles that may hypothetically sustain a collection of networks in a subgame perfect manner, in a way such that the collection satisfies the self-consistency property demanded by renegotiation-proofness. The issue then is that to show that some network is not renegotiation-proof we must be sure that none among the large number of the different strategy profiles that could sustain it, actually work. Moreover, showing that some network is renegotiation-proof involves first showing that some other networks are not in the set. The way in which we tackle this difficulty is by arguing that we can avoid the vast set of potential equilibria that could be used to sustain networks and can focus on a nicely behaved set of strategy profiles. The details appear in the appendix. CHAPTER 2. SOCIAL QUILTS 66

Figure 2.4: Two renegotiation-proof networks when m 2 and n 9: a non robust one and a robust one. 2.4 Robustness

We now turn to our criterion of robustness. The idea is that a network is robust if it relies only on local “damage” due to a failure to provide a favor, rather than more global sorts of damage. That is, failure to provide a favor will require some lost links and there is a question of how far that loss of links propagates. We begin with a simple observation.

Observation 1 If (A.1) holds for m ¥ 2, g is a renegotiation-proof network, and ij P g, then g ¡ ij is not a renegotiation proof network.

The observation follows since otherwise the continuation from i or j failing to do each other a favor would only result in the loss of one link, and so they would not do each other favors and g would not be sustainable. The important implication of the observation is that beginning from a network that is renegotiation-proof, if a link is deleted then the network will necessarily further degrade in terms of what is sustainable. There may be diﬀerent ways in which things could degrade. Here is where the idea of robustness comes into play. Robust- ness against social contagion seeks to minimize the extent to which the loss of links propagates beyond the original deviator(s) in the network.

Example 3 Robustness CHAPTER 2. SOCIAL QUILTS 67

Suppose that (A.1) holds for m 2, and there are n 9 agents. Figure 2.4 lists two renegotiation-proof networks: one is a single cycle containing all agents, t12, 23, 34, 45,..., 91u; the other one is a tree union of four triads, t12, 23, 13, 34, 45, 35, 46, 47, 74, 58, 59, 89u. Note that if any link is deleted from the ﬁrst network, then it completely collapses. If any link is deleted from the second network, only two other links are deleted and they are limited to a local neighborhood of the original link that is deleted.

The second network in Example 3 is more “robust” than the ﬁrst one in the sense that the damage by the deletion of a link is more “local” in a sense that we now discuss.

2.4.1 Robustness Against Social Contagion

We say that a network g is robust against social contagion if it is renegotiation-proof and sustained as part of a pure strategy subgame perfect equilibrium with g0 g such that in any subgame continuation from any renegotiation proof g1 g, and for any i and ij P g1, if i does not perform the favor for j when called upon, then the 2 2 1 1 continuation leads to g such that if h` R g then h P Nipg q Y tiu and ` P Nipg q Y tiu. Robustness requires that a network be sustainable as part of an equilibrium such that in any continuation starting from some renegotiation proof (sub-)network, if some link is deleted then the only other links that are deleted in response must only involve the agent deleting the link and his or her neighbors. In a well-defined sense this localizes the damage to society.22 We now characterize the networks that are robust against social contagion, or robust for short. 22An alternative approach would be to explicitly allow nodes or links to fail with given probabilities, and then to look for networks that are still sustainable as equilibria in the face of such failures. If failure probabilities are high enough, then that will result in the definitions that we examine. However, for lower failure rates, other equilibria would also be sustainable. As the contagion effects of such probabilities become intractible quickly in a network setting, we work with the definitions as presented here, but it could be interesting to investigate other approaches. CHAPTER 2. SOCIAL QUILTS 68

Social Quilts

The networks that are robust against social contagion have a particular structure to them that is described as follows. An m-clique is a complete network with m 1 nodes so that every node has exactly m links. m-cliques are an important class of critical networks.23 We can build robust renegotiation proof networks by putting cliques together as long as we do not end up generating any cycles involving more than m 1 nodes when we construct the network out of more than one clique. Thus, a social quilt is a “tree union” of networks.24 We say that a network g is an m-quilt if g is a union of m-cliques25 and there is no cycle in the network involving more than m 1 nodes. Figure A.3 shows a 2-quilt. The following example shows a non-tree union of cliques that is not a social quilt and is not robust against social contagion.

Example 4 A union of m-cliques that is not an m-quilt and is not robust.

23Note that a clique g with m 2 nodes (each having m 1 links) is not renegotiation-proof. To see this, suppose the contrary and have some i delete a link ij. In order for this not to be a valid deviation, it must be that i loses all links in the continuation, so that the continuation is such that there are at most m 1 agents in the network and each has just m links. This is Pareto dominated by a network g1 with all m 2 agents such that all agents have m links except for possibly one agent. There is such a network that is a subset of g ¡ ij (which is such that all but i and j have m 1 links and i and j each have m links). It also follows that g1 is critical and thus renegotiation-proof. Thus, we have a contradiction. 24 A union of several networks g1, ..., gK is a tree union if the networks can be ordered in a way g1, ..., gK such that successive unions ¤ U1 g1,...,Uk Uk¡1 Y gk,...,UK gk k1...K are such that each additional network has at most one node in common with the preceding union: |NpUk¡1q X Npgkq| ¤ 1. 25That is, for any ij P g there exists a subnetwork g1 g with ij P g1 and such that g1 is an m-clique. CHAPTER 2. SOCIAL QUILTS 69

Figure 2.5: A union of m-cliques that is not an m-quilt. presence of this cycle makes it not robust against social contagion: g1 t12, 26, 64, 41u is a subnetwork of g and is a critical network and so is renegotiation-proof, however g1 is not robust since any deleted links leads to its total collapse and so more than local contagion. It then follows from the deﬁnition of robustness, which requires that any subnetwork that could be reached and sustained in a continuation be robust itself.

Here are some useful properties of m-quilts, where m ¡ 1:

• There are no bridges.

• The removal of a link does not change the distance between any two nodes except the two losing the link, and that distance increases just by 1.

• The removal of any link increases the diameter by at most 1, so there are no “long-distance” links.

Theorem 2 A network is robust against social contagion if and only if it is a social quilt.

Given our previous discussion of critical networks, it is a simple extension to see that social quilts are renegotiation-proof, and the critical cliques limit contagion to be local in nature. The subtle and diﬃcult part of the proof of Theorem 2 is in showing that only social quilts are robust. For example, why is a complete network not robust? This requires an involved argument, which draws upon both the renegotiation-proofness and the local aspect of punishments. Roughly, the intuition CHAPTER 2. SOCIAL QUILTS 70

is as follows. First, any robust network must contain some cliques, as an agent who cheats must lose some number of links, which must all be local. In terms of continuation equilibria, any smallest sustainable subnetwork of a given network must be a clique. This follows since any deviation must lead to the loss of all its links since it is the smallest, and by locality the agents must all be neighbors. Moreover, it must be of minimal size by renegotiation-proofness as otherwise the society could renegotiate to keep a minimal sized clique which would contradict this being the smallest sustainable subnetwork. The proof then works by using some graph theoretic reasoning to show that any network that is not a social quilt has some subnetwork that is a critical network, and hence a smallest sustainable subnetwork, but is not a clique. Thus, if a network is not a social quilt, there is some way in which it could be broken down so that the eventual contagion in a last stage of destruction would necessarily be non-local.

2.4.2 The Relative Number of Robust Networks Compared to Subgame Perfect Equilibria

We now present some results that contrast the set of subgame perfect networks with the set of robust networks. Whereas almost all networks are subgame perfect equilibria, a fraction going to 0 of networks are robust. Thus, robustness is a very discriminating reﬁnement of the set of equilibrium networks providing pointed predictions.26

Proposition 9 Fix m and let n grow.

• The fraction of networks that are sustainable as subgame perfect equilibria goes to 1.

• The fraction of social quilts (and thus robust networks) goes to 0.

26See the Supplementary Appendix for some calculations concerning the number of renegotiation- proof networks (including non-robust ones). CHAPTER 2. SOCIAL QUILTS 71

2.5 Asymmetric Payoﬀs

Before we examine data concerning favor exchange settings, we extend the model to allow for asymmetries in payoffs. Given the heterogeneity in characteristics of agents in the societies we examine, it is clear that they may face different costs and benefits from favor exchange, and so this extension is needed to provide predictions to take to the data. In particular, suppose that the probabilities of favors, and their values and costs are specific to relationships. Moreover, let the probability that i needs a favor from j depend on the degree of agent i, dipgq. Suppose that doing a favor for an agent j costs an agent i an amount cji ¡ 0 and the value of the favor to an agent i from an agent j is an amount vij. Let pijpdipgqq denote the probability that i needs a favor from j. Moreover, this also allows us to discuss directed networks, as pij 0 and pji ¡ 0 suggest that only j needs favors from i and i never needs favors from j.

Agents discount over time according to a factor 0 δi 1. Agents’ expected utilities are similarly as that in the symmetric case. An agent i’s expected future utility from being in a network g where all favors are provided in perpetuity is ° p pd pgqqv ¡ p pd pgqqc jPNipgq ij i ij ji j ji uipgq δi . 1 ¡ δi

Thus, if agent i currently provides a favor to agent j with ij P g, i’s current expected discounted utility stream is ¡cji uipgq, whereas if agent i is called to receive a favor from agent j it is vij uipgq. Let us consider settings such that

pijpdiqvij ¡ pjipdjqcji cji ¡ δi . (2.2) 1 ¡ δi for each ij and di ¡ 0 and dj ¡ 0. Thus, no relationship is sustainable on its own. Our deﬁnitions of renegotiation-proof networks and robustness are exactly as previously stated. A link ij P g is supported if there exists agent k diﬀerent from i and j such that ik P g and jk P g. CHAPTER 2. SOCIAL QUILTS 72

Figure 2.6: The contrast between support (left) and a standard clustering or transitivity measure (right).

So a link is supported if it is part of a triad. Support is a necessary condition for robustness in the general heterogeneous case where bilateral relationships are not sustainable by themselves.

Theorem 3 If (2.2) holds and a network g is robust against social contagion, then all links in g are supported.

Support is an important prediction since it diﬀers from standard clustering measures, as illustrated in Figure 2.6. For example, it is possible that a standard measure of clustering27 of a network is close to 0 while support is close to or even equal to 1.28 In fact, as we shall see below, the support measure in the observed networks is quite high while standard clustering measures are much lower.

2.5.1 A Special Heterogeneous Case

27See Jackson (2008) [53] for various deﬁnitions of clustering and transitivity. 28 For example, consider an agent i who has many friends: Nipgq pj1, k1, j2, k2, . . . , jM , kM q such that jm and km are linked to each other for each m but such that there are no other relationships between the friends. The support measure here is 1 since every link is part of a triad. Yet, the M 2 clustering for i is very small: MpM¡1q{2 , which simpliﬁes to M¡1 and goes to 0 as M grows. CHAPTER 2. SOCIAL QUILTS 73

these values are not dependent upon to whom agents are linked and also where the favor probabilities are not agent dependent. In that case, each agent is characterized 29 by his or her own mi such that

δippvi ¡ ciq δippvi ¡ ciq mi ¡ c ¡ pmi ¡ 1q . (2.3) 1 ¡ δi 1 ¡ δi

For this case, our previous results have analogs. The deﬁnition of social quilts is slightly more complicated, but intuitively related to the previous deﬁnition. We provide a full characterization in the supplementary appendix.

2.6 Support vs. Clustering in Favor Networks in Rural India

Along with the degree distribution, and the distribution of distances between nodes, the clustering coefficient is one of the standard network statistics most often reported in network analysis. As mentioned in the introduction, it has often been thought that clustering coefficients reflect the ability of social groups to address social dilemmas. Our analysis has suggested a different and new measure, support, as being the necessary condition for a society to enforce cooperation, at least in the robust manner that we have defined here. This is a distinct measure, and a close scrutiny reveals that clustering and support are conceptually very different. Clustering is a property of the neighborhood of an agent while support is an edge property. While networks with very high levels of clustering will necessarily display a high fraction of supported links the converse is not true. For example, the social quilt on the right hand side of Figure 2.4 has a support measure of 1 (all of its links are supported) and yet only has an overall clustering of 1/2 (as only half of the pairs of links ij and ik lead to a completed triad -for instance only 1/3 of node 3’s pairs of neighbors are linked to each other). More generally, in any society in which agents tend to have multiple disjoint sub-neighborhoods (different cliques of friends), for instance their colleagues

29Again, we rule out indiﬀerence. CHAPTER 2. SOCIAL QUILTS 74

at work, the geographic neighbors at home, their hobby friends, etc., the fraction of supported links would be close to 1, while the overall clustering could be very low. Given that the theory predicts support to be high, but does not predict the same for clustering, we investigate whether this is true empirically. In particular, we now analyze a large data set of a variety of networks that include various forms of favor exchange. We find that support is quite high and significantly higher than the clustering coefficient. As mentioned in the introduction we have data that are particularly well-suited for our study: they document network structures for a variety of different sorts of relationships, including various sorts of favor exchange, and moreover have this for many separate villages. We are not aware of any other data set having these attributes.

2.6.1 Description of the Data

The data are from 75 rural villages in Karnataka, an area of southern India within a few hours from Bangalore. The average population per village is 926.48. The survey was designed as part of a study of the deployment of a micro-finance program (see Banerjee et al. (2010) [1]). Only half of the households were surveyed, which could bias our support measures (downwards) as we discuss below. Households were selected by a stratified random sample in order to control for selection biases; with households stratified by religion (Hindu, Muslim, Christian) and also by geographic sub-locations based on a full census of the villages that was conducted just prior to and in conjunction with the survey. Each surveyed individual was asked to name the people that he or she has various sorts of relationships with. The relationships that were queried in the survey are listed in Table 2.1.30 There are several potential sources of measurement error in these data. First, not all people were surveyed and so there are missing nodes and links in the data

30In the borrowing and lending relationships, ﬁfty Rupees are roughly a dollar and the per capita income in the areas surveyed is currently on the order of three dollars per day or less, although a precise income census is not available. CHAPTER 2. SOCIAL QUILTS 75

Relationships in Survey Codename Question in Survey Friends Name the 5 non-relatives whom you speak to the most. Visit-go In your free time, whose house do you visit? Visit-come Who visits your house in his or her free time? Borrow-kerorice If you need to borrow kerosene or rice, to whom would you go? Who would come to you if he or she needed to borrow Lend-kerorice kerosene or rice? If you suddenly needed to borrow Rs. 50 for a day, whom Borrow-money would you ask? Whom do you trust enough that if he or she needed to borrow Lend-money Rs. 50 for a day you would lend it the him or her? Advice-come Who comes to you for advice? If you had to make a diﬃcult personal decision, whom would Advice-go you ask for advice? If you had a medical emergency and were alone at home, Medical-help whom would you ask for help in getting to a hospital? Name any close relatives, aside those in this household, Relatives who also live in this village. Plus people in the same household. Temple-company Whom do you go to temple with?

Table 2.1: The Contents of the Survey set.31 Without any particular selection of which nodes are missing (given the random selection of households), the missing data would bias the measure of support downwards, since support looks at observed links ij and then asks whether i and j have a common neighbor. If that neighbor is missing from the data, then support can be underestimated. Second, there are the usual measurement issues with survey data: people may forget to mention some of their connections, people get fatigued by interviews, and the survey did not allow individuals to name more than ﬁve or

31Nodes in the networks that we construct from the surveys are individuals who were surveyed. Surveyed individuals could name non-surveyed individuals as friends, relatives, etc., but we omit such links unless both individuals were surveyed. Thus, the networks we work with are sub-networks of the true networks. CHAPTER 2. SOCIAL QUILTS 76

eight other people depending on the categories (although the cap was reached in a negligible number of cases).32 Out of the 12 survey questions described in 2.1 the caps were only ever reached in those questions with codenames Visit-come, Borrow-money, Lend-money and Rel- atives. The caps were 8, 5, 5 and 8 respectively, and they were reached in less than 0.6 in 10000, 2 in 10000, 0.5 in 10000 and 2 in 10000 of the total number of surveys.33 We provide some analysis of measurement error in the supplementary appendix. Thus, in building networks, we say that agent i borrows money from agent j if and only if either agent i reports borrowing money from j or agent j reports lending money to agent i, and we do the same with respect to lend-money, borrow-kerorice and lend- kerorice, visit-come and visit-go and advice-come and advice-go relationships. Note that these are all directed relationships. In the case of friends, medical-help, relatives and temple-company we define that agent i has a relationship of the type in question with agent j if and only if at least one of them acknowledge so. One can also work with other variations on these definitions, and in the supplementary appendix we report on some of these variations, which do not significantly alter the conclusions.34 The supplementary appendix contains a variety of statistical measures of these networks.

2.6.2 Measuring Support and Clustering

The data include a variety of types of relationships. The way in which we categorize relationships is captured in Table 2.2. In our deﬁnitions of these relationships, although we call certain relationships “hedonic,” it could be that those relationships involve various sorts of interactions, including things like favor exchange or risk-sharing, and so forth. Indeed, as we shall see, although we ﬁnd more support in various “favor” relations, there is still a high

32Note that the questions were worded in ways to avoid basic perception issues that are associated with questions such as “who are your friends?” Based on wording that is more explicit about particular interactions (borrowing rice, asking for medical advice, etc.) the relationships are more concrete. 33The Supplementary Appendix shows the cumulative distribution of the number of reported relations for each relationship type. 34In particular, we re-analyze all of the data where we exclude “relative” links, among other things. Whether or not relatives are included does not signiﬁcantly change the results. CHAPTER 2. SOCIAL QUILTS 77

$ ' ' Relatives ' ' Temple-company" ' Visit-go & Visit-come &' Hedonic Friends All $ " ' ' Borrow-money & Lend-money ' &' Physical Favors ' Borrow-kerorice & Lend-kerorice ' Favors " ' ' Advice-come & Advice-go %' %' Intangible Favors Medical-help

Table 2.2: Network Definitions Note: In relationship a&b, i and j are related if and only if they are related in a and in b. level of support among the “hedonic” relationships. In order to capture the possible combinations of various relationships, we enrich our measures of support and clustering. In particular, we can ask whether relationships of one type are supported through relationships of another type, and analogously for clustering, whether a pair of neighbors of a given node in a network of one type have a relationship of a second type. To this end, we define the support of a network g1 relative to another network g 1 as Spg , gq, ° P 1 1tDk,ikPg,kjPgu Spg1, gq ij g ° ijPg1 1 which is the proportion of links in g1 whose nodes have common neighbors in g. Similarly we can define the clustering of a given network g1 relative to some other network g, Cluspg1, gq, as ° P 1 P 1 1tjkPgu Cluspg1, gq ij°g ,ik g ijPg1,ikPg1 1

Note that g g1 is allowed and this reduces these two measures to self-support, and to the standard clustering coeﬃcient respectively. We refer to g1 as the base network and g as the context network. The reason for considering variations on the support measure is that it is quite CHAPTER 2. SOCIAL QUILTS 78

possible that exchange of one type of favor is supported via relationships involving exchange of some other sort of favor or some other valuable interaction. The corresponding variations on the standard clustering coeﬃcients are supplied in order to have appropriate benchmarks.

2.6.3 Support in the Data

Among the relationships described in Table 2.1, we identified those that can be accurately described as favor relationships. In what follows we focus on the average support and clustering measures setting g1 F avors and g All where F avors and All are defined as in Table 2.2. We provide a similar analysis for all sorts of combinations of relationships in the Supplementary Appendix. Figure 2.7 shows the inverse cumulative distribution function of support in the favor networks in our sample of 75 villages along with the plots of the fraction of links supported by exactly k other links in the marginal village. The support measure is generally well above fifty percent, and ranges from more than 50% to over 80% depending on the village. We note that this measurement is likely to be biased downwards by the measurement error of missing nodes, as we detail in the supplementary appendix. Moreover, when we look at certain kinds of favor relationships, the support measure exceeds 85% on average across the villages (again, see the supplementary appendix). Our theory suggests that in robust favor exchange networks, that any relationship that is not self-sustaining on a bilateral level requires support. In the data we see support that is less than 100 percent. Of course, this could be due to some relationships having frequent enough interaction to be self-sustaining. It could also be due to various forms of measurement error, such as missing nodes and also potentially missing links even within the observed networks. We discuss the measurement error possibility in more detail in the Supporting Appendix. Given that the data does not include information allowing us to determine which CHAPTER 2. SOCIAL QUILTS 79

Figure 2.7: The inverse cumulative distribution function of support levels in the villages: The horizontal axis is the fraction of villages having support no more than the amount listed on the vertical axis. The upper-most curve is the inverse CDF of the fraction of supported favor relationships in the All network. The ﬁve curves below list the breakdown of the fraction for the marginal village by various levels of support: “by k” indicates the fraction of links in that village that are supported by exactly k other nodes (so that i and j have k friends in common), and so the ﬁve lines below sum to the curve above. CHAPTER 2. SOCIAL QUILTS 80

relationships are bilaterally self-sustaining, we need to look at other things to determine whether support is simply incidental or reflects the enforcement of cooperative behavior as in our theory. One way to do this is to examine pairs of nodes i and j and then examine the extent to which they have a friend in common, and to see whether this happens more frequently when i and j are exchanging favors compared to situations where they are not. Formally, we do so by extending the definition of support from linked pairs of agents to arbitrary pairs of agents. We define the support of a collection of pairs of agents P N ¢ N relative to a network g, denoted SpP, gq, by ° t uP 1tDk,ikPg,kjPgu SpP, gq i,j °P . ti,juPP 1

So, the support measures we considered previously were those where P was the set of pairs of agents who exchanged favors with each other. This more general measure allows us to also measure support for the situation where P is the set of agents who do not exchange favors. Thus, we can see whether support is something correlated with favor exchange. Figure 2.8 shows the inverse cumulative distribution function of support in the favor networks in our sample of 75 villages along with the plots of the support of the pairs of agents that do not share a link in the favors network in the marginal village. The diﬀerence is statistically signiﬁcant beyond the 99.9 percent level for all villages (based on a one-sided t-test).35

35Such a test assumes independence between the support of pairs of agents that exchange favors and pairs of agents that do not exchange favors, which is violated as some such pairs of pairs overlap, but fraction of overlapping pairs goes to 0 in n. The largest standard error for the estimated diﬀerence of support levels across the 75 villages computed under this independence assumption is 0.084. The analysis relies only on the agents that were surveyed in each of the 75 villages. The sizes of the surveyed populations range from 95 to 395. The village with the smallest number of linked pairs of agents has 115 linked pairs and 4350 pairs that do not exchange favors. The village with the highest number of linked pairs has 843, and 76972 pairs that are not linked. CHAPTER 2. SOCIAL QUILTS 81

Figure 2.8: The inverse cumulative distribution function of support levels of present and absent favor links in the villages. The horizontal axis is the fraction of villages having support no more than the amount listed on the vertical axis. The upper curve reports the fraction of favor relationships that are supported in the All network. The lower curve is the same fraction for the case of pairs of agents who do not have a favor relationship. CHAPTER 2. SOCIAL QUILTS 82

Figure 2.9: The inverse cumulative distribution function of support and clustering levels in the villages: The horizontal axis is the fraction of villages having support/clustering no more than the amount listed on the vertical axis. The upper- most curve is support and the lower-most is the clustering coeﬃcient of the marginal village.

2.6.4 Comparing Support to Clustering

As mentioned before, our measure of support provides a new network characteristic. We now compare it to the clustering coeﬃcient in the villages. We calculate the clustering in each village and compare it to the corresponding support measure. In both cases we work with the base of favor relationships and the context of all relationships. Similar comparisons and conclusions hold for other variations of comparisons as shown in the Supplementary Appendix. Figure A.13 shows that support levels are not only higher than clustering, but by an order of magnitude. There are various ways in which we can see that support is signiﬁcantly higher than clustering in these villages. The comparison of having it be higher in 75/75 CHAPTER 2. SOCIAL QUILTS 83

villages has a p-value that is effectively 0. Moreover, the average ratio of support over clustering across the villages is 2.94 with a standard error of 0.38 and this has a p-value of effectively 0 of being significantly higher than 1. We can also see how the two compare for various types of relationships, as seen in Table 2.3.

G Favors Physical Favors Intangible Favors Hedonic All

CluspG, Allq 0.234 0.257 0.249 0.236 0.222 SpG, Allq 0.717 0.72 0.733 0.661 0.696

Table 2.3: Clustering and Support Measures for various types of relationships. In every case the All network serves as context.

Thus, we see that support signiﬁcantly exceeds clustering. This suggests that support should be a useful measure of network cohesion more generally, one that is complementary to clustering, and especially useful in situations where social pressures may be needed to provide incentives. Moreover, the comparison of support over clustering could also be a useful statistic in more general social network analysis. If the ratio is close to 1, then eﬀectively a typical agent would just have one group of friends, while a higher ratio would suggest that a typical agent has several disjoint groups of friends.

2.6.5 Comparing Support in Diﬀerent Sorts of Relationships

The data also allow us to compare the support measures of favor networks with those of hedonic networks (HR). We can test whether there are statistically significant differences in support among networks of different types. To do this, we compare the values of any given support measure village by village. If there were no difference in support between relationships of the types being compared, then in any given village each one would have a 50% chance to be larger than the other. Thus, under the null hypothesis that there is no difference in support, the number of villages where one has a higher support than the other should have a binomial distribution. CHAPTER 2. SOCIAL QUILTS 84

When we examine the data we see significant differences between the support of different types of relationships. For example, in 72 out 75 villages, the support of intangible favors SpIF avors, Allq is higher than that of social relationships SpHedonic, Allq (both relative to the all network). Of course the probability of a binomial random variable realizing a one in 72 out of 75 trials is effectively 0. Table 2.4 shows the comparison of support measures of various relationships (based on the context of All relationships). The entry in the table is how many times out of 75, the support measure in the row is higher than that in the column. The Supporting Appendix has comparisons for other contexts with similar patterns.

Network g1 Favors Physical Favors Intangible Favors Hedonic All Favors – 30¦¦ 24¦¦¦ 72¦¦¦ 60¦¦¦ Physical Favors 45¦¦ – 38 69¦¦¦ 56¦¦¦ Intangible Favors 51¦¦¦ 37 – 72¦¦¦ 57¦¦¦ Hedonic 3¦¦¦ 6¦¦¦ 3¦¦¦ – 6¦¦¦ All 15¦¦¦ 19¦¦¦ 18¦¦¦ 69¦¦¦ – *** significant difference at 1% level ** significant difference at 5% level

Table 2.4: Comparison of Support Measures. Entry i, j is the number of villages for p 1 q ¡ p 1 q which S gi, All S gj, All

In terms of the observed diﬀerences between hedonic and explicit favor networks, a hypothesis for support in explicit favor networks emerges from our theory. With regards to hedonic relationships, there might also be some informal favor exchange as well, which would result in similar predictions. However, hedonic relationships may also have some mutual gain aspects to them (engaging in activities that beneﬁt both parties), which can lead them to be self-enforcing in the absence of support.36

36For a new ﬁeld experiment that includes some discussion of how behavior varies with relationship type see D’Exelle and Riedl (2010) [5]. CHAPTER 2. SOCIAL QUILTS 85

2.6.6 How Observed Support Compares to that Expected in a Random Network

Next, we look at whether observed support is statistically significant, even when one corrects for geography. This answers whether support patterns are something that arose simply because of geography, or whether the patterns really reflect social structure. For example, underlying social processes are likely to be influenced by geography as costs of interacting will often decrease with proximity. It is plausible that the likelihood of relationships increases with the physical proximity of the agents.37 And, since physical proximity has some transitive features, relationships that are geographically correlated may display completed triads and therefore high support levels. To address this, we estimate exponential random graph models (“ergms”) using the observed networks. In particular, we estimate ergms with the likelihood of a link being present depend on (i) whether the link is supported, (ii) the observed density of the network, and (iii) the physical distance of the agents involved (measured using the GPS coordinates of their respective households)38 As should be expected, in every network (i.e., every

37This could be for two reasons, both having links emerge because of proximity, and having people likely to be linked locate close to each other. 38The exponential random graph model that we estimate is: ¸ ¸ ¸ p p qq p 1q p q log P r G g β0 β1 gij β2 gijs g, g ij β3 d i, j gij i j i j i j where, G is a random variable taking values in the space of all possible graphs on N nodes, g is a particular network, spgq is the associated indication of whether the link ij is supported or not (which may be calculated relative to a related “all network” g1), and dpi, jq denotes the physical distance between i and j as measured by the GPS coordinates of their households, and normalized by the mean distance between surveyed households in the village. In order to estimate such a model, the exponential of both sides must be a probability, and so must normalize to sum to one, which puts restrictions on the coefficients (especially β0), and estimating these coefficients cannot be accomplished without proper normalization. However, since the number of networks on n nodes is exponential in n, and impossible to sum over even for very small n, one has to estimate those coefficients via random sampling of networks. The standard method (and used in the program that we used for our estimation) is via MCMC (Markov chain Monte Carlo) estimation, as outlined by Snijders (2002) [80]. More specifically this method simulates the exponential random graph model using Gibbs or Metropolis-hastings sampling and the Robins-Monro algorithm for approximating a solution to the likelihood function maximization problem. A problem with this methodology is that these models can be multi-modal, and the search space is too vast (the collection of graphs on n CHAPTER 2. SOCIAL QUILTS 86

village) the coefficient associated to physical distance was negative and significant at well less than the 1% level. Nonetheless, the coefficients on support are still large and statistically significant.39 Figure A.14 shows the support coefficient estimates with 99% confidence bars when analyzing g1 F avors with g All as a context. The Supplementary Appendix contains the full set of ergm estimates for many other base-context pairs. The ergm estimates suggest that the observed support levels are not merely an artifact of geographic proximity. More specifically support of a link in the All network is a significant predictor (in the statistical sense) of the presence of that link in the favors network, when controlling for density and for geographic proximity of the agents involved.

2.6.7 The Relation of Support to Characteristics of Links, Households, and Individuals

In our analysis, whether or not a link needs to be supported as part of an equilibrium depends on whether bilateral favor exchange is enforceable in isolation (e.g., our definition of m in the symmetric case). The payoffs to different relationships depend on the value of favors, the cost of favors, interaction probabilities, discount factors, and various other characteristics of the nodes. In this section we take advantage of the richness of our dataset to relate support to various social variables of interest. This is exploratory in nature, since one can think of many hypotheses of how the relative value of favors and the frequency of interaction would vary with demographics. Nonetheless, this information will still be valuable in developing models that relate demographics to patterns of favor exchange; and also provides new insights into how demographics relate to a specific measure of social capital.

pnq nodes is of size 2 2 ) to reliably estimate, even with the most advanced algorithms. The standard errors are subject to similar noise, and should be interpreted with appropriate caution. 39To make sense of the size of the coeﬃcient on support being roughly 2, from the ergm speciﬁcation some simple algebra shows that

r p | q{ p | qq p 1q p q log Pr gij 1 g¡ij Pr gij 0 g¡ij β1 β2s g, g ij β3d i, j , so that having a link be supported roughly increases by 2 or more the log odds ratio that the link will be present in the network. CHAPTER 2. SOCIAL QUILTS 87

Figure 2.10: The coeﬃcients associated to the support terms in exponential random graph models of the “favors” networks in the 75 villages. The value of a link’s support binary variable is 1 if and only if the link is supported in the All network.

Village Level Predictors of Support

To start, we look at how the average support of a network in a village is correlated with the characteristics of the village. In particular, we examine how the average support level in a village relates to the village’s: population size, average education level, average wealth level, and microﬁnance take-up rate.40 As indicators of wealth, we consider the average number of rooms in villagers’ homes, and whether villagers own, share, or don’t have electricity, and similarly for latrines. Table 2.5 provides results for OLS regressions of the fraction of links that are supported (separately for favor networks and hedonic networks) as a function of these village average characteristics. We see that education is signiﬁcantly positively related to favor support, while population size is negatively related. The patterns are similar for both favor and hedonic networks (which have nontrivial overlap).

40Microﬁnance was only available in a subset of 38 villages, and thus the results of microﬁnance are reported for a subset of the villages. CHAPTER 2. SOCIAL QUILTS 88

Favor Support Hedonic Support Population ¡.00008¦¦ ¡.00005 Education .02723¦¦ .02669¦¦ Rooms per person .1401 .3626¦¦¦ Electricity ¡.01986 ¡.1025 Latrine ¡.06683 ¡.09043¦ Microfinance ¡.06485 ¡.01758 (***) Significant at 1%, (**) Significant at 5%, (*) Significant at 10%

Table 2.5: OLS coeﬃcients for the village level regressions of favor support and hedonic support. Population is in total number of people; the education scale is from 0 to 15 representing years in school; in electricity and latrine, 0 means none, 1 means shared and 2 means owned; and microﬁnance is an indicator of participation. Village-level variables are averages of the values of these variables associated with surveyed individuals in the village.

The village level data provides some insights, but given the relatively aggregated nature of the data some variation is obscured. Thus, we next proceed to analyze things at the link and individual levels.

Link Level Predictors of Support

The next analysis examines link-level support as a function of link demographics. We examine the likelihood that a given link is supported conditional on the two individuals involved in the link being similar/dissimilar according to various demographics: education,41 caste, gender, and participation in microﬁnance. In order to keep track of how high the support level is, we report the ratio of support for linked individuals compared to support for pairs of individuals who are unlinked. (See the Supplementary Appendix for the full breakdown of all support levels.) So, a ratio of 5 indicates that the frequency of support of linked pairs of agents in a given category is 5 times higher than the average support level for pairs of agents that are in the same category but who are not linked. The categories here relate to whether pairs of individuals both have the same characteristic or diﬀerent characteristics. For example, in Table 2.6 the ratio of 5.91 in the gender category of

41For education, we categorize individuals as either above or below the median education level. CHAPTER 2. SOCIAL QUILTS 89

female/male means that when we look at pairs of agents with one being female and one being male, the fraction of favor-linked pairs are supported (in the All network) 5.91 times more frequently than favor-unlinked pairs that involve one female and one male.

Education Subcaste both below median 5.09 diﬀerent castes 6.35 above/below median 5.49 same caste 3.16 both above median 4.79

Gender Microﬁnance both male 3.48 neither belong 6.76 female/male 5.91 belong/not belong 6.91 both female 6.09 both belong 3.99

Table 2.6: Ratios of the fractions of pairs of agents that have at least one friend in common in the All network comparing situations where the pairs have a link in the favors network (numerator) to those where they do not have a link in the favors network (denominator). The ratios are split by the similarity/dissimilarity of the pairs of agents according to various characteristics. The education scale has 15 diﬀerent levels, ranging from no formal education to graduate degree, with the median level of education in the scale is 5 and so above/below refers to being above or below the median level. The participation in the microﬁnance program is restricted to the subsample of pairs of agents in which both individuals were eligible for the program.

We see that support levels are (relatively) much more pronounced when individuals fall in different castes compared to when they are in the same caste. We also see lower levels of (relative) support when individuals both participate in microfinance than when they do not. Both-male links are less supported than cross-gender or both-female links. Given that support is related to incentives in our theory, this may be suggestive that some of these situations require more external incentives to enforce exchange than others. The next table, Table 2.7, presents a different angle on how support differs depending on whether the individuals have the same or different characteristics. In particular, the table shows the number of villages in which a given categorization (row) is the one with the highest fraction of pairs of agents who are supported among CHAPTER 2. SOCIAL QUILTS 90

all categorizations. The columns present the results for unlinked pairs and linked pairs separately, and each columns should sum up to the total number of villages, 75.

Favors Favors Education Subcaste Not Linked Linked Not Linked Linked both below median 26 25 diﬀerent castes 0 3 above/below median 3 25 same castes 75 72 both above median 46 25

Favors Favors Gender Microﬁnance Not Linked Linked Not Linked Linked both male 75 16 neither belong 7 14 female/male 0 30 belong/not belong 5 8 both female 0 29 both belong 26 16

Table 2.7: The row of each table shows the number of villages in which the corresponding similarity/dissimilarity class is the one with the highest fraction of pairs of agents that are supported across the 3 classes (or 2 in the case of subcaste). The ﬁrst column considers pairs that are not linked in the favor network, while the second column considers pairs who have a favor link.

The differences between the columns suggests that the patterns of support differ for unlinked versus linked pairs of agents not only in terms of overall likelihood (as we saw earlier in Figure 2.8), but also in terms of how support depends on the characteristics of the agents. The differences across columns are significant in all cases except caste.42

Individual Level Predictors of Support

In this section we examine the relation between an agent’s characteristics and the extent to which that agent’s links are supported, as well as the extent to which that agent’s non-links are supported. Table 2.8 presents the coeﬃcients associated to two probit regressions relating the likelihood of having a friend in common in the All network to gender, education, 42Similar patterns hold for hedonic networks, and more details are presented in the supplementary appendix. CHAPTER 2. SOCIAL QUILTS 91

age, rooms per person in the agent’s household and the household’s size. In the ﬁrst one an observation corresponds to a randomly chosen friend in the favors network of a randomly chosen agent in one of the 75 villages in our sample. In the second regression an observation corresponds to a randomly chosen individual unrelated to a randomly chosen agent in one of the 75 villages.

Linked Not Linked Gender (Female) 0.075¦¦ ¡0.136¦¦ Age 0.001¦ 0.005¦¦ Education 0.0016¦¦ 0.0017¦ Rooms per person ¡0.058¦¦ 0.031 Household size ¡0.017¦¦ 0.0005¦ Intercept 0.399¦¦ ¡1.18¦¦ (**) Signiﬁcant at 5%, (*) Signiﬁcant at 10%

Table 2.8: Probit regressions for an individual’s support as a function of the individual’s characteristics, for both the individuals links and non-links. With the exception of education, the differences between the coefficients in the two columns are significant at the 5% level in the case of age and at the 1% level in all the remaining ones.

More of an agent’s links are supported if the agent is female, older, educated, and from a smaller, poorer household. And again, we see very diﬀerent patterns of support as a function of characteristics for links versus non-links. We present similar probit regressions done at the household level in the supplementary appendix.

2.7 Conclusion

Our analysis of favor exchange provides various insights. We have shown that renegotiation results in specific critical structures and that robustness involves social quilts and, more generally, supported links. Support provides a new local characteristic of networks and insight into closure and an operationalization of a sort of social capital which emphasizes social structure. Our empirical analysis finds high levels of support in favor networks in rural Indian villages. We also find that support levels are much higher than clustering, and that CHAPTER 2. SOCIAL QUILTS 92

support for favor networks is higher than that of more “hedonic” networks.

2.7.1 Information and Robustness

In closing, we discuss some issues regarding the information observed by the agents in the society. We have deliberately looked at a complete information setting for two reasons. First, in many applications, including the Indian villages we look at empirically, word of mouth communication travels much faster than actions and so if someone behaves badly other people hear about it quickly. Gossip serves a strong purpose. Second, much of the previous literature has focused on the information as the driver of network structure in providing incentives and so our analysis is completely complementary. Nonetheless, there is an important observation that comes out. Our analysis ends up yielding social quilts which end up having strong informational robustness properties in addition to the properties that we have investigated. In particular, agents only need to know what the agents whom they are linked to directly are doing, and for all of those agents they also have common friends - so they are both directly, and indirectly connected at a short distance through an independent channel to all of the agents whose behavior they have to be aware of in order to best respond. Thus, even with very limited communication, the robust social networks that we have uncovered can be sustained. Our analysis ends up yielding networks that might otherwise be justiﬁed for their informational properties from a completely diﬀerent perspective.

2.8 Appendix: Proofs of Results

Proof of Proposition 8: Let g1 be a smallest nonempty network (in the sense of set inclusion) that is a subset of g and lies in Gpmq. Such a network exists (possibly g itself) and is thus critical by deﬁnition. The second statement is easily veriﬁed by construction, but also follows as a corollary to Theorem 1 which is proven below.

Proof of Theorem 1: CHAPTER 2. SOCIAL QUILTS 93

We first show that if g P RPNk then g P TCk. Given a network g P RPNk, by the definition it follows that g is sustainable on the equilibrium path. So for any i and ij P g, if i is called upon to do a favor for j and does not, then at least one 43 1 1 possible continuation must lead to a network g g ¡ ij such that g P RPNk1 1 2 2 TCk1 , dipg q ¤ dipgq ¡ m, and there is no g g ¡ ij such that g P RPNk2 and Dpg2q ¡ Dpg1q. If this were not the case, then if i did not perform the favor, he or she would save the cost c and lose at most m ¡ 1 links in any continuation. Thus, i would benefit from deviating and not performing the favor since by the definition of m, (A.1) holds and so the cost of the favor outweighs the loss in future payoffs from losing no more than m ¡ 1 links, which contradicts the fact that g is sustained as an 1 1 equilibrium. Thus, for every i and ij, there exists g g ¡ ij such that g P TCk1 1 1 2 2 for some k , dipg q ¤ dipgq ¡ m, and there is no g P TCk2 such that g g ¡ ij and 2 1 Dpg q ¡ Dpg q. Therefore g P TCk.

Next, we show that if g P TCk then g P RPNk. We do this by induction on the number of links in a network. In order to establish the result, we also need to be careful about what happens starting at subgames that are oﬀ the equilibrium path. As such, we work with a stronger induction hypothesis, with the induction indexed by k.

The induction hypothesis is that starting from any node and any g0 P Gk, there exists a pure strategy subgame perfect equilibrium continuation such that

P ¤ • (i) there is a unique network g1 RPNk1 for some k1 k that is reached in the

continuation, with g1 g0 if g0 P TCk,

• (ii) on the equilibrium continuation path a favor is performed if and only if it

corresponds to a link in g1, and

1 1 2 • (iii) in any subgame starting with some network g P Gk1 with k ¤ k if g is 2 played in perpetuity with some probability in the continuation then g P RPNk2 2 3 1 3 for some k and there does not exist any g g such that g P RPNk3 and 3 2 uipg q ¥ uipg q for all i with strict inequality for some i.

43Even though agents use pure strategies, nature picks which favors are asked in the future, so there are potentially many continuations from any given node. CHAPTER 2. SOCIAL QUILTS 94

As a first step in the induction note that it follows directly from the definitions that RPN0 tHu TC0. Note also that starting from g0 H there is a unique subgame perfect equilibrium continuation (no favors can be supplied and no links can be maintained) and so it follows directly that conditions (i)-(iii) are satisfied. So, let us presume that the induction hypothesis holds for all k1 k. We show that the same is true for k.

Begin with the case such that g0 g P TCk. On the equilibrium path, have all agents maintain all links (so Lipgtq Nipgtq whenever gt g0 g) and perform all favors. The off the equilibrium path strategies are described as follows. If an agent i is called upon to provide a favor for an agent j such that ij P g and does not do the favor, then the continuation is as follows. Given that g P TCk, by the definition of transitive 1 1 1 criticality, there exists g g¡ij such that g P TCk1 RPNk1 , dipg q ¤ dipgq¡m and 2 2 2 2 1 there is no g P TCk2 RPEk2 for any k such that g g ¡ ij and Dpg q ¡ Dpg q. Denote this network by gpi, jq g1. Following i’s failure to provide a favor to j, have the continuation be such that L`pg ¡ ijq Nkpgpi, jqq for all `. This results in the network gpi, jq P RPNk1 following the link announcement phase, and so from then on there is a pure strategy subgame perfect equilibrium sustaining gpi, jq and satisfying (i) - (iii) by the induction step, and so have agents play the strategies corresponding to such an equilibrium in that continuation. At all other nodes off the equilibrium path for which strategies are not already specified we are necessarily at a network with fewer links, and so pick a pure strategy equilibrium continuation that satisfies (i) - (iii), which is possible by the induction hypothesis. This satisfies (i)-(iii) by construction. To check that this a subgame perfect equilibrium, by the specification of the strategies above, we only need to check that no agent wants to deviate from the equilibrium path, and also that following some i’s failure to provide a favor to j, no agent ` wants to deviate from L`pg ¡ ijq N`pgpi, jqq. We only need to check these sorts of deviations since all other continuations were specified to be pure strategy subgame perfect equilibrium continuations. By construction, an agent i who is called upon to do a favor for an agent j who deviates will end up losing at least m links, and so by (A.1) this cannot be an improving deviation.

Next, consider, some agent `’s incentive to deviate from L` N`pg0q if g0 is still in CHAPTER 2. SOCIAL QUILTS 95

play, or else from L`pg ¡ ijq N`pgpi, jqq following some i’s failure to provide a favor to j. By not deviating the agent gets the payoﬀ from g0 or gpi, jq in perpetuity. By 2 deviating, the agent ` will end up with a continuation starting from a network g g0 or g2 gpi, jq, respectively, where the agent has not gained any links and may have lost some links. Since each link has a positive future expected value, this cannot be an improving deviation.

Next, let us show that from any node in the continuation from some initial g0 R

TCk there exists a pure strategy subgame perfect equilibrium continuation satisfying (i)-(iii). There are two types of nodes to consider. One is a node at which some agent i is called upon to provide a favor for an agent j such that ij P g0, and another is a node where agents announce the links they wish to sustain.

First, consider starting at g0 and a node where agents announce the links that they wish to sustain. Find some g1 that has the maximal k1 k of links such that 1 1 1 1 1 P 1 p q g RPNk and g g0. For each ` set L` N` g and then from g play a continuation satisfying (i) - (iii) (by the induction step). If any agent deviates, to L 1 such that L` L, then play the same continuation as this will not aﬀect the network formed. Otherwise, the continuation will lead to some g2 with strictly fewer links for ` and the continuation will necessarily result in a lower expected continuation payoﬀ. This establishes the claim for this sort of node. Next, let us consider a node at which some agent i is called upon to provide a favor for an agent j such that ij P g0. There are two cases that can follow: one where i performs the favor and so the resulting network is then g0. In that case, we have just shown that there is a pure strategy subgame perfect equilibrium continuation satisfying (i) to (iii). Let g1 be the network sustained on the equilibrium path in one of these that has the most links for i. If i does not perform the favor, then g ¡ ij P Gk¡1 is reached. By the induction hypothesis again there is a pure strategy subgame perfect equilibrium continuations satisfying (i)-(iii), and let g2 be a network sustained by one of these that has the most links for i. Now, based on those two continuations, have i choose a pure strategy best response. The claim follows. CHAPTER 2. SOCIAL QUILTS 96

Proof of Theorem 2: We ﬁrst prove that m-quilts, or tree unions of m-cliques,44 are robust against social contagion. The proof proceeds by induction on the size k of the tree union. When k 1, it is a single m-clique, and so it is renegotiation proof since it is critical. Note that also that the only subnetwork of a clique that is in Gpmq is the empty network. Thus, it follows that any equilibrium continuation in (any) equilibrium supporting the clique is the empty network. Thus, a clique is robust against social contagion. Suppose it is true for all k1 k. We show that a tree union of k m-cliques is robust against social contagion. To do this, we show that tree unions of m-cliques and some nonempty strict subnetworks of m-cliques cannot be renegotiation-proof. This is enough to establish that tree unions of m-cliques are robust against social contagion, since it shows that link deletions can be punished by deleting all links in the particular m-clique which the deleted link belonged to and that will not be Pareto dominated by any continuation renegotiation-proof equilibrium.

Begin with¢ a tree union of ¢k m-cliques, g1, . . . gk. 0 Y 0 ¤ 0 0 @ ¥ Let g gh gh , with m0 k, gh gh, gh gh h m0 h1...m0¡1 hm0...k 0 where at least one gh in the union is nonempty. So this is the tree union of m-cliques and some nonempty strict subnetworks of m-cliques. Suppose to the contrary that it is renegotiation-proof. 0 Note that gh is a tree union of networks, and it must therefore have some hm0...k 0 0 leafs. Pick one such leaf and denote it gh¦ . Since gh¦ is a strict subset of the m- clique gh¦ and a leaf of the subtree, there is some agent i0 who has a positive number of links, less than m, in the subtree. Suppose this agent were to fail to provide a 0 0 P favor on a link i0j0 in gh¦ . Since by assumption g RPN, agent i0 would have to lose at least m links if he or she failed to provide a favor on any link i0j0 in the subtree. Since the agent does not have enough links to lose in the subtree, he or she 1 would have to lose links in gh. Denote the continuation by g which must ¡ h 1...m0 1 1 be renegotiation-proof. Note that g cannot be a strict subset of gh, since h1...m0¡1

44It is straightforward to verify that a union of m-cliques is an m-quilt (has no simple cycles of more than m 1 nodes) if and only if it is a tree union of m-cliques. CHAPTER 2. SOCIAL QUILTS 97

1 by the inductive hypothesis gh P RPN. Therefore g must have some links h1...m0 ¢ ¢ 0 1 Y 1 1 from gh. In particular g gh gh , where gh gh, hm0...k h1...m1¡1 hm1...k g1 g @h ¥ m and m m . This last inequality results from the fact that i lost h h 1 1 0 0 1 links in gh. Again, any agent who has fewer than m links in gh must h 1...m0 h m1...k 2 1 have links in gh. We the derive a subnetwork g from g analogously to the h1...m1¡1 way we derived g1 from g0. Proceeding in this fashion we produce a ﬁnite sequence 0 1 ` of renegotiation proof networks g , g , ..., g , with m m ¡ at each iteration and x x 1 x there is always at least one link in gh. Continue until m` 0. Using the same hmx...k argument with which we found i0, it can be seen that we would ﬁnd some node with less than m links in total, contradicting g` P RPN.

We now prove that robustness against social contagion implies that a network must be a social quilt. Suppose that g is robust against social contagion. If there is an m-clique, gc g, that has at most one node i connected with nodes outside of the clique then g ¡ gc is also robust against social contagion. This follows since if any agent j i who is in gc does not perform a favor, then in order for g to have been sustainable j must expect to lose all of his or her links in the continuation (as the continuation must be in Gpmq to be sustainable and j will have fewer than m links). Then since all agents other than i in gc will have fewer than m links in the continuation, they must all lose all links in order for the continuation to be sustainable. Thus, the clique must disappear in the continuation, but by robustness against social contagion no other links can be deleted. So, eliminate gc and continue with the network g ¡ gc. If repeating this process leads to an empty network, then g must have been a social quilt. Suppose instead, that this elimination process leads to some nonempty g1 (and note that g1 is robust by the deﬁnition of robustness) such that g1 contains no m-clique where at most one agent has links outside of the clique. By the above process, any remaining m-cliques that are subnetworks of g1 must satisfy the condition that the clique has at least two agents who have links outside of the clique. So, identify some remaining m-clique that is a subnetwork of g1 and delete a link between a pair of agents, say i and j, who have links outside of the clique. The CHAPTER 2. SOCIAL QUILTS 98

remaining network is still in Gpmq since i and j necessarily had more than m links in g1. Note also that this removal of the link ij does not change the fact that each other clique in g1 has two agents with links outside of the clique (as if either of those agents is i or j then they still maintain m ¡ 1 links each in the original clique). Repeat this process once for each clique in the original network such that at least two agents have links outside of the clique until there are no m-cliques left in the resulting network, and let this nonempty resulting network be g2 P Gpmq. Thus, a direct extension (literally word for word) of the proof of Proposition 8 implies that there is a subnetwork of g2 that is “minimally” critical45 and nonempty. Let that network be g3. It follows from our derivation of g2 that g3 cannot be a clique. However, since g3 is minimally critical, then by robustness and the renegotiation- proofness of g, there is an equilibrium where in any subgame starting from g3, it is sustained. However, starting from g3 if any link is deleted then all links must be deleted in any equilibrium continuation by the deﬁnition of minimal criticality. This contradicts the robustness of g, since g3 is not a clique and hence there is some link ij and some k who is not linked to both i and j who loses a link as a result of the deletion of the link ij.

Proof of Proposition 9: First, we analyze the fraction of subgame perfect equilibria.

Denote by SPEn the set of subgame perfect networks on n nodes. A network is a subgame perfect network if and only if no node in the network has at least 1 and no more than m ¡ 1 links. We set a very loose upper bound on the number of networks in which at least one node has at least 1 and no more than m ¡ 1 links by ﬁrst picking any node and its links and then allowing the remaining n ¡ 1 nodes to link among themselves however they want, and then multiplying by n to allow npn¡1q for any starting node. A lower bound on the cardinality¢ of SPEn is 2 2 minus m¸¡1 npn¡1q n ¡ 1 pn¡1qpn¡2q npn¡1q this bound. Therefore: |SPE | ¥ 2 2 ¡ n 2 2 ¥ 2 2 ¡ pm ¡ n k ¢ k1 n ¡ 1 pn¡1qpn¡2q 1qn 2 2 , m ¡ 1

45A critical network is called minimally critical if any deletion of a link leads the collapse of the whole network, that is if any subnetwork of the original network that lies in Gpmq is the empty network. CHAPTER 2. SOCIAL QUILTS 99

where the inequality on the right holds for any m such that n ¡ 1 ¡ 2pm ¡ 1q. This implies that p ¡ qp ¡ q ¡ n 1 n 2 p ¡ q p n 1 q 2 |SPEn| m 1 n m¡1 2 ¥ 1 ¡ p ¡ q |Gn| n n 1 2 2 p ¡ qp ¡ q ¡ n 1 n 2 pm¡1qnp n 1 q2 2 ¡ m¡1 1 ¡ p ¡ qp ¡ q n°1 ¡ n 1 n 2 pn 1q 2 k 2 k0 ¡ pm¡1qnp n 1 q ¡ m¡1 1 n°¡1 pn¡1q k k0 pm¡1q2p n q ¡ m¡1 Ñ 1 n°¡1 1 as n goes to infinity pn¡1q k k0 Next, we find an upper bound on the fraction of social quilts that goes to 0 as n grows. There are two possible sets of degrees that an agent can have in a network: A tkm n|k is a nonnegative integeru and B tk n|k is an integeru ¡ A. In a social quilt g, every agent i has a degree dipgq P A. We show that the number of 1 n¡1 networks such that all agents have degrees in A is a fraction of no more than 2 of all possible networks. To do this we catalog networks g by first forming a network among the agents 1 to n ¡ 1, and then considering links between those agents and agent n. We show that regardless of the starting network g0, there is at most one configuration of links for agent n (out of 2n¡1 possible) that will allow all agents to have degrees in A. The result then follows directly. Beginning with the network g0 among agents 1 to n ¡ 1, if dipg0q P A then it must be that there is no link between i and n in g. In contrast, if dipg0q R A then in order for dipgq to be in A it would have to be that i and n are linked in g. Thus, if there is some configuration of links between n and the other agents that results in the correct configuration of degrees, there is at most one such configuration out of the 2n¡1 possible configurations.

Proof of Theorem 3: Suppose to the contrary that g is robust against social contagion and ij is not supported. Consider h R ti, ju and delete a link of h (and there is at least one such h who has a link, as the single link ij is not sustainable independently as part of a subgame perfect equilibrium). This leads to a continuation g1 that is robust against social contagion and such that ij P g1, as otherwise by CHAPTER 2. SOCIAL QUILTS 100

robustness both i and j would have to be neighbors of h which would contradict the fact that ij is not supported. Iterate on this argument. Eventually, we reach the empty network which is a contradiction since ij cannot be deleted as the result of deletion of a link of just a neighbor of one of the two nodes (by the deﬁnition of robustness against social contagion). Chapter 3

The Structure of the Sets of Pure Strategy Nash Equilibria in Binary Games of Social Inﬂuence

3.1 Introduction

A game theoretic model of social influence is a game in which one of the main primi- tives is some notion of social structure defining the way in which the behaviors of the players are intertwined. These models have a number of applications in economics and other social sciences including understanding information diffusion, the adoption of new technologies, the production of locally non-excludable goods, fads and fashion, and peer effects in a variety of social environments such as schools, gangs and cults. It is also the case that in some of these applications the agents just decide over a few options: whether to adopt an innovation or not, which version of a product to buy or whether to join or not join a group. As an example consider the task of modeling the choice of what newspaper to read in a small community. Suppose that the only data to inform the modeling exercise is the collection of different social circles to which a person belongs and an understanding of whether each individual is a conven- tionalist or an agitator. While agitators tend to prefer the least common option in their social circles, conventionalists always align themselves with the majority. This

101 CHAPTER 3. BINARY GAMES OF SOCIAL INFLUENCE 102

example showcases three common distinguishing features of situations that are apt to be described by discrete models of social influence: 1) The modeler has very good data about the way in which behaviors among players are intertwined yet no solid grounds for making precise statements about payoffs. 2) The choice spaces of the agents are strongly discrete in the sense that acceptable continuous approximations are unlikely. 3) The data specifies an underlying social structure governing the way in which choices are entangled and many questions of interest involve understanding how behavior may change with variations in this social structure. These characteristics have three major implications over the game theoretic frameworks that are likely to be useful. First, best response correspondences are a more immediate primitive for the models than individual payoff functions. Second, mixed strategies and mixed strategy equilibria are of limited use to the extent that payoffs can’t be credibly specified. This deems many of the standard game theoretic tools for studying existence and analyzing the properties of equilibria inappropriate as they often rely on the continuity of best response correspondences. And finally, the formulation should fa- cilitate the analysis of the connections between the properties of the set of equilibria and social structure. This paper studies the properties of the set of pure strategy Nash equilibria of one of the simplest classes of discrete games of social influence: complete information, simultaneous move two action games of strategic complements with no indifference, under a variety of assumptions on the underlying social structure that entangles players’ payoffs. While there are many situations of interest involving approximately complete information and choices between just two options, most have a significant dynamic component. The significance of this study stems from the idea that simple simultaneous move games along with the concept of pure strategy Nash equilibria capture a basic necessary condition of equilibrium behavior in dynamic settings that meet two conditions. The first one is that agents can frequently and repeatedly revise their choices, and the second one is that the agents disregard their own influence over the social group.1 Under these conditions, the more refined models based on the

1These two requirements are the deﬁning characteristics of many situations in small communities in which social pressures are important. CHAPTER 3. BINARY GAMES OF SOCIAL INFLUENCE 103

complicating features of a specific motivating situation would select their predictions from the set of equilibria of the simple games studied in this paper. The main overall contribution of the paper is to contextualize graphical games of strategic complements, which have had a wide array of applications in economics in the recent years (see for example Jackson and Yariv [54], Calvo-Armengol et. al. (2005) [19]) within the broader class of complete information, simultaneous move two action games of strategic complements. One potential advantage of the network approach over the standard peer effects literature2 lies in the fact that it allows us to capture the way in which global patterns of behavior are affected by the local neighborhood structure. There does not seem to exist a most suitable concept of equilibrium for these settings. The “right” definition is often suggested by the particular features of the application, for instance the depth of the agents’ rationality, the nuances of communication, or the ability of groups of players to form coalitions. At least in part however, the choice of the equilibrium notion is also be driven by mathematical considerations: i.e. existence, uniqueness, tractability. Finally, a key consideration is whether the predictions stemming from a given definition choice seem to correspond well to what is actually observed, or at the very least there is a good explanation for why they fail to do so. It turns out that in most cases, this definition leads to a complex and possibly large set of equilibria, and while constructing one equilibrium is easy, navigating the whole set is in general quite difficult: there are not many straightforward structural relations between different equilibria. This multiplicity and complexity, however, is often consistent with what one intuitively expects in many motivating real world situations, and therefore despite its intractability and lack of parsimony it may well be the adequate one (or maybe precisely for those reasons). Moreover, we can expect the sets of equilibria implied by other definitions to be subsets of the set of Nash equilibria, since the latter in some sense captures the minimal requirements that one would demand from any other notion. It is therefore useful to understand the mathematical properties of the these sets of equilibria. The paper is divided into five sections. Section 3.2 introduces the definitions

2Manski (1993) [66] and Lee (2004) [40] provide instances of models relying on the standard “average interactions” assumption whereby agents are partitioned into groups, within which every agent interacts with every other agent. CHAPTER 3. BINARY GAMES OF SOCIAL INFLUENCE 104

that are used throughout. Section 3.3 characterizes the sets of equilibria of general complete information, simultaneous move two action games of strategic complements with no indifference, to which we refer throughout as increasing games of influence. The main result of this section provides conditions on the structure of a lattice that are necessary and sufficient for it to be the set of equilibria of some game in our class. Using the groundwork laid out in Section 3.3, Sections 3.4 and 3.5 study subclasses of increasing games of social influence that arise from imposing restrictions on the best response behavior of the agents stemming from the social structures that they reflect: 1) Simple games in which each agent is influenced by only one core group of other agents. 2) Nested games in which each agent can only be influenced by agents who are in turn influenced by other agents that influence him. 3) Hierarchical games in which the agents can be embedded in a hierarchy that respects the influence structure. 4) Clan-like games, which are simple games with the additional property that the influence is always mutual and therefore partitions the community. 5)Games of thresholds, games in which the way in which agents influence each other can be represented by a network. Most of the analysis in these sections relies on the analytic approach and insights of the literature of games of strategic complements in lattices (see Topkis (1988) [84]). We build the analysis of each of the classes of games that we consider around the question: What collections of sets are expressible as the set of equilibria of some game in the class? Section 3.6 concludes by studying the algorithmic complexity of the problem of deciding whether a given lattice is the set of equilibria of some increasing game of social. The analysis of this section is based on (Echenique 2007 [34]), which provides the most efficient algorithm available for finding all the equilibria of general finite games of strategic complements. Kempe and Tardos (2003) [57] provide algorithms and complexity bounds for the related problem of identifying targets of “infection” in order to maximize influence.

3.2 General Games of Social Inﬂuence

A peer inﬂuence structure on a set of agents N t1, 2, 3, ..., nu is a collection of N functions Ij : 2 Ñ t0, 1u, one for each j P N, satisfying the property that @j P N, CHAPTER 3. BINARY GAMES OF SOCIAL INFLUENCE 105

3 N Ijpxq Ijpx Y tjuq. For a given x P 2 . Ijpxq 1 is interpreted as meaning that when all agents in j P x set aj 1 then i strictly prefers to set ai 1. Similarly,

Iipxq 0 is interpreted as meaning that when all agents in j P x set aj 1 then i strictly prefers to set ai 0. The idea behind the only requirement in the definition of influence structure is to preclude self reference in action; that is, an agent is influenced by what other agents do, but not by what he himself does.4 For the purpose of making comparisons between influence structures we rely on the partial order ¨ on the 1 ¨ 1p q ¤ p q collection of all influence structures on N defined by letting I I if Ii x Ii x @x, i. Note that if I and I1 are influence structures then so is I _ I1 defined by _ 1p q t p q 1p qu ^ 1 ^ 1p q t p q 1p qu I Ii x max Ii x , Ii x and I I defined by I Ii x min Ii x , Ii x . In this way the collection of influence structures forms a lattice.

A game of social influence induced by a peer influence structure I is a simulta- x tt uun t uy neous move game ΓI N, 0, 1 i1, Ri in which each agent has strategies 0 and 1, and the unique strict preferences induced5 by the influence structure I. This paper is only concerned with strategic behavior relying on pure strategies, and from that perspective it suffices to specify the ordinal properties of preferences; that is we do not require utility functions. A pure strategy Nash equilibrium of a game of influence is a strategy profile σ1 ¢ σ2¢, ..., ¢σn, where σi P t0, 1u has the property that @i P ti P N : σi 1u, Iipti P N : σi 1uq 1 and @i R ti P N : σi 1u,

Iipti P N : σi 1uq 0. Note that any strategy profile in a game of social influence can be described succinctly by the set ti P N : σi 1u. In this way and for convenience, throughout the paper we will be thinking of strategy profiles, and in particular of equilibria as subsets of N. We will denote the set of Nash equilibria of game ΓI as NEpΓI q. An influence structure I is said to express a collection of sets N N of N, C 2 if C NEpΓI q. Similarly, we say that a collection C 2 is expressible by a given family of influence structures tIu if there exists some structure I¦ in the

32N denotes the power set of N. 4Formally, influence structures are best response correspondences (functions) in games for n players, with two actions, in which players are never indifferent. The terminology is useful as the paper studies statements about “social structure restrictions” related to how individual players are affected by other groups of players. 5The induced preferences for each agent are unique as orderings on 2N . CHAPTER 3. BINARY GAMES OF SOCIAL INFLUENCE 106

family such that NEpΓI¦ q C. When talking about a collection of sets, we think of it as endowed with the partial order induced by the weak set containment relation ().

A nonempty collection of subsets of N, C 2N is expressible by some inﬂuence structure if and only if @x P I and i P x xztiu R L. (1)

This condition follows immediately from the deﬁnition of social inﬂuence structure.

If I is an influence structure that expresses C it must be the case that @x P C, Iipxq 1 for all i P x and Iipxq 0 for all i R x. If for some x P C it is the case that xztju P C for some j P x we would have Ijpxq 1 and Ijpxztjuq 1, which violates the definition of social influence structure. Given any collection C satisfying the condition in (1) we can define I that expresses it as follows: Begin by letting Iipxq 0 @x and @i, and then for each x P C and i in x set Iipxq 1. If HP C we are done; otherwise let x be a 1 1 minimal element of C, and set Iipx q 1 @i P x and x x. Given that C satisfies the condition in (1) this collection of functions will be a well defined influence structure. It is clear that there are many influence structures which express a given collection of subsets of N, C 2N . Note that if HR C then the influence structure that we just constructed is minimal in the sense that there does not exist I1 expressing C such that I1 I. The following is a list of properties which are useful to characterize different kinds of social influence structures. We have chosen them because they reflect features of various applications rather than for their performance as axioms6.

1 1 (P1) Increasing if @j P N x x and Ijpxq 1 ñ Ijpx q 1. The elements of the

set Bi minimaltx N : Iipxq 1u are called the bases for action of i.

(P1-1) Simple if for each i P N, the collection Bi contains only one set, which

we denote bi.

(P1-11) Nested if it is simple and has the property that j P bi ñ bj bi.

6As it can be seen they are not independent. It would not be diﬃcult to reconstruct these properties in terms of an independent set of axioms. CHAPTER 3. BINARY GAMES OF SOCIAL INFLUENCE 107

(P1-12) Clan-like if it is simple and has the property that j P bi ñ i P bj.

This is equivalent to saying that the collection tbi Y tiu : i P Nu forms a partition of N 7.

(P1-2) Hierarchical if the agents can be partitioned into a hierarchy

H tH ,H ,H , ..., H u such that if i belongs to level H p q P H then 1 2 3 m h i D 1 2 r p q ñ s P t u bi , bi , ..., bi Hk such that Ii x 1 bi x, s 1, 2, ..., r . k¥hpiq (P1-3) Admits a network representation if for each player i there exist weights tw un , w ¥ 0, and a threshold t ¥ 0 such that I pxq 1 if and only °ij j1 ij i i if wij ¥ ti. jPx

3.3 Increasing Inﬂuence Structures

One very useful result that applies to our setting from the general theory of games of strategic complements is the following:

If I is an increasing inﬂuence structure (P1), then NEpΓI q forms a lattice (with respect to the set containment () partial order). (2)

A proof is presented below, as it showcases an argument that arises frequently when working within this class of games of inﬂuence. 1 1 1 Consider x P NEpΓI q and x P NEpΓI q. Let x _ x minty : y x and y x u. To see that this minimum exists, suppose it doesn’t. Then it must be the case that the set ty : y x and y x1u has at least two minimal elements z and z1. Then some 1 1 1 1 set w, x Y x w z X z must be an equilibrium: If i R z X z then Iipz X z q 0, as either i R z or i R z1. Without loss of generality suppose that it is z. Then, as 1 z is an equilibrium it mus be the case that Iipzq 0 and because z X z z and 1 1 I is increasing we must also have Iipz X z q 0. On the other hand if i P x Y x 1 1 then Iipx Y x q 1, because i belongs to either x or x , I is increasing and both x

7 Formally, bi and biztiu are always valid bases for tiu; bi Y tiu is the same whichever convention is being used. CHAPTER 3. BINARY GAMES OF SOCIAL INFLUENCE 108

and x1 are equilibria. So starting at z X z1 we can iteratively remove from the set the elements that would rather not belong. Due to the fact that I is increasing, at each iteration it will be the case that elements not belonging to the set prefer not to belong. And at some point, before reaching xYx1 or at xYx1 , it will also be the case that all elements in the set will prefer to be in the set. This set is the equilibrium W that we were looking for, and as w z and w z1, its existence contradicts the minimality of these two elements. We conclude that the ty : y x and y x1u only has one minimal element; the minimum we were looking for. With an analogous argument we can show the existence of the meet x ^ x1.

Given a lattice L, how can we know whether it is expressible by an increasing influence structure? Providing necessary and sufficient conditions for expressibility is much harder than in (1). It is not difficult to see what is needed in order to produce an influence structure which expresses some superset L1 of L. Making sure that L1zL H takes some care, as by trying to rule the extra equilibria from the game one might destroy wanted equilibria or give rise to new ones. So to begin with it is not clear whether a simple characterization is possible. As the following proposition shows, thanks to the lattice structure of the set we can get rid of unwanted equilibria one by one using a simple rule.

Proposition 1 A lattice L P 2N is expressible by an increasing inﬂuence structure I if and only if:

(1) i P y P L ñ @x P L such that yztiu x, we have i P x. (2) x R L ñ a) Dj P x such that @y P L, y x, we have that j R y or b) Dj R x and j P x1 @x1 P L such that x x1. (3) HR L ñ Dj P x, @x P L.

Proof of Proposition 1: Necessity: Suppose condition (1) does not hold. That is, assume that there exists x P L and i P N such that x P L, i R x and i P y for some y P L such that y ¡ tiu x . But then the peer inﬂuence structure inducing the CHAPTER 3. BINARY GAMES OF SOCIAL INFLUENCE 109

game satisﬁes Iipy ¡tiuq Iipyq 1 and Iipxq 0, contradicting I being increasing.

Suppose that we have a game ΓI such that every element x of L is an equilibrium of the game. Let y R L. If 2a) fails then due to the fact that I must be increasing,

Iipyq 1 @i P y. So for y not to be an equilibrium of ΓI it must be that there exists ¦ 1 1 i R y with Ii¦ pyq 1. But that 2b) does not hold means that Dy P L, y y and ¦ 1 1 1 i R y . As I is increasing we must have Ii¦ py q 1 but then y is not an equilibrium of

ΓI . And ﬁnally note that if HR L then there must exist j P N such that IjpHq 1,

But as I is increasing this implies IjpHq 1 @x P L, which in turn requires j P x.

Suﬃciency: We begin by setting Iipxq 0 and modify them in the following steps.

Induce all required equilibria:

S1) Let Ijpxq 1 @x, @j such that @y P L, j P y. 1 1 S2) For each x P L and i P x let Iipx q 1 for x xztiu.

Note that in steps S1) and S2) we construct an increasing peer influence structure and by virtue of p1q each x P L is an equilibrium of the game that it induces. Specif- ically note that by construction if i P x P L then Iipxq 1. Moreover if i R x P L then Iipxq 0. To see this, note that to have Iipxq 1 it needs to be the case that yztiu x for some y P L such that i P y, but by p1q this would imply i P x, which by assumption is false. The issue is that the game based upon the influence structure that we have so far I may have other equilibria. To see this in a simple example consider the influence structure that results from the application of S1) and S2) to the lattice satisfying p1q ¡ p3q depicted on the left in figure 3.2, and note that it induces actually expresses the lattice shown on the right.

As it generally seems to be the case when attempting to construct games that express a given set of equilibria, it is easy to guarantee that the members of the set are all equilibria of the game, and a lot harder to guarantee that those are the only equilibria. S3) removes the unwanted equilibria, and this can be done thanks to the fact that the set in question is a lattice. So let I0 denote the inﬂuence structure constructed in S1) and S2) and ΓI0 the game that it induces. CHAPTER 3. BINARY GAMES OF SOCIAL INFLUENCE 110

{1,2,3,4,5,6,7,8,9,10} {1,2,3,4,5,6,7,8,9,10}

{1,2,3,4,5,6,7,8} {1,2,3,4,5,6,9,10} {1,2,3,4,5,6,7,8} {1,2,3,4,5,6,9,10}

{1,2,3,4,5,6} {1,2,3,4,5,6}

{1,2,3,4}

{1,2} {3,4} {1,2} {3,4}

{ } { }

Figure 3.1: The ﬁgure on the right is the lattice expressed by the game induced by the inﬂuence structured constructed by applying S1) and S2) on the lattice on the left.

Remove all unwanted equilibria: Having constructed I0 as above, we will show by induction that for all t ¥ 0 given an t c increasing inﬂuence structure I such that L NEpΓIt q, if NEpΓIt q X L H then t 1 we can produce an increasing inﬂuence structure I such that L NEpΓIt 1 q and c c such that NEpΓIt 1 q X L is a strict subset of NEpΓIt q X L .

The base case: c c Suppose that L NEpΓI0 q, NEpΓI0 q X L H and pick x P NEpΓI0 q X L . Since it must be the case that Iipxq 1 @i P x, the condition in 2aq does not hold. So 2b) implies that for each i P x there exists y P L such that i P y x. Therefore i i 0 x yi yi, where yi is the join in NEpΓI0 q of all the equilibria yi. As I is iPx iPx iPx an increasing inﬂuence structure and for all i yi P NEpΓI0 q, (2) guarantees that the join exists. So let: CHAPTER 3. BINARY GAMES OF SOCIAL INFLUENCE 111

$ & 1 : y x and j P p yiqzx 1p q iPx Ij y % 0p q Ij y : otherwise

1 We will now show that L NEpΓI1 q. The deﬁnition of I only makes adjustments 0 to I on sets y x. So if y P L and y x, then y P NEpΓ 1 q (as y P NEpΓ 0 q). I I So consider some y P L, such that y x. Any such y must contain yi, as any set iPx 1 0 containing x yi, and the only possible diﬀerence between I and I can be on iPx 1 0 components (Ij vs. Ij ) involving elements in yi. As these elements also belong iPx 0p q to y which is an equilibrium of ΓI0 , the images Ij y where already 1 to begin with. So actually no change was really made to the function in these sets.

c c 1 We now show that NEpΓI1 q X L NEpΓI0 q X L . By construction of I , c x R NEpΓI1 q, and we will show that no other element y P L that was not in NEpΓI0 q can now be part of NEpΓ 1 q. As above we only need be concerned with sets y x. If I 1 0 y yi then as seen above I and I are identical, so the only possible occurrence iPx 1 of a a new equilibrium in ΓpI q must involve sets y such that x y, y yiq. Note iPx however that any such set must lack some element in j P p yiqzx, and for such an iPx 1p q element Ij x 1, so y can’t be an equilibrium of ΓI1 .

The Inductive step: Now make the inductive hypothesis that this procedure can be consistently repro- c duced k ¡ 1 times on each occasion m k taking the set NEpΓIm¡1 q X L to a strict c c subset NEpΓIm q X L . If NEpΓIk q X L H then we are done. Otherwise, suppose P p q X c kp q @ P that there exists x NE ΓIk L . It must be the case that Ii x 1 i x, however it is not immediate that condition 2a) does not hold (which was a key part P 0p q of our argument in the base case), as it might be that for some j x Ij x 0 but m¡1p q mp q that at some iteration m k, there was an update from Ij x 0 to Ij x 1. m c So suppose that this is the case. Denoting by x the element of NEpΓ m¡1 q X L re- I m m P p mqz m moved in iteration m it must have been that x x yi and j yi x . iPxm iPxm CHAPTER 3. BINARY GAMES OF SOCIAL INFLUENCE 112

P pp mqz mqz mp q R So there must have been some l yi x x. But then Il x 1 and l x, iPxm c which means that x R NEpΓIm qXL , which in turn implies by virtue of the inductive c hypothesis that x R NEpΓIk q X L , a contradiction. So we can conclude that @i P x 0p q q R Ij x 0. But this in turn means that 2a does not hold for x L. From this point k 1 on we can proceed just as in the base case to construct I such that L NEpΓIk 1 q c c and NEpΓIk 1 q X L NEpΓIk q X L .

So now we are ready to state the last step in our construction:

S3) Let I0 be the increasing inﬂuence structure constructed in S1) and S2). If c NEpΓI0 q X L H then we are done. Otherwise we use the algorithm depicted t muT above to produce a sequence of increasing inﬂuence structures I m1, where for c c each m T L NEpΓIm q and H NEpΓIm q X L NEpΓIm¡1 q X L ; and for m T , L NEpΓIm q.

One nice property of increasing inﬂuence structures that also follows immediately from the theory of games of strategic complements, is that if in structure I1 every agent is at leas as willing to take action under any pattern of activity than under I then the equilibria of ΓI1 are larger than the equilibria of I in the following sense.

p q 1p q P p q If for all i and x, Ii x 1 implies Ii x 1, then y NE ΓI implies 1 1 Dy P NEpΓI1 q such that y y. (3)

As in the case of p2q above, the proof exhibits a way of reasoning that was exten- sively used in the proof of Proposition 1 and which we will continue to use throughout this paper.

Proof: Consider some y P NEpΓI q, then Iipyq 1 for all i P y which by assumption 1p q P implies that Ii y 1 for all i y. This means that the only reason it may be the case that y does not belong to NEpΓI1 q is that for some some elements j R y we 1p q Y t R 1p q u have that Ii y 1. Then let y0 y and yi 1 y j y : Ii y 1 . As the 1 ¦ ¦ number of agents is ﬁnite and I is increasing for some i we will have yi¦ 1 y . CHAPTER 3. BINARY GAMES OF SOCIAL INFLUENCE 113

¦ ¦ By construction y y0 y and y is an equilibrium of ΓI1 .

3.4 Simple, Nested, Clan-like and Hierarchical In- ﬂuence Structures

Simple influence structures are useful to capture situations in which each agent i is influenced by a single group of agents bi: that is, he prefers to be active if and only if every agent in bi is active. The following definitions will help us to characterize these influence structures. Given a set s N, its completion which we denote by sp, is the smallest superset of s containing all the bases of its elements. That is:

sp minta s : a bj, @j P au

Notice that given any increasing influence structure we can characterize the map- t um ping Ii in terms of a finite number of bases for action Bi bi k1 for each agent i P N. And given a set s we can therefore think of the collection of minimal supersets of s which contain at least one base for each of its elements. In the case of simple influence structures however, there is always a unique minimal element, reason for which the above definition makes sense. An agent i only finds it optimal to be active if all the agents in his base for action bi are active, and each of these in turn requires all the agents in his base for action to be active and so on. This means that any equilibrium to which an agent i belongs must contain a base for action for each of its elements, and therefore the set of equilibria of the game induced by an increasing influence structure I must be a subset of the set of equilibria of the game induced by the influence structure generated by all the possible completions of the bases of I.

Example 1 Completion of bases when the inﬂuence structure is not simple.

Consider the increasing inﬂuence structure on t1, 2, 3, 4, 5u in which agent 1 ﬁnds it optimal to be active if either a superset of t2, 3u or a superset of t4, 5u are active. Assume also that each of the sets t2, 3, u and t4, 5u makes activity optimal for its elements. Then tt1, 2, 3u, t1, 4, 5uu is the collection of supersets of t1u with the property CHAPTER 3. BINARY GAMES OF SOCIAL INFLUENCE 114

that they contain bases for all its elements. This collection does not have a minimum element. It is clear that this can not happen when the inﬂuence structure in question is simple. In the case of simple inﬂuence structures we can take advantage of the fact that the base for action of each element has a unique completion, and make a much stronger statement.

p Lemma 1 Let I be the influence structure defined by the collection of completions p tbiu of the bases for action of some simple influence structure I, in which bi H for P p q p q each i N. Then NE ΓI NE ΓIp .

Proof of Lemma 1: Suppose that x P NEpΓI q and consider some i P x. Then p p bi x and bj x for each j P bi. This implies that bi x and therefore Iipxq 1. p p p Suppose that i R x, then bi x. As bi bi this implies that bi x and Iipxq 0. P p q R p q We therefore have that x NE ΓIp . Now suppose that x NE ΓI . This can be the case for one of two reasons, either (1) for some i P x Iipxq 0 or (2) for some i R x p Iipxq 1. If (1) is the case, then it must be that bi x, which in turn implies bi x p and Iipxq 0. If (1) does not hold yet (2) is the case, then we have that bi x and bj x for all j P x (as the statement in (1) does not hold for any j P x). But then p p bi x by definition of completion and therefore Iipxq 1. So in both cases we can R p q conclude that x NE ΓIp . This simple lemma is very powerful as it allows us to easily characterize the set of equilibria of games induced by simple influence structures. Given an increasing influence structure I and some set s let ÒI s denote the smallest set larger than s such that conditional on everyone in s being active, no agent not in s would prefer to be active. That is, ÒI s mintx s : i R x ñ Iipxq 0u. Note that ÒI S is well defined for any set s given that we are just focusing on increasing influence structures. When C is a collection of sets then ÒI C tÒI s : s P Cu. Given a collection of lattices S tL1, L2, ..., Lku, their product, denoted L is LPS the collection of all possible unions of elements of S. That is: CHAPTER 3. BINARY GAMES OF SOCIAL INFLUENCE 115

{1,2,3,4,5,6,7} {1,2,3} {4,5,6,7}

{1,2,3,4,5} {4,5,6,7}

{4,5} {1,2,3}

{4,5}

{ } { } { }

Figure 3.2: The ﬁgure on the right represents the product of the lattices on the left.

¡ ¤k L tx : x yj where yj P Lju LPS ji

Proposition 2 Let I be a simple inﬂuence structure, such that bi is nonempty for p each i P N. Then NEpΓI q Ò p Lq where S ttH, bi Y tiuu : i P Nu. It does not LPS matter whether the operator Ò is applied with respect to I or Ip, so for simplicity we omit the subscript.

The following Lemma, regarding the completions of bases is very useful for the proof of Proposition 2, as it allows us exploit Lemma 1 very proﬁtably.

p Lemma 2 Let I be a simple inﬂuence structure. Then its completion I is nested. p p p That is j P bi ñ bj bi.

p Proof of Lemma 2: Suppose that j P bi, then by deﬁnition of completion it must p p be the case that bi bj, which in turn implies that bk bj for all k P bi. As bi is the p p smallest set wit this property we must have that bi bk. Proof of Proposition 2: Let Ip be deﬁned as in Lemma 1. By the Lemma we just p q Ò p q P p q P need to show NE ΓIp L . Suppose that x NE ΓIp and i x. Then it must LPS CHAPTER 3. BINARY GAMES OF SOCIAL INFLUENCE 116

p p be the case bi Y tiu x. We therefore have that pbi Y tiuq x and this in turn iPx p 8 implies Ò p pbi Y tiuqq x given that x is a Nash equilibrium of ΓI . We therefore iPx p have that x Ò p pbi Y tiuqq and therefore x PÒ p Lq. iPx LPS p Now suppose that x PÒ p Lq. Then x Ò p pbi Y tiuqq by virtue of the fact LPS iPx p p p that by Lemma 2 if k P bi then bk bi. So suppose that i P x. Then either p p p i P pbj Y tjuq in which case we automatically have bi x and Iipxq 1. Otherwise jPx p p i P pÒ pbj Y tjuqzp pbj Y tjuqq and by definition of the Ò operator, it must be the jPx jPx p p case that Iipxq 1. If on the other hand i R x then it must be the case that Iipxq 0 Ò P p q by definition of . We can conclude that x NE ΓIp . The key element in the proof of Proposition 2 is the fact that the completions of the bases are nested as shown by Lemma 2. So a slightly more general version of the Proposition holds for all nested influence structures, in which the relevant collection p for the product lattice is S ttH, bi Y tiuu : i P Nu. Example 1 shows that the key property for Lemma 1 and therefore for Proposition 2 is not that I is simple but rather that unique completions of bases for action of each i P N can be defined. Given some increasing influence structure I the bases for action of i can be defined as Bi minimaltx : Iipxq 1u, and given some set s its set of completions by

Cpsq minimalta s : a bj for some bj P Nj, @j P au. We can then easily generalize Lemma 1 and Proposition 2 to the class of influence structures with the p q P p q p 1 q property that C bi is a singleton for each bi Bi and C bi C bi whenever 1 P bi, bi Bi for all elements i. These influence structures are essentially simple in the sense that regarding their sets of equilibria, they can be equivalently represented by simple influence structures. The more general versions of the results above would be more cumbersome notationally, yet they would not be any more enlightening.

Among simple inﬂuence structures, clan-like inﬂuence structures, in which the bases for action partition the set of agents into equivalence classes have many applications, as they represent well the ways in which families, closely-knit groups of

8 Ò 1 1p q Ñ p q Note that we could be taking with respect to I or I , as Ii x Ii x for all i. CHAPTER 3. BINARY GAMES OF SOCIAL INFLUENCE 117

friends, maﬁas or clans act. As we see below the set of equilibria of clan-like inﬂuence structures have a very simple structure. Proposition 3 If I is clan-like, then NEpΓI q L LPB where B ttH, bi Y tiuu : i P Nu.

Proof of Proposition 3: Let x P L, then by definition of clan-like influence LPB structure I pxq 1 @i P x. If i R x then b x, as the sets b Y tju partition N, and i i j therefore Iipxq 0. So we have that x P NEpΓI q. If x R L then x must be the LPB union of some members of the lattices in B and a non-empty set which has some but not all elements from bj Y tju for some j. Then any such element k of bj Y tju which it does contain would rather take action 0. That is Ikpxq 0, so x R NEpΓI q. Besides clan-like structures, there are other simple influence structures which give rise to games with sets of equilibria that are the product of chains such that their maximum elements are disjoint. In what follows we refer to these lattices as simple p lattices. A simple influence structure is sub-clan-like if its bases tHuYtbi Ytiu : i P Nu form a collection of chains9 whose maximum elements partition N.

Proposition 4 NEpΓI q is a simple lattice containing N when I is sub-clan-like. On the other hand if L is a simple lattice containing N and satisfying condition (1) then there exists a sub-clan-like inﬂuence structure I such that NEpΓI q L.

Proof of Proposition 4: Suppose that I is sub-clan-like. Let C be the collection of maximal chains10 that can be formed using elements from tbp Y tiuu : i P Nu Y tHu. i We will show that NEpΓI q C. Suppose that x P C. Then by deﬁnition x CPC CPC p is the union of some collection of sets tbk Y tku : k P I Nu, therefore for all i P x, p bi x and as a result Iipxq 1. Now suppose that i R x and that Iipxq 1. Then it p must be the case that bi x. By the fact that I is sub-cyclic, for each k P I either p p p p p p bi Y tiu bk Y tku, bk Y tku bi Y tiu or pbi Y tiuq X pbk Y tkuq H. As i R x it

9A chain is a lattice with the property that x, y P C Ñ x x or y x. 10A chain C is maximal in a collection of chains C if C C1 for any C1 P C. CHAPTER 3. BINARY GAMES OF SOCIAL INFLUENCE 118

can’t the case that pb Y tiu pb Y tku for some k P I, so there must exist I1 I such i k p p 1 p p that bk Y tku bi Y tiu for k P I and bi bk Y tku. Due to the fact that I tkPI1Iu p 1 is sub-clan-like it must be the case that all the sets bk Y tku for k P I belong to the same maximal chain (as otherwise the intersection of the maximum elements of at p p ¦ least two such chains would have to be non-empty). But this implies bi bk¦ Y tk u ¦ 1 p p ¦ for some k P I , but then pbi Y tiuq X pbk¦ Y tk uq H and therefore it must be the p p ¦ case that bi Y tiu belongs to the same chain as bk¦ Y tk u and we therefore have i P x, a contradiction. It must therefore be the case that Iipxq 0. Now suppose that L is a simple lattice containing N, then deﬁne I by letting

Iipxq 1 if and only if x minimalty P L : i P xu. Consider some x P L, then by construction Iipxq 1 for all i P x. On the other hand suppose that i R x. If

Iipxq 1 then it must be the case that x minimalty P L : i P xuztiu. The fact that L is simple and that it satisﬁes condition (1) however, implies that all sets containing ty P L : i P xuztiu must contain piq, a contradiction, so it must be the case that

Iipxq 0, and we can conclude that L NEpΓI q By construction, all the equilibria of ΓI are unions of sets of the form minimalty P L : i P xu so it is also the case that NEpΓI q L. The fact that I is sub-clan-like follows from the fact that as L is a simple lattice, for each i, the set minimalty P L : i P xu is always the smallest element containing i of the only chain to which it belongs in the collection of chains whose product is L.

3.4.1 Hierarchical Inﬂuence Structures

The sets of equilibria of hierarchical influence structures exhibit well a theme that arises repeatedly across binary games of social influence: As the individual action spaces are so simple, all the complexity of equilibria arises from the social structure, and more specifically from indirect self reference in action. It is difficult to compute and to understand the structure of the sets of equilibria in a game of thresholds because they are often irreducible in the sense that they cannot be broken apart into smaller games which can solved more easily. Given a hierarchical influence structure CHAPTER 3. BINARY GAMES OF SOCIAL INFLUENCE 119

I on a hierarchy H tH1,H2, ..., Hmu, an active set of agents x Hk induces an k¡r r,x increasing inﬂuence structure I on the agents in Hr given by

r,xp q p Y q P P Ii y 1 if and only if Ii y x 1, where y Hr and i Hr

m Letting I I we can compute the NEpΓI q by a kind of backward induction, beginning by computing the Nash equilibria of the game restricted to the top level of the hierarchy NEpΓIm q, taking advantage of the fact the actions of agents in a given level of the hierarchy can only inﬂuence other agents at the same level or at lower levels:

m E NEpΓIm q r r 1 E ttxu ¢ NEpΓIrpxqq : x P E u

1 We can proceed this way down to the base of the hierarchy, having NEpΓI q E . The original problem is reduced to coping with the inﬂuence structures induced at each level of the hierarchy, by the higher levels.

3.5 Increasing Inﬂuence Structures that Admit a Network Representation

For the most part throughout this section we will maintain the convention of thinking of a strategy profile as the subset x N of agents that are active. In some parts however it will be convenient to rely on matrix notation to represent incentives, and binary vectors ~x P t0, 1u|N| to represent strategy profiles. To avoid confusion we consistently denote strategy profiles using the vector notation ~x when thinking of 11 them as binary vectors . In this notation ~xi 1 if and only if agent i chooses to be active. An influence structure I admits a network representation when for each player

11Formally in moving back and forth between the set notation and the vector notation we have that i P x ô ~xi 1. CHAPTER 3. BINARY GAMES OF SOCIAL INFLUENCE 120

i there exist weights tw un , w ¥ 0, and a threshold t ¥ 0 such that I pxq 1 ° ij j1 ij i i if and only if wij ¥ ti. We can group all the individual weights in a matrix W jPx ÝÑ 1 and let t pt1, t2, ..., ti, ..., tnq denote the vector of thresholds. As W and ~t fully capture an inﬂuence structure we will directly denote it by pW,~tq, instead of using I. The games induced by increasing structures that admit network representation are well known in the literature as graphical games of strategic complements or games of thresholds. Throughout this section we refer to the game induced by pW,~tq as a game of thresholds and denote it by ΓpW,~tq. Given a lattice L we say that is is expressible by a game of thresholds if there exist some weights W and a vector of thresholds ~t p q such that NE ΓpW,~tq L. Using this notation we can represent the equilibrium condition using a system of P t un inequalities. A vector ~x 0, 1 is an equilibrium of game ΓpW,~tq if and only if:

ÝÑ ÝÑ ÝÑ ÝÑ rDiagp¡tiq W s x ¥ 0 and rDiagpti ¡ 1qq W s x t

The problem of ﬁnding all the equilibria of a graphical games of thresholds is therefore a subclass of the problem of ﬁnding all solutions to systems of linear inequalities, as disregarding for an instant the strictness of the second inequality, we can write the above as12 £ £ ÝÑ Diagp¡tiq W ÝÑ 0 x ¥ ÝÑ ¡Diagpti ¡ 1qq ¡ W ¡ t

In general the problem of ﬁnding all solutions to linear systems of inequalities is a very hard one, but we can hope to make some general statements about it by exploiting the additional structure of the matrices and vector involved stemming from the fact that they represent graphical games of strategic complements. There are two features of games of thresholds which make them very appealing: 1) Their structure summarizes social interaction in a variety of settings, and 2) They can be described very succinctly: Their representations are not more complex than those of simple

12The problem involving the strict inequality can be approached by solving the weaker version involving the weak inequality and then testing one by one the individual solutions to determine the subset in which the strict inequality holds. CHAPTER 3. BINARY GAMES OF SOCIAL INFLUENCE 121

games of social inﬂuence, but they have a much greater expressive power. As seen in Example 2 there are some increasing inﬂuence structures that do not admit network representations.

Example 2 An inﬂuence structure that does not admit a network representation.

Let I be an an increasing inﬂuence structure on t1, 2, 3, 4, 5u such that I1pxq 1 if and only if x t2, 3u or x t4, 5u. This inﬂuence structure does not admit a pt uq ¥ t1 network representation: The reason is that I1 2, 3 1 implies that either w12 2 ¥ t1 ¥ 1 ¥ 1 or w13 2 . Similarly, it must be the case that either w14 2 or w15 2 . But this in turn implies that at least one of I1pt2, 4uq, I1pt2, 5uq, I1pt3, 4uq or I1pt3, 5uq must also be 1.

Example 2 shows that in depicting agent 1’s social incentives using weights and a threshold, if we want him to be triggered by groups of agents t2, 3u and t4, 5u then it must be the case that he is also triggered by at least one other group of agents that is not a superset of either of these. In general, this is the only kind of limitation that we encounter when constructing network representations of influence structures: It is straightforward to assign the weights and pick the threshold in order to have an agent prefer to be active when every element in a specified collection of subsets of N is active, what can be difficult and sometimes impossible is choosing them in order to ensure that those are the only triggers. That is, it is always possible to construct a network approximation of an influence structure containing all the triggers of an agent, but in general in any such approximation, agents will be strictly more sensitive to social influence. As shown in statement p3q in section 3.3, the equilibria induced by these approximations will be weakly larger than those of the games induced by the original structures.

We now turn to the question of which sub-collection of lattices expressible by increasing influence structures are also expressible by games of thresholds. There are two different kinds of problems that we may face in trying to express a given lattice. The first one is that due to the approximation limitations seen in Example 2, we may CHAPTER 3. BINARY GAMES OF SOCIAL INFLUENCE 122

gain some equilibria which we cannot destroy without compromising the equilibria that we do want to include.. The other possibility, is that none of the approximations expresses supersets of the lattice in question. We do not have much to say regarding the question of expressibility in the general class of inﬂuence structures admitting a network representation. In what follows we provide some result and intuition related to a few special cases.

3.5.1 A Few Special Cases ¡ Given a graph W , a cycle is a a subset of N, i0, i1, ..., ik, such th such that wi0i1 ¡ ¡ ¡ 0, wi1i2 2, ..., wik¡1ik 0 and wiii0 0. The ﬁrst result is very simple and puts forth an important idea, which is that all the complexity of the lattice of equilibria is closely related to the existence of cycles. A graph that has no cycles is called a tree. In general the issue is the same as the one highlighted in Section 3.4.1 in relation to Hierarchical inﬂuence structures. It becomes a lot more clear in the context of networks.

Claim 1 If W has no cycles, the game has a unique equilibrium.

Proof of Claim 1: If W has no cycles there must exist at least an agent i such p q ¦ that wij 0. By virtue of A2 all such agents have a single optimal action xi . In ¦ ¡ ¦ particular xi 0 if ti 0 and xi 1 if ti 0. We can therefore create a simpler and 1 equivalent game Γ ÝÑ1 by removing all such agents from the graph and adjusting the W 1, t the thresholds of the remaining agents. Formally let H ti P N : wij 0 @j P Nu. 1 1 1 Let W rw s where w w if i R H and j R H and w 0 ij i 1,2,...,n;j 1,2,...,n ij ° ij ij P 1 ¡ t u otherwise. And for each j N let tj tj 1 ti 0 wji. As the resulting graph iPH is also a tree, we can continue this process, each time removing a positive number of agents and assigning them their optimal actions until we are left wit a single agent, ÝÑ and a complete assignment of the unique equilibrium actions x .

If we allow inﬂuence structures to be correspondences (letting agents be indiﬀerent between playing 0 or 1) games on trees will in general have multiple equilibria. The spirit of the result nevertheless continues to hold: The set of equilibria can be found CHAPTER 3. BINARY GAMES OF SOCIAL INFLUENCE 123

in its entirety by a form of backward induction starting in the leaves.

We now examine the problem of expressing L using games of thresholds on the complete graph, that is the graph in which the weights wij w are the same for all pairs of agents i j. This special case is of interest because it is equivalent to many models of peer effects in the applied literature, in which each agent chooses her activity level in response to some measure of mean activity level in the environment. In these models, the agents may have different sensitivities to the environment (in our language, different thresholds) but the environment is usually the same for everyone, which is analogous to all the weights being equal.

Claim 2 If x and y are equilibria of a game of thresholds in the complete graph then either x y or y x. That is, the set of equilibria must be a chain.

Proof of Claim 2:

Suppose that we have two equilibria x and y. Without loss of generality assume that ° ° |x| ¤ |y| and let i P x. Then ti ¤ wij |x|w and therefore ti ¤ |y|w wij. So jPx jPy it must be the case that i P y.

The converse of the claim above is also true, any chain which satisﬁes the strictness condition p1q of Section 3.2 is expressible by a game of thresholds on the complete graph.

Claim 3 If L is a chain with the property if x P L then xztiu R L @i P x13 then its is expressible by a game of thresholds on the complete graph.

Proof of Claim 3:

Let C tx1, x2, ..., xmu be a chain satisfying the condition of the claim, and without 1 loss of generality suppose k k implies xk xk1 . Then it must be true that | |¡ | |¡ | ¡ | ¡ x1 1 @ P xk 1 @ P z xk 1 xk 1. So let, let ti n , i x1 and in general ti k , i xk xk¡1, 13Condition p1q of Section 3.2. CHAPTER 3. BINARY GAMES OF SOCIAL INFLUENCE 124

¤ @ R 1 @ for each k m. Finally, let ti 1, i xm. Let wij n i j. By construction, the set of equilibria of this game of thresholds is precisely C tx1, x2, ..., xmu. The simple structure of the lattices that are expressible by games of thresholds on the complete network means that we are able to count the number of diﬀerent (up to relabeling of the agents) possible sets of equilibria of these games. In enunciating this counting result we restrict attention to sets of equilibria which always include the complete set N.

Corollary 1 There are Fn¡1 diﬀerent chains including the set N that can be expressed as equilibria of games of thresholds on the complete network on the set N th t1, 2, ..., nu, where Fn¡1 denotes the n ¡ 1 Fibonacci number.

Proof of Corollary 1: Since we are counting the number of different chains up to relabeling of the agents, two chains C1 and C2 are different if and only if one of them contains a set different in size from all the sets contained in the other chain. So in what follows, when referring to a givem set we only speak of its size. We can partition the set of expressible chains on |N| players into two groups: (1) Those that contain an element of size |N ¡ 2| and (2) Those that do not contain an element of size |N ¡ 2|. Each of the chains in (1) corresponds to a unique chain expressible with |N ¡ 2| elements. A bijection is given by fpcq czN. On the other hand, each of the chains in (2) corresponds to a unique chain expressible with |N ¡ 1| elements. A bijection is given by gpcq czN Y N ¡ 1. Therefore, if we denote the number of expressible chains on |N| players by cN , we have that cN cN¡1 cN¡2. Moreover, c1 0 F0, c2 1 F1.

Claims 2 and 3 are not very helpful in the sense that it is easy to come up with examples of threshold games with equilibria that are not nested. In general, network architectures which have many components, or have a number of highly intra-connected islands, only inter-connected by a few bridges will tend to have pairs of equilibria not comparable by set inclusion. These claims, however do show quite succinctly that threshold games are interesting precisely due to the interplay between CHAPTER 3. BINARY GAMES OF SOCIAL INFLUENCE 125

{1,2,3,4,5,6,7}

{1,2,3,4,5} {1,2,3,6,7}

{1,2,3} {1,4,5} {1,6,7}

Figure 3.3: A lattice L that can only be expressed by games of thresholds on graphs with weighted links. thresholds and network structure. No matter how much freedom we have to play with the thresholds, we will never be able to abstract away from the network structure. As seen in example 3, allowing the network to be weighted also adds expressive power to games of thresholds.

Example 3 The expressive power added by weights.

The lattice L shown in Figure 3.3 cannot be expressed by a game of thresholds on a network in which all links have the same weight. To see this, suppose that there p ~q p q existed a a game W, t , in which wij 0 or wij w and such that NE ΓW,~t L. We begin noting that the agents in t1, 2, 3u need to be connected as otherwise one of them would need to have threshold 0, but this can’t be the case since HP L. Whoever among 2 and 3 is linked to 1 must have a threshold of at most 2w, and therefore cannot be linked to 4, 5, 6, 7 since otherwise at least one of t1, 4, 5u or t1, 6, 7u would not be an equilibrium. If j among 3 and 3 is not directly linked to 1 then j must have a threshold of at most w, and just as before it cannot be linked to 4, 5, 6 or 7, since as above at least one of t1, 4, 5u or t1, 6, 7u. would not be an equilibrium. We can therefore conclude that neither 2 nor 3 can be linked to 4, 5, 6 or 7. Therefore t1, 4, 5, 6, 7u has to be an equilibrium as well, This contradicts the existence of an ~ p q unweighted network W and a vector of weights t satisfying NE ΓW,~t L. CHAPTER 3. BINARY GAMES OF SOCIAL INFLUENCE 126

3.6 Finding All Equilibria and Deciding Express- ibility

(Echenique 2007 [34]) sets forth the fastest known algorithm for computing all equilibria in general finite games of strategic complements. In what follows we present a version of Echenique’s algorithm for our games taking advantage of the notation that has been introduced in sections 3.3 through 3.4. The main idea behind the algorithm is to traverse the space of of the 2n subsets of N efficiently by taking advantage of the fact that the influence structure is increasing.

H tpH Hqu H 1) Initialization: Let E0 , S0 , and C0 . H We start with an empty set of equilibria E0 , a stack of sets (the seeds) to be inspected S0 tHu just containing the empty set, and an empty stack of already- inspected elements (Checked elements) C0 H.

2) Create Et, St and Ct from Et¡1, St¡1 and Ct¡1:

Select one (any) of the elements of px, zq P St¡1zCt¡1 (If the set is empty, go to 3).

a) If IipÒ px Y zqq 1 for all i P z then:

Ct Ct¡1 Y tpy, zq : such that y ¨ x ¨Ò px Y zqu

St St¡1 Y tpÒ px Y zq, tjuq : j R x Y zu Y tÒ p Y qu Et Et¡1 x z . Go back to the beginning of step 2).

b) If IipÒ px Y zqq 0 for some i P z then:

Ct Ct¡1 Y tpy, zq : such that y ¨ x ¨Ò px Y zqu

St St¡1 Y tpx, z Y tjuq : j R x Y zu Et Et¡1. Go back to the beginning of step 2).

¦ p q ¦ 3) Let NE ΓI Et where t represents the stopping time of the iterative procedure presented in 2q. CHAPTER 3. BINARY GAMES OF SOCIAL INFLUENCE 127

The algorithm starts at the bottom of the lattice. At the beginning of each iteration the set x represents agents that for sure want to be active given that everyone else in x is active. The set z is a group of agents held artificially active by the algorithm, in order to be able to rely on the influence structure in order to navigate the power set of N. The algorithm evaluates the best response of the agents that do not belong to the set x Y z, looking for the smallest superset Ò px Y zq with the property that no agents not belonging to it would rather be active. Resolving whether Ò px Y zq is an equilibrium just requires checking whether the agents in z are willing to be active, as by construction we know that all other agents in the set —the agents in x, and those that are added by the Ò operator— are best responding. Note that regardless of whether Ò px Y zq is an equilibrium or not, at that set the algorithm cannot take advantage of the fact that the influence structure is increasing to continue navigating the power set of N. So we must add new seeds to the stack St. While adding any immediate successor of Ò px Y zq would suffice to re-start the process, in order to make sure that we traverse the entire power set of N all the successors ought to be added. Note that the stack Ct of already-inspected sets is only kept for efficiency, as the re-seeding process may eventually lead to considering a given pair px, zq in 2q more than once.

Provided that the Ò can be applied efficiently, this algorithm is a huge improvement over evaluating the best response function of each agent at each subset of N, which is the only general algorithm for finding all pure strategy Nash equilibria of arbitrary discrete games. The efficiency in the evaluation of the Ò operator in turn, depends on the extent to which we can efficiently evaluate the best response function of the agents. If this is the case , the algorithm will terminate quickly to the extent that the gaps between the the sets x Y z and Ò px Y zq are large, and therefore traversing the space does not essentially rely on the reseeding process. As the next example shows, the worst case performance can be exponential on n, and moreover its halting time is unrelated to the size of the set of equilibria of the problem at hand.

Example 4 Worst case performance. CHAPTER 3. BINARY GAMES OF SOCIAL INFLUENCE 128

Consider the increasing inﬂuence structure in which every agent in N ﬁnds it optimal to remain inactive regardless of what the other agents do. Then the algorithm evaluates the best response function of every single agent at each of the 2n subsets of N: It is forced to reseed after every single iteration.

Example 4 shows that the worst case performance of the algorithm is exponential on n. Formally, an instance of the problem of computing all the pure strategy Nash equilibria of one of our games is a description of the influence structure. There are many classes of increasing influence structures of interest whose representations are of polynomial size on n. Consider for example the class in which each agent has at most k bases for action (k can be any number). Then, provided that the description of each subset of N is of size Opppnqq, where ppnq is some polynomial of n, the description of the whole problem is Opknppnqq, so polynomial on n. Note that he class of influence structures in which each agent has at most k bases includes important subclasses of the special cases studied in this paper: For example all simple structures, and all all structures admitting a network representation in which the number of neighbors of each agent is bounded. So Example 4 shows that within the class of problems whose description are of polynomial size on n, the worst case halting time of the algorithm is exponential on n. The problem of deciding whether a lattice L is expressible by an arbitrary influence structure can be decided in polynomial time on the size of the lattice, by just applying p1q, we just need to verify that for each x P L we have xztiu R x. On the other hand the problem of deciding whether a lattice is expressible by an increasing influence structure14 is in the complexity class NEXP based on the algorithms available. That is, given a true15 instance of the problem and a certificate16 of its truth, the best algorithm available has worst case performance which is exponential on the size of the certificate. This follows because verifying whether a given lattice is indeed the

14Within the class of efficiently expressible structures: with representations of polynomial size on n. 15A lattice which in fact can be expressed by some increasing influence structure. 16A certificate is a proof of expressibility: in this case an influence structure inducing a game whose set of equilibria is the lattice in question. CHAPTER 3. BINARY GAMES OF SOCIAL INFLUENCE 129

whole set of equilibria of an influence structure (the certificate of expressibility of the lattice), is essentially not any easier than finding all the equilibria of the lattice. The issue is that it requires testing not only whether each element of the lattice is an equilibrium of the game proposed, but also whether the game proposed has no other equilibria. In order to substantially improve on this appalling performance we would need a characterization of expressibility that only depended on the elements of L, as is the case with the problem of expressibility by arbitrary influence structures—. In terms of the algorithmic complexity of the problem, Proposition 1 is of little use as it requires verifying statements not only about those sets that are members of the lattice, but also about all the sets that do not belong to it. Appendix A

Social Capital and Social Quilts: Supplementary Results

A.1 Critical Networks and Renegotiation-Proofness

Here, we provide more discussion of critical networks and suﬃcient conditions for renegotiation-proof networks. Recall that m is deﬁned by

δppv ¡ cq δppv ¡ cq m ¡ c ¡ pm ¡ 1q . (A.1) 1 ¡ δ 1 ¡ δ and

Gpmq tg | @i, dipgq ¥ m or dipgq 0u is the set of networks in which each node has either at least m links or 0 links.

A.1.1 Critical Networks

Recall that a network g is m-critical if

• g P Gpmq

1 1 • for any i and ij P g, there is no subnetwork g g ¡ ij such that dipg q ¡ 1 dipgq ¡ m and g P Gpmq.

130 APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 131

Figure A.1: A critical network where two agents have an excess number of links

As the following example shows, it is also possible to have more than one node have more than m links, as long as those two nodes are not adjacent.

Example 5 A critical network such that two nodes have more than m links.

Consider the following network, which is pictured in Figure A.1: t12, 13, 14, 15, 23, 45, 26, 36, 46, 56u. In this network nodes 1 and 6 have degree four. This is critical and is a renegotiation- proof network when (A.1) holds for m 3. If any node, including 1 or 6, drop a link, then some node’s degree drops below 3 and there is no subnetwork that is sustainable.

Unions of Critical Networks

Let us explore the extent to which one can build richer classes of networks that are renegotiation-proof by agglomerating critical networks. A ﬁrst question is, “Are unions of critical networks renegotiation-proof networks?” The ﬁrst point is that one has to be careful as to how one builds a union of networks. To see this, suppose that m 2 and we consider a union of two critical networks which are two triads. If the “union” is two disjoint networks with no intersecting nodes, then it is clear that the resulting network will be renegotiation-proof. However, if the union involves duplication of a link, then the resulting network might not be. For example, consider the network in Figure A.2. This can be seen as the union of two triads where the link 13 is shared by the triads. We already know that APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 132

Figure A.2: A ﬁve link network that is not sustained as a renegotiation-proof equilibrium this is not renegotiation-proof. Thus, we need to be careful that if the networks intersect, then they do not share links. Recall that we deﬁned “tree unions” of networks in the footnote of the main paper.

A union of several networks g1, ..., gK is called a tree union if the networks can be ordered in a way g1, ..., gK such that successive unions ¤ U1 g1,...,Uk Uk¡1 Y gk,...,UK gk k1...K are such that each additional network has at most one node in common with the preceding union: |NpUk¡1q X Npgkq| ¤ 1. One thing to note is that a tree union of critical networks is not necessarily critical, as illustrated in the following example.

Example 6 A Tree Union of Critical Networks

Let m 2 and consider the network of three linked triads pictured in Figure A.3. This is not critical since if 1 cuts the link 12, then all nodes in the sub-network still have at least 2 links. Nonetheless, (as we will verify below) this network is renegotiation-proof.

Although Example 6 shows that it is possible to have a network that is a tree union of critical networks not be critical, and yet still renegotiation-proof, that is not true of all tree unions, as the following example shows. APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 133

Figure A.3: A tree union of critical networks that is not a critical network, but is still renegotiation-proof.

Example 7 A tree union of critical networks that is not renegotiation-proof.

Let m 3 and consider the three critical networks ga, gb and gc shown in Figure 1 A.4. Let ga be a network with the same structure as ga but with a diﬀerent group of t 1 1u 1 2 3 agents 1a , ..., 9a , and deﬁne gb, gb and gb similarly. Consider two tree unions of these critical networks:

Y 1 Y 1 • U1 ga ga gc. intersecting at the node 1a 1a 1c;

Y 1 Y 2 Y 3 1 2 3 • U2 gb gb gb gb , intersecting at the node 1b 1b 1b 1b .

Structurally, U2 ¡ t6b, 7b, 8b, 9b, 10bu is the same as U1. The claim is U1 and U2 cannot both be both renegotiation proof networks. Otherwise, starting from U2 if agent 1b refuses a favor to agent 10b, the network played in the continuation has to be U2 ¡ t6b, 7b, 8b, 9b, 10bu since it is renegotiation-proof (having the same structure as U1) and noting that the nodes t6b, 7b, 8b, 9b, 10bu must lose their links in any continuation. Thus, agent 1b only loses one link and would prefer not to do a favor for 10b, contradicting the supposition that U2 is a renegotiation-proof network. One special character of networks in this example that potentially prevents the unions to be renegotiation-proof is there are some “critical” nodes in the networks such as 1a, 1b and 1c. A node is called critical if deleting this node increases the number of components in the network. APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 134

Figure A.4: A tree union of critical networks that is not renegotiation-proof.

In other words, a node is critical if it plays an essential role in connecting diﬀerent agents who will be in diﬀerent components without the critical node. For example, without 1a, agents 2a and 6a won’t be connected in ga. Another way to present this character is by noting that any path between 2a and 6a has to contain 1a such that there is no simple cycle containing agents 2a and 6a. The lemma below implies the equivalence of these two presentations of the special character of networks in Example 7.

Lemma 3 Consider a path-connected network involving links among at least three nodes. For each pair of path-connected nodes in the network there is a simple cycle containing them if and only if there is no critical node in the network.

Proof of Lemma 3: Let us ﬁrst argue that if there is a critical node, then there are at least two nodes that are path-connected but that do not lie on a simple cycle. Suppose that there is a critical node i in the path connected network g, such that deleting i results in at least two separate components. Pick nodes j in one of those components and k in another component. It follows that i lies on all paths connecting APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 135

j and k or else deleting i would not have resulted in these nodes falling in separate components. Thus, there could not have been a simple cycle containing these two nodes in the original network. For the other direction of the lemma, we consider any two path-connected nodes i and j that are embedded in a network with at least 3 nodes that has no critical nodes. We show that there exists a simple cycle containing i and j. We proceed by induction on the distance between i and j (with the standard deﬁnition of distance being the number of links of the shortest path between them). For the base case let the distance between i and j be 1 so that i and j are direct neighbors. There must exist some other node k that is a neighbor of either i or j since the network involves at least 3 nodes and is path connected. Without loss of generality assume that it is adjacent to i. Since there are no critical nodes in the graph, k and j remain path-connected if we delete node i. Thus, let P be a path that goes from k to j without passing through i. There is a simple cycle containing i and j given by j ¡ i ¡ k, and then taking P from k to j. For the inductive step, suppose that the claim is true for any pair of nodes of distance n or less, and consider some pair of nodes i and j at distance n 1, and let S be a path of length n 1 between i and j. There is a unique node k adjacent to i on S, at distance n from j and by the inductive hypothesis there exists a simple cycle containing k and j. Let P1 be a path from k to j contained in this simple cycle, and

P2 a path from j to k disjoint from P1. Since the graph has no critical nodes there exists some path P3 from i to j that does not go through k. If P3 is disjoint from P1 or P2 we are done, since we then have a simple cycle given either by i ¡ k ¡ P1 to j and then back to i via P3, or by i ¡ k ¡ P2 to j and then back to i via P3. So assume that P3 intersects both P1 and P2 and without loss of generality that it intersects P1

first, at some node m (since P1 and P2 are disjoint, this first-to-be-intersected order is strict). We now have a simple cycle including i and j given by i Ñ m (via P3), then m Ñ j (via P1) and then from j to k (via P2), and then k ¡ i finally back to i.

So in the following, we look at a nice subclass of critical networks that don’t have these critical nodes. And it turns out that tree unions of networks in this subclass APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 136

Figure A.5: A critical network with a bridge are renegotiation-proof.

Simply Critical Networks

A useful subclass of critical networks is the class in which any two nodes are connected via a simple cycle. Such networks can be agglomerated quite nicely.

A network g is called simply critical if dipgq equals m or 0 for every i, and for any pair of nodes i and j there is a simple cycle containing them. Clearly a simply critical network is critical. An obvious example of a simply critical network is a clique of m 1 nodes.1 To get a deeper feeling for what simplicity implies, see the network pictured in Figure A.5 which is critical but not simply critical.

Example 8 A Critical Network with a Bridge.

Consider the network pictured in Figure A.5: g t12, 13, 24, 27, 35, 36, 45, 46, 67, 75, 18, 89, 8¡10, 9¡11, 9¡14, 10¡12, 10¡13, 11¡ 12, 11 ¡ 13, 13 ¡ 14, 14 ¡ 12u In this network every node has exactly 3 links. There is a bridge: the link 18. This network is renegotiation-proof when (A.1) holds for m 3. If any node drops a

1A clique is a completely connected (sub-)network APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 137

link, then all links are dropped since there is no proper subnetwork where each node in the subnetwork has at least 3 links. However, this network is not simply critical

Thus, the idea of simply critical networks is that each each agent has exactly m links and is cyclicly tied to every other agent. A nice feature of simply critical networks is that they make nice “building blocks” in that they can be agglomerated via tree unions to create renegotiation-proof networks.

Proposition 10 A tree union of simply critical networks is renegotiation-proof.

Proof of Proposition 10: The proof proceeds by induction on the size of the tree union. When k 1, it is a single critical network, and so it is renegotiation proof. Suppose it is true for all k1 k. We show that a tree union of k simply critical networks is renegotiation proof. To establish the proposition, we show that tree unions of simply critical networks and some nonempty strict subnetworks of simply critical networks cannot be renegotiation-proof. This is enough to establish that tree unions of simply critical networks are renegotiation-proof, simply by deleting all links in any particular simply critical subnetwork of the tree union if some agent fails to perform a favor in that subnetwork.

Begin with¢ a tree union of ¢k simply critical networks, g1, . . . gk. 0 Y 0 ¤ 0 0 @ ¥ Let g gh gh , with m0 k, gh gh, gh gh h m0 h1...m0¡1 hm0...k 0 and at least one gh in the union is nonempty. So this is the tree union of simply critical networks and some nonempty strict subnetworks of simply critical networks. Suppose to the contrary that it is renegotiation-proof. 0 Note that gh is a tree union of networks, and it must therefore have some hm0...k 0 0 leafs. Pick one such leaf and denote it gh¦ . Since gh¦ is a strict subset of the simply critical network gh¦ and a leaf of the subtree, there is some agent i0 who has a positive number of links, less than m, in the subtree. Suppose this agent were to fail to provide 0 0 P a favor on a link i0j0 in gh¦ . Since by supposition g RPN, agent i0 would have APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 138

to lose at least m links if he or she failed to provide a favor on any link i0j0 in the subtree. Since the agent does not have enough links to lose in the subtree, he or she 1 would have to lose links in gh. Denote the continuation by g which must ¡ h 1...m0 1 1 be renegotiation-proof. Note that g cannot be a strict subset of gh, since ¡ h 1...m0 1 1 by the inductive hypothesis gh P RPN. Therefore g must have some links h1...m0 ¢ ¢ 0 1 Y 1 1 from gh. In particular g gh gh , where gh gh, hm0...k h1...m1¡1 hm1...k g1 g @h ¥ m and m m . This last inequality results from the fact that i lost h h 1 1 0 0 1 links in gh. Again, any agent who has fewer than m links in gh must h 1...m0 h m1...k 2 1 have links in gh. We the derive a subnetwork g from g analogously to the h1...m1¡1 way we derived g1 from g0. Proceeding in this fashion we produce a finite sequence 0 1 ` of renegotiation proof networks g , g , ..., g , with m m ¡ at each iteration and x x 1 x there is always at least one link in gh. Continue until m` 0. Using the same hmx...k argument with which we found i0, it can be seen that we would find some node with less than m links in total, contradicting g` P RPN. Before moving on, we note one useful observation for identifying simply critical networks. A Hamiltonian network 2 is a network with a simple cycle visiting all nodes. So a Hamiltonian network is sufficient, but not necessary, for there to be a simple cycle containing any given pair of nodes in a network. Thus, any critical network that is a Hamiltonian where each node has m links is simply critical, but not vice versa.3

We end this section with an example showing a non-tree union of cliques that is not a social quilt and is not robust against social contagion.

Example 9 A union of m-cliques that is not an m-quilt and is not robust.

2See Jackson (2008) for more background. 3Consider the network g t13, 34, 35, 45, 46, 56, 62, 17, 78, 79, 89, 8¡10, 9¡10, 10¡2, 1¡11, 11¡ 12, 11 ¡ 13, 12 ¡ 13, 12 ¡ 14, 13 ¡ 14, 14 ¡ 2u and m 3. g is critical since every node has exactly m links and any pair of nodes has a cycle containing them. However, g is not a Hamiltonian network since there are three “highways” connecting 1 and 2 such that there is no way a simple cycle can contain all nodes. APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 139

Figure A.6: A union of m-cliques that is not an m-quilt.

Let m 2 and consider the network g t12, 23, 13, 14, 15, 45, 26, 27, 67, 46, 68, 84u as in Figure A.6. It is a union of four linked 2-cliques (triads) and any two of these cliques intersect in at most one node. However, it is not a 2-quilt since there is a simple cycle C t12, 26, 64, 41u involving 4 nodes which is more than m 1. The presence of this cycle makes it not robust against social contagion: g1 t12, 26, 64, 41u is a subnetwork of g and is a critical network and so is renegotiation-proof, however g1 is not robust since any deleted links leads to its total collapse and so more than local contagion. It then follows from the deﬁnition of robustness, which requires that any subnetwork that could be reached and sustained in a continuation be robust itself.

A.2 Some Renegotiation-Proof Networks

The equivalence between renegotiation proof and transitively critical works provides us with a straightforward algorithm for deciding whether a given network is renegoti- tation proof or not. Implementing this algorithm for large numbers of vertices (n), and high values of m however, is currently not feasible due to the sheer size of the space of non-isomorphic graphs that must be traversed. Table A.1 shows the number of non-isomorphic renegotation proof networks for a few small values of m and n, along with the corresponding number of non-isomorphic subgame perfect networks. Note that the numbers in the ﬁrst column (m 1), also correspond to the number of non-isomorphic networks on the number of vertices associated to each row. Fig- ure A.7 shows the non-isomorphic renegotiation proof networks corresponding to the APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 140

RNP m SP m n 1 2 3 4 5 n 1 2 3 4 5 1 1 1 1 1 1 1 1 1 1 1 1 2 2 1 1 1 1 2 2 1 1 1 1 3 4 2 1 1 1 3 4 2 1 1 1 4 11 3 2 1 1 4 11 5 2 1 1 5 34 7 3 2 1 5 34 16 5 2 1 6 156 16 7 3 2 6 156 78 24 6 2

Table A.1: Number of non-isomorphic renegotiation proof and subgame perfect networks for some values of n and m. values of n and m shown in the table.

A.3 A Special Heterogeneous Case

An interesting case that generalizes the homogeneous case and yet is not as fully general as the heterogeneous case examined above is one where agents may have idiosyncratic values and costs to favors vi and ci, and discount factors δi, but where these values are not dependent upon to whom agents are linked and also where the favor probabilities are not agent dependent. In that case, each agent is characterized 4 by his or her own mi such that

δippvi ¡ ciq δippvi ¡ ciq mi ¡ c ¡ pmi ¡ 1q . (A.2) 1 ¡ δi 1 ¡ δi

For this case, our previous results have analogs. We deﬁne transitively critical networks as before, simply changing the reference number of links to be agent speciﬁc.

Given m pm1, . . . , mnq, let TCkpmq Gk denote the set of transitively critical networks with k links.

• Let TC0pmq tHu.

4Again, we rule out indiﬀerence. APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 141

n=3,m=2

n=4,m=2

n=5,m=2

Figure A.7: Non isomorphic networks for n=1-6, m=2 and m=3 APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 142

n=6,m=2

Figure A.7 continued. APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 143

n=4,m=3

n=5,m=4

n=6,m=3

Figure A.7 continued. APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 144

• Inductively on k, TCkpmq Gk is such that g P TCkpmq if and only if for any i 1 1 1 and ij P g, there exists g g ¡ ij such that g P TCk1 pmq, dipg q ¤ dipgq ¡ mi, 2 2 2 1 and there is no g P TCk2 pmq such that g g ¡ ij and Dpg q ¡ Dpg q.

Next, in order to define the analog of social quilts we need to define an analog of a minimal clique. In the fully symmetric case a critical clique was simply one where each agent had m links. Now, however, different agents may have different critical numbers of favor relationships that they must fear losing in order to give them incentives to exchange favors. There cannot be too much asymmetry in these critical numbers across the members of a clique or else some subset of the clique could sever some relationships and still have it be sustainable. For example if mi 2 for two agents and mi 3 for another two agents, making a clique from these four agents will not be renegotiation-proof. The first two agents could sever the link between them and end up with a (transitively) critical network. 1 1 A critical clique with m nodes is a clique that has m nodes and such that mi ¤ 1 1 m ¡ 1 for each i in the clique and mi m ¡ 1 for at most one i.

Thus, a critical clique has all but one agent having identical mi’s and the remaining agent’s mi being the lowest. Next, in order to deﬁne social quilts we also need to be careful about how cliques are pieced together whenever there is a node involved in two of the cliques. A (tree) union of critical cliques is not always robust, as the following example illustrates.

Example 10 A tree union of critical cliques that is not robust.

Consider two critical cliques with agents t1, 2, 3, 4u and t1, 5, 6, 7u, respectively, where mi 3 for all i except m4 m7 2. The tree-union of these two cliques is denoted g as in Figure A.8. Then deleting links 14 and 17 leads to a subnetwork of g that is critical and renegotiation-proof. Indeed, all agents end up with exactly their critical number of links except agent 1 who has one extra link. It then follows that g is not robust, since this subnetwork would violate a local contagion condition.

Thus, we have to be careful in deﬁning tree unions when some of the cliques have asymmetries in the agents’ respective numbers of critical links. When we unite two APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 145

Figure A.8: A tree union of critical cliques that is not robust

Figure A.9: A tree union of critical cliques that is robust critical networks at some agent like agent 1, we add extra links for that agent and some of them might become non-critical.

The example above suggests that if some agent has a lower mi than other agents in a clique, and we piece cliques together, then it should be that lowest agent who is the common agent in two cliques. Exactly how this works when there are various heterogeneities across cliques is somewhat subtle as the following example shows.

Example 11 A tree union of heterogeneous critical cliques that is robust.

Consider two critical cliques with agents t1, 2, 3u and t1, 4, 5, 6u, respectively, where m1 m2 2, m3 1 and m4 m5 m6 3. The tree-union of these two cliques is denoted g as in Figure A.9. Here, agent 1 has the lowest mi in one of the cliques, but not the other. This network is still robust, since as long as the “connecting” agent is minimal in at least one of the two cliques, then that clique APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 146

remains completely critical, and so then has no interaction with the adjacent clique. Here it is impossible to remove any link in t1, 4, 5, 6u without it losing all links.

The insights from these two examples lead to the following deﬁnition.

Given a proﬁle m pm1, . . . , mnq, we say that g is an ordered tree union of networks g1, ..., gK if the networks can be ordered in a way g1, ..., gK such that

• successive unions ¤ U1 g1,...,Uk Uk¡1 Y gk,...,UK gk k1...K

are such that each additional network has at most one node in common with

the preceding union: |NpUk¡1q X Npgkq| ¤ 1, and

• in each step of the union Uk Uk¡1 Y gpkq the node in common (in NpUk¡1q X

Npgkq if the intersection is nonempty) is the node with the smallest mi in gpkq.

Of course, an ordered tree union is the same as a tree union in the case where all agents have the same critical number. Now we define a social quilt to be a ordered tree union of critical cliques. Social quilts thus defined are sufficient and necessary for robustness.

Theorem 4 In cases where each agent has an idiosyncratic mi deﬁned by (A.2), a network is renegotiation-proof if and only if it is transitively critical, and a network is robust against social contagion if and only if it is a social quilt.

Given our previous discussion of critical networks, it is a simple extension to see that transitive criticality characterizes renegotiation-proofness, and social quilts are renegotiation-proof. The critical cliques limit contagion to be local in nature. The subtle and diﬃcult part of the proof of Theorem 4 is in showing that only social quilts are robust. For example, why is a complete network not robust? This requires an involved argument, which draws upon both the renegotiation-proofness and the local aspect of punishments. Roughly, the intuition is as follows. First, any robust network must contain some cliques, as an agent who cheats must lose some number of links, APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 147

which must all be local. In terms of continuation equilibria, any smallest sustainable subnetwork of a given network must be a clique. This follows since any deviation must lead to the loss of all its links since it is the smallest, and by locality the agents must all be neighbors. Moreover, it must be of minimal size by renegotiation-proofness as otherwise the society could renegotiate to keep a minimal sized clique which would contradict this being the smallest sustainable subnetwork. The proof then works by using some graph theoretic reasoning to show that any network that is not a social quilt has some subnetwork that is a critical network, and hence a smallest sustainable subnetwork, but is not a clique. Thus, if a network is not a social quilt, there is some way in which it could be broken down so that the eventual contagion in a last stage of destruction would necessarily be non-local.

Proof of Theorem 4: We only prove that robustness implies that a network must be a social quilt, since the converse is an easy analog of the proof of Proposition 10, adapted to strong tree unions.

Suppose that g is robust against social contagion. If there is a critical clique gc g that has at most one node i connected with nodes outside of the clique and mi is the smallest in gc, then g ¡ gc is also robust against social contagion. This follows since if any agent j i who is in gc deletes a link, he or she must lose all of his or her links, and then so must all other agents except i in the clique, but by robustness no other links can be deleted. So, eliminate gc and continue with the network g ¡ gc. If repeating this process leads to an empty network, then g must have been a social quilt. Suppose instead, that this elimination process leads to some nonempty g1. Note that since g is robust, g1 is then also robust and hence sustainable. By the above process, g1 contains no critical cliques where at most one agent has links outside of the clique and that agent has the smallest mi. By the above process, any remaining cliques that are subnetworks of g1 and are such that each agent i in the clique has at least mi links, must belong to at least one of the following sets:

(1) Cliques that are not critical.

(2) Cliques that are critical but such that some agent j connected to another part APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 148

of the network is not the agent i with the smallest mi in the clique.

(3) Cliques that have at least two agents who have links outside of the clique.

So, identify some remaining clique that is a subnetwork of g1 and is such that each agent i in the clique has at least mi links, and identify the ﬁrst case of (1) to (3) that applies. Next, do the following depending on which case applied: In case (1) delete a link between the agents with the two smallest mis. In case (2) delete the link ij. In case (3), delete a link between a pair of agents who have links outside of the clique. The remainder of the proof of the result follows the the logic in the end of the proof in Theorem 2 in the main body of the paper.

A.4 Weak Robustness

It also turns out that we can weaken the definition of robustness and still end up with exactly the class of social quilts. In particular, we can weaken the notion of renegotiation-proofness in the definition. This is useful because it allows us to define robustness in a way that is not inductive, and can thus be easier to implement. Let us say that a network g is weakly renegotiation-proof if it is sustainable by a pure strategy subgame perfect equilibrium and at any subgame that starts with g1 g that is critical, g1 is sustained in the equilibrium continuation.

Let WPRNk denote the set of all networks that have exactly k links and can be sustained in perpetuity as part of a pure strategy weakly renegotiation-proof equilibrium.

Lemma 4 All renegotiation-proof networks are weakly renegotiation-proof; that is,

RPNk WRPNk for all k.

Weakly renegotiation-proof networks are a richer set than renegotiation-proof networks. This is not obvious, since the latter deﬁnition is inductive. Both deﬁnitions require that in any subgame starting with a network g1 that cannot degrade further without collapsing (thus being critical), g1 be played in perpetuity. Otherwise, APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 149

weak renegotiation-proofness puts no additional restrictions on the networks in continuation whereas renegotiation-proofness does. Weak renegotiation-proofness allows richer punishments than renegotiation-proofness, while it still rules out things like grim trigger. While it may be on less solid ground as a solution concept, it is useful in proving some results, which then also hold a fortiori for the stronger concept of renegotiation-proofness. We can also use weak renegotiation-proofness as a basis for a robustness deﬁnition. We say that a network g is weakly robust against social contagion if it is weakly renegotiation-proof and sustained by a pure strategy subgame perfect equilibrium with g0 g such that in any subgame continuation from some weakly renegotiation proof g1 g, and for any i and ij P g1, if i does not perform the favor for j when called 2 2 1 upon, then the continuation leads to g such that if h` R g then h P Nipg q Y tiu and 1 ` P Nipg q Y tiu. Weak robustness turns out to be equivalent to robustness.

Proposition 11 A network is weakly robust against social contagion if and only if it is robust against social contagion.

Proposition 11 follows easily from the observation that weak robustness against social contagion implies that a network must be a social quilt, which is then in turn robust against social contagion.

A.5 Maximal Equilibria

In this part, we consider a slight variation on the concept of renegotiation-proof equilibrium, called maximal equilibrium that is similar but does not require that a deviating agent be considered in the Pareto calculations. Thus, it allows agents to ostracize some deviating agent even when there would be some continuation that would make that agent better oﬀ without hurting any of the other agents. Maximal equilibria are a subset of pure strategy subgame perfect equilibria and are deﬁned as follows. Let Gk denote the set of all networks that have k links.

• Let ME0 tHu APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 150

• Let MEk denote the subset of Gk such that g P MEk if and only if beginning

with g0 g implies there exists a pure strategy subgame perfect equilibrium such that

– on the equilibrium path g is always sustained, and

1 1 2 – in any subgame starting with some network g P Gk1 with k k if g is played in perpetuity with some probability in the continuation then 2 2 3 2 3 1 g P MEk2 for some k and there does not exist any g , g g g such 3 that g P MEk3 .

These equilibria still embody a sort of renegotiation-proofness. For instance, they still rule out a full grim-trigger strategy where once any link is cut then all links are cut. At any point in time, if a given network is reached and that network can be sustained via some equilibrium (with the inductively deﬁned continuations satisfying the same maximality condition), then it is sustained. However, maximal equilibria allow agents to ostracize other agents, which is not always the case in renegotiation-proof equilibria as we illustrate in Example 12. Thus maximal equilibria might be appropriate in some social settings: they permit a society to punish an agent who does not abide by a social norm, and at the same time the society does not resort to arbitrarily drastic punishments but instead limits itself to punishments that minimize the damage to other agents.

Example 12 Maximal Equilibria

Let there be 4 nodes. Consider a case where

δppv ¡ cq δppv ¡ cq 2 ¡ c ¡ 1 ¡ δ 1 ¡ δ

Here, no link is sustainable in isolation, since the value of providing a favor c is greater than the value of the future expected stream of giving and receiving favors: δppv¡cq 1¡δ . APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 151

However, if an agent risks losing two links by not performing a favor, then links δppv¡cq could be sustainable depending on the conﬁguration of the network, since c 2 1¡δ . What do the maximal equilibrium networks look like in this case?

Here, ME1 H since no isolated links are sustainable.

Similarly, ME2 H since any agent who only has one link will never perform a favor.

ME3 tg tij, jk, iku : for some distinct i, j, ku. Thus triads are sustainable, since if any agent severs a link, then that will lead to a two-link network which is not sustainable, and so becomes an empty network. Thus, not performing a favor leads to an empty network, and so it is a best response to perform a favor, anticipating favors by other agents.

ME4 tg tij, jk, k`, ìu : for some distinct i, j, k, ù. This is an obvious extension of the logic from three-link networks. The interesting difference between maximal and non-maximal equilibria come with k 5 or more links. Consider the network g t12, 23, 34, 41, 13u as pictured in Figure A.2. So, agents 1 and 3 each have three links and agents 2 and 4 have two links. There is a subgame perfect equilibrium sustaining this network: if any link is ever cut, then all agents cut every link in the future. However, there is no maximal equilibrium sustaining this network. To see this, suppose that agent 1 is called upon to do a favor for agent 3. If agent 1 does not do the favor, then the resulting network is g1 t12, 23, 34, 41u. 1 Note that g P ME4, and so there is a maximal equilibrium continuation sustaining g1. Thus, under any maximal equilibrium, g1 would be sustained in the equilibrium continuation. Thus, agent 1 can cut the link 13 and still expect the network g1 to endure, and so this is the unique best response for agent 1 and so g is not part of any maximal equilibrium: g R ME5. Thus, ME5 H.

Next, note that ME6 G6, i.e., the complete network. To see this, consider an equilibrium as follows. If an agent i severs a link, then the remaining agents never perform a favor for i again and sustain the triad that excludes i. This continuation satisfies the requirements of maximal equilibrium, and is a subgame perfect equilibrium (with a fuller specification of all the off-equilibrium-path behaviors, which we APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 152

provide in more detail in the results below). Here we also point out the difference between maximal equilibria and renegotiation- proofness. The complete network is not renegotiation-proof. If agent 1 deletes the link 12, the above equilibrium calls for agents to then go to the triad that excludes agent 1. However, that equilibrium continuation is Pareto dominated by a continuation of the four link network 12, 23, 34, 41. Thus, it is a Pareto improvement for the society to forgive agent 1 and go to a four link network instead of the three link network. Interestingly, the other agents are indifferent and it is only the deviating agent who is helped. This is the aspect that if the other agents move to the other equilibrium where they are equally well off, they will help sustain the better six link network with a threat to ostracize agent 1.

A.5.1 Characterizing Maximal Networks

Before moving to the complete characterization of maximal networks, we consider various critical networks ﬁrst. Recall the deﬁnition of criticality such that a network g is m-critical, if

• g P Gpmq

1 1 • for any i and ij P g, there is no subnetwork g g ¡ ij such that dipg q ¡ 1 dipgq ¡ m and g P Gpmq.

Any critical network g is also sustainable as a pure strategy maximal equilibrium. Because criticality is defined such that if any agent i delete a link, i expects to lose at least m links in the sequel. So criticality is sufficient, but not necessary for pure strategy ME. The network in Example 6 is sustainable in pure strategy ME, but it is not critical. We also keep the definition of simply critical networks. Since criticality is not related to the utility of the agents. All the results on them can be generalized to maximal equilibria directly. Such as the following useful proposition, which ensures the social quilts are maximal networks:

Proposition 12 A tree union of simply critical networks is maximal. APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 153

Transitively Critical Networks

We deﬁne transitively critical networks for maximal equilibria as follows.

M p q 1 Given a whole number m, let TCk m Gk denote the set of transitively critical networks with k links, where m satisﬁes (A.1).

M p q H • Let TC0 m .

M p q P M p q • Inductively on k, TCk m Gk is such that g TCk m if and only if for any P 1 ¡ 1 P M p q p 1q ¤ p q ¡ i and ij g, there exists g g ij such that g TCk1 m , di g di g m, 2 P M p q 2 1 2 ¡ and there is no g TCk2 m for any k such that g g g ij.

Even though this is also an inductive definition (not surprisingly, given that maximal equilibria are so defined), it does not involve any incentive descriptions and is effectively an algorithm that can be run on any graph without any knowledge of agents, payoffs, etc. Now let us examine maximal equilibria.

Theorem 5 Let m satisfy (A.1). A network is sustainable as a pure strategy maximal equilibrium if and only if it is transitively critical.

Proof of Theorem 5 is very similar to the Proof of the corresponding theorem in the paper. So we omit it here.

A.5.2 Social Quilts

Recall a m-clique is a complete network with m 1 nodes so that every node has exactly m links. m-cliques are an important class of critical networks. Note that different from pure strategy renegotiation-proof equilibria, a clique (completely connected subnetwork) g with m 2 nodes (each having m 1 links) is sustainable as a pure strategy maximal equilibrium. To see this, have some i delete a link ij. The continuation network is a m-clique g1 with all agents but i. It is not a valid continuation in pure strategy RPE since we can find some network g2 Pareto dominating it. However, it is a valid continuation in pure strategy ME. By the defi- 2 2 1 2 nition, there should not exist g such that g P MEk2 and g g g ¡ ij. Suppose APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 154

such g2 does exist, i should be in g2 with at least m links. So the only possible g2 g ¡ ij, which is not sustainable in pure strategy ME. To see this, if any agent j i refuses a favor to i, the network in continuation should be g1 and j only loses one link. So g2 is not sustainable. We keep the deﬁnition that a network g is an m-quilt if g can be written as the union of m-cliques, such that any two of these cliques intersect in at most one node and there are no simple cycles involving more than m 1 nodes.

A.5.3 Robustness

Note the following observation:

Observation 2 If (A.1) holds for m ¥ 2, g is sustainable as a pure strategy maximal equilibrium, and ij P g, then g ¡ ij is not sustainable as a pure strategy maximal equilibrium.

Thus, beginning from some ME network, if a link is deleted then the network will necessarily further degrade in terms of what is sustainable. The deﬁnition of robustness remains for ME such that a network g is robust against social contagion if it is renegotiation-proof and sustained as part of a pure strategy subgame perfect equilibrium with g0 g such that in any subgame continuation from any renegotiation proof g1 g, and for any i and ij P g1, if i does not perform the favor for j when called upon, then the continuation leads to g2 such that if h` R g2 1 1 then h P Nipg q Y tiu and ` P Nipg q Y tiu. Thus all theorems discussing the structures of the robust networks work for pure strategy maximal equilibria as well. Especially the following two:

Theorem 6 A network is robust against social contagion if and only if it is a social quilt.

Theorem 7 In the asymmetric payoﬀs case, if a network g is robust against social contagion then all links in g are supported. APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 155

A.6 Background Statistics on the Indian Village Networks

A.6.1 Descriptive Statistics

In this section we present some snapshots of the network data discussed in Section 6. The graphs shown in Figure A.10 summarize the distributions of the normalized degree, betweenness and eigenvalue centralities in each of the networks of relationships described in Tables 1 and 2 of Section 6.2. The distributions were computed by considering the network defined among all the surveyed people in the 75 villages in our sample by each relationship type5. Mores specifically, the aggregate sample is comprised by all the people that were surveyed and who reported at least one relationship of the types described in Table 1 of Section 6.2 with some other surveyed individual. The total number of people in the sample thus defined was 16855. The first graph for each relationship type in Figure A.10 shows the inverse cumulative distribution functions of normalized degree, betweenness and eigenvalue centralities. The second graph shows the inverse cumulative distribution function of normalized degree and the betweenness and eigenvalue centralities of the marginal village according to the normalized degree. Figure A.11 exhibits the distribution of the number of reported relationships by surveyed individuals for each relationship type described in Table 1 of Section 6.2. As shown in these graphs and discussed in Section 6.2, only a very small fraction of the surveyed population reported a number of relationships reaching the limits of 5 or 8 implied by the survey design.

A.6.2 Support in the Data

Figure A.12 shows the inverse cumulative distribution functions of support Spg1, gq in our sample of 75 villages for a number of combinations of the networks g and g1 deﬁned in Table 2 of Section 6.1. Each graph also includes the plots of the fraction

5Note that relationships were restricted to lie within each village. APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 156

Figure A.10: Centrality Measures APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 157

Figure A.10 (continued) APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 158

Figure A.10 (continued) APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 159

Figure A.10 (continued) APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 160

Figure A.10 (continued) APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 161

Figure A.10 (continued) APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 162

Figure A.10 (continued)

Figure A.11: Distribution of the number of reported relationships by surveyed individuals. APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 163

Figure A.12: The inverse cumulative distribution function of support levels in the villages: The horizontal axis is the fraction of villages having support no more than the amount listed on the vertical axis. The upper-most curve is the inverse CDF of the fraction of supported g1 relationships in the g network. The five curves below list the breakdown of the fraction for the marginal village by various levels of support: “by k” indicates the fraction of links in that village that are supported by exactly k other nodes (so that i and j have k friends in common), and so the five lines below sum to the curve above. of links supported by exactly k other links in the marginal village (ordered according to their aggregate support levels). Note that the context network g that defines in each case whether a given link in g is supported or not, is always by construction a superset of g1.

A.6.3 Comparing Support to Clustering

Figure A.13 shows graphs comparing clustering and support in the various networks deﬁned in Table 2 of Section 6.1. APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 164

Figure A.12 (continued) APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 165

Figure A.12 (continued) APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 166

Figure A.12 (continued) APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 167

Figure A.13: The inverse cumulative distribution function of support and clustering levels in the villages: The horizontal axis is the fraction of villages having support/clustering no more than the amount listed on the vertical axis. The upper-most curve is support and the lower-most is the clustering coeﬃcient of the marginal village. APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 168

Figure A.13 (continued) APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 169

Figure A.13 (continued) APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 170

Figure A.13(continued) APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 171

A.6.4 Comparing Support in Diﬀerent Sorts of Relationships

In the main paper, we worked with a definition of an “All” network that included relatives. We also now include various calculations that include definitions that simply look at the union of “hedonic” and “favor” networks, H Y F in what follows. Tables A.3-A.5 show the comparison of support measures of various relationships. p 1 qq ¡ The entry ij of the first table reports the number of villages for which S gi, All p 1 qq S gj, All . The entry ij of Table A.4 reports the number of villages for which p 1 p q qq ¡ p 1 p q qq p q S gi,H or F S gj,H or F , where H or F is the union of the Favor network and the Hedonic network. Finally, Table A.5 presents the comparison of self support p 1 1qq ¡ measures. That is, the entry ij reports the number of villages for which S gi, gi p 1 1 q6 S gj, gj .

Support Measure Self H Y F All Physical Favors 0.2807 0.6002 0.7200 Intangible Favors 0.2587 0.5721 0.7198 Hedonic 0.3795 0.5569 0.6530 H Y F – 0.5556 – All – – 0.6931

Table A.2: The Average Support Measures

Network g1 Favors Physical Favors Intangible Favors Hedonic Favors – 30¦¦ 24¦¦¦ 72¦¦¦ Physical Favors 45¦¦ – 38 69¦¦¦ Intangible Favors 51¦¦¦ 37 – 72¦¦¦ Hedonic 3¦¦¦ 6¦¦¦ 3¦¦¦ – *** significant difference at 1% level ** significant difference at 5% level

Table A.3: Comparison of Support Measures. Entry i, j is the number of villages for p 1 qq ¡ p 1 qq which S gi, All S gj, All .

6The entries the self support of the Favors network with that of the Physical Favors and Intangible Favors networks are left blank because since they are 75 by deﬁnition (Since each of the latter are subnetworks of the favors network). APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 172

Network g1 Favors Physical Favors Intangible Favors Hedonic Favors – 15¦¦¦ 40¦¦¦ 46 ¦¦ Physical Favors 60¦¦¦ – 57¦¦¦ 63¦¦¦ Intangible Favors 35 18¦¦¦ – 38 Hedonic 29 ¦¦ 12¦¦¦ 37 – *** significant difference at 1% level ** significant difference at 5% level

Table A.4: Comparison of Support Measures. Entry i, j is the number of villages for p 1 Y qq ¡ p 1 Y qq which S gi,H F S gj,H F .

Network g1 Favors Physical Favors Intangible Favors Hedonic Favors – – – 55¦¦¦ Physical Favors – – 48¦¦¦ 8¦¦¦ Intangible Favors – 27¦¦¦ – 11¦¦¦ Hedonic 20¦¦¦ 8¦¦¦ 64¦¦¦ – *** significant difference at 1% level ** significant difference at 5% level

Table A.5: Comparison of Support Measures. Entry i, j is the number of villages for p 1 1qq ¡ p 1 1 qq which S gi, gi S gj, gj .

A.6.5 How Observed Support Compares to that Expected in a Random Network

One way to get a feeling for how much support we observe is to compare the observed level of support with that which would arise if the same number of links were instead distributed purely at random. In a random network with a the probability p of a link and n as the population of the network, the expected support measure can be approximated as follows: the average degree is D p¤pn¡1q, and the chance any given link is supported is roughly S 1 ¡ p1 ¡ pqpD¡1q, since the chance a link ij is not supported is that none of the other D ¡ 1 friends of agent i are friends of agent j which is p1 ¡ pqpD¡1q.7 From the

7This approximates things since not all nodes are expected to have degree D. Slightly more accurate estimates could be obtained by working with the speciﬁc degree distribution that would be generated, or some other degree distribution. We do not pursue those, since we provide a more rigorous test with the geographic based networks in any case. APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 173

data, we can calculate p and n for each village and estimate what the support measure S would be if the network were generated uniformly at random. Table A.6 shows the average support measure in real networks are substantially larger than those expected in random networks.

S(PF, PF) S(IF, IF) S(H, H) S(HYF, HYF) S(All, All) Observed network 0.2807 0.2587 0.3795 0.5556 0.6931 Random network 0.0137 0.0172 0.0400 0.0960 0.1487

Table A.6: Average Support Measures for Observed and Random Networks

While these numbers are suggestive, we now provide a more detailed statistical test to see if the support is higher than would be generated at random, where random also allows for geographic biases.

Geography and Support

We proceed in two different manners. First, we build an explicit random network model that incorporates geography directly. Next, we work with an exponential random graph model. Some of the relationships in these data are bound to be at least partly correlated by geographic closeness, since it is natural to expect some sorts of favor exchange among geographic neighbors , and geographic closeness is a transitive relation. Therefore networks that we observe may inherit some support from this geographic determinacy in a manner unrelated to the network structure based favor exchange that we have examined. In order to address this issue, we examine a geographically-biased-random network formation model and then see whether the support measures from that model differ in a statistically significant way from the observed support measures. We proceed as follows.

• For each village we decomposed the observed links of each type into deciles according to the geographic proximity of the members of the pair in question, as measured by the households’ GPS coordinates. Based upon this decomposition APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 174

we constructed an empirical link distribution for each speciﬁc relationship and each village.

• We then carried out 50 simulations for each village and relationship. In each case we constructed a random graph based on the corresponding empirical distribution of links by geographic location. In order to guarantee that each simulated base network was a subset of the context network, we produced the context network by augmenting the simulated base network with the appropriate number of randomly drawn links according to the appropriate conditional distribution.

• We measured the support of each simulated network, and by comparing it to the observed support for the corresponding village and relationship, created a realization of a random variable with value 1 if the simulated support measure exceeded the observed support and 0 otherwise.

• Pooling all the random variables generated according to this method for a given relationship across all villages, we performed a one sided test of the null hypothesis that the random variable was binomially distributed with equal probability of being 1 or 0.8 As shown in Table A.7, for every one of relationships, the observed support is signiﬁcantly greater (with p-values smaller than 0.0001 in all cases) than the one generated by the geographically biased random graph models.

A.6.6 How Observed Support Compares to that Expected via an Exponential Random Graph Model Incorporat- ing Geography

Table A.8 shows the estimated coefficients and the standard errors in the collection of models: 8The variances of the random variable are likely to be different for different villages. Not taking into account heteroskedasticity biases the test against rejection of the null hypothesis. APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 175

Base-Context p-value PF-PF 0.000 PF-HR 0.000 PF-All 0.000 IF-IF 0.000 IF-HR 0.000 IF-All 0.000 HR-HR 0.000

Table A.7: Binomial one sided test

¸ ¸ ¸ p p qq p 1q p q log P r G g β0 β1 gij β2 gijs g, g ij β3 d i, j gij (A.3) i j i j i j

We have ﬁt one such model for each of the Favors networks in each of the 75 p 1q villages in the sample. The indicator function s g, g ij is 1 if and only if link ij is supported in the All networks of the corresponding village.

We used version 2.1-1 of the R package statnet developed by Handcock et.al. to estimate the exponential random graph models. Figure A.14 shows the support coeﬃcient estimates with 99% conﬁdence bars when analyzing g1 F avors with g All as a context.

A.6.7 Bounding Measurement Error

Another thing that we do is examine how much measurement error there would have to be in order to see support measures of the level that we observed if the true support level were really 100 percent. In particular, what fraction of links would have to be missing to get the observed relationships?9 Speciﬁcally, the types of errors that are likely to arise in our data are one-sided: while people are quite likely to forget relationships, it is less likely that they “imagine” ones given the way in which these

9This test is biased against us since we are not considering missing nodes. APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 176

ˆ ˆ ˆ ˆ ˆ ˆ Village β1 β2 β3 Village β1 β2 β3 1 -3.535 1.951 -2.64591 18 -4.167 2.366 -1.39615 (0.103) (0.094) (0.15248) (0.089) (0.082) (0.08668) 2 -3.411 2.031 -2.99944 19 -5.203 2.964 -0.1163 (0.108) (0.102) (0.18072) (0.103) (0.112) (0.03037) 3 -5.741 3.228 -0.0239 20 -5.411 2.786 -0.03094 (0.055) (0.066) (0.00903) (0.079) (0.094) (0.01605) 4 -5.416 3.138 -0.06672 21 -5.716 3.273 0.01757 (0.064) (0.077) (0.01876) (0.066) (0.076) (0.00286) 5 -5.283 3.287 -0.10529 22 -5.295 2.635 -0.0024 (0.098) (0.111) (0.02784) (0.074) (0.088) (0.00612) 6 -4.562 2.435 0.02029 23 -5.233 2.597 -0.17919 (0.099) (0.126) (0.01198) (0.055) (0.071) (0.03192) 7 -3.831 2.5 -2.26155 24 -5.364 3.144 0.02495 (0.143) (0.129) (0.14617) (0.107) (0.12) (0.00619) 8 -4.6 2.305 -0.00219 25 -4.921 2.371 0.00219 (0.109) (0.125) (0.01419) (0.073) (0.097) (0.00424) 9 -5.57 2.845 -0.01145 26 -5.811 3.006 -0.00087 (0.071) (0.086) (0.00866) (0.048) (0.064) (0.00364) 10 -4.599 2.256 -0.28231 27 -5.615 3.11 -0.00811 (0.161) (0.163) (0.09628) (0.058) (0.07) (0.00394) 11 -5.045 2.58 0.02389 28 -5.119 2.658 0.00057 (0.1) (0.114) (0.01656) (0.085) (0.099) (0.00611) 12 -4.975 2.435 -0.07023 29 -5.011 2.67 -0.04409 (0.074) (0.088) (0.03028) (0.071) (0.084) (0.02404) 13 -5.047 2.684 -0.03246 30 -5.284 2.303 0.01208 (0.094) (0.11) (0.01584) (0.05) (0.069) (0.00419) 14 -5.027 2.509 -0.14136 31 -4.469 2.508 -1.15936 (0.072) (0.084) (0.03267) (0.106) (0.093) (0.10727) 15 -5.218 2.84 -0.09215 32 -3.564 2.173 -2.78001 (0.09) (0.101) (0.02527) (0.114) (0.11) (0.18851) 16 -5.06 2.613 -0.08345 33 -5.523 3.35 0.01532 (0.077) (0.087) (0.03251) (0.08) (0.089) (0.01093) 17 -5.573 3.062 -0.01536 34 -4.566 2.498 -1.5863 (0.061) (0.076) (0.00996) (0.081) (0.079) (0.07984)

Table A.8: Estimated Coeﬃcients of the ergm model for the° Favors network. The model is logpP rpG gqq β0 β1 gij ° ° i j p 1q p q β2 gijs g, g ij β3 d i, j gij i j i j APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 177

ˆ ˆ ˆ ˆ ˆ ˆ Village β1 β2 β3 Village β1 β2 β3 35 -2.929 2.064 -2.42283 52 -2.565 1.231 -2.87712 (0.139) (0.122) (0.13827) (0.121) (0.118) (0.18388) 36 -4.69 2.003 -0.01248 53 -5.638 3.609 -0.00381 (0.062) (0.091) (0.00822) (0.064) (0.073) (0.00457) 37 -5.593 2.937 -0.00046 54 -3.705 2.396 -2794.07224 (0.047) (0.06) (0.00348) (0.113) (0.105) (230.7352) 38 -5.627 3.448 0.00809 55 -3.626 1.914 -2.00195 (0.067) (0.076) (0.00535) (0.089) (0.084) (0.10752) 39 -2.806 2 -2.84171 56 -4.961 2.273 0.00665 (0.108) (0.093) (0.13161) (0.059) (0.082) (0.00469) 40 -3.439 2.12 -2.23149 57 -5.879 3.36 0.00788 (0.099) (0.09) (0.12461) (0.051) (0.063) (0.00269) 41 -5.035 2.717 -0.0257 58 -5.17 2.816 -0.46967 (0.062) (0.072) (0.01106) (0.05) (0.055) (0.04399) 42 -5.359 2.951 0.00402 59 -4.598 2.793 -0.65429 (0.063) (0.071) (0.00778) (0.119) (0.118) (0.08063) 43 -5.436 2.862 -0.00495 60 -4.517 2.522 -1.20137 (0.061) (0.08) (0.00828) (0.09) (0.083) (0.11496) 44 -5.381 2.601 0.00614 61 -4.762 2.937 -0.53211 (0.055) (0.078) (0.0034) (0.093) (0.095) (0.08401) 45 -4.982 2.596 0.02517 62 -4.097 2.544 -2.07602 (0.081) (0.1) (0.00517) (0.077) (0.073) (0.12048) 46 -4.915 2.327 -0.14076 63 -5.718 3.244 0.00255 (0.065) (0.08) (0.02759) (0.055) (0.064) (0.00411) 47 -5.042 2.584 -0.08957 64 -4.123 2.399 -1.59247 (0.085) (0.092) (0.03094) (0.117) (0.106) (0.13163) 48 -5.274 3.073 -0.31429 65 -5.59 2.95 -0.03898 (0.077) (0.077) (0.0453) (0.078) (0.094) (0.01971) 49 -5.536 2.857 -0.00476 66 -5.064 2.771 -0.17221 (0.057) (0.065) (0.00529) (0.094) (0.105) (0.0409) 50 -4.595 2.495 -1.49029 67 -3.857 2.363 -1.96435 (0.063) (0.06) (0.06625) (0.102) (0.091) (0.10493) 51 -3.623 1.986 -1.67765 68 -4.909 2.281 -0.16078 (0.097) (0.097) (0.10073) (0.061) (0.072) (0.02437)

Table A.8 (continued) APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 178

Figure A.14: The coefficients associated to the support terms in exponential random graph models of the “favors” networks in the 75 villages. The value of a link’s support binary variable is 1 if and only if the link is supported in the All network. questions were designed.10 To address this issue, there are various ways in which one might proceed, and here we followed a fairly simple one where we simulated the survey process 100 times, proceeding as follows in each iteration: For each village in our sample and type combination of base-context networks in question we consider the closest network to the context network that leads the base network to have full support; where closest is defined as having the least number of additional links.11 We then remove each link in the augmented network with a measurement error probability, and calculate the support of the resulting base- context pair. Figure A.15 shows for each measurement error in the x axis the mean fraction of villages in the sample that ended with a support fraction of at most the level observed in the survey. It should be noted that the number of simulations (100) is such that any differences in expectation for different measurement errors in a given

10By asking questions regarding actual actions (borrowing or lending money or rice, visiting someone’s home, etc.) rather than asking about perceived relationships (who is your friend), we eliminate many problems with misperceived or asymmetric sorts of relationships. 11We randomly draw one network from the set of closest networks. APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 179

ˆ ˆ ˆ ˆ ˆ ˆ Village β1 β2 β3 Village β1 β2 β3 69 -5.692 3.216 -0.00852 73 -5.331 2.899 -0.01296 (0.062) (0.072) (0.00561) (0.076) (0.088) (0.01131) 70 -5.503 2.947 -0.00275 74 -5.403 3.251 -0.0077 (0.072) (0.084) (0.00429) (0.059) (0.071) (0.00875) 71 -5.491 3.029 -0.00194 75 -4.654 2.59 -0.20904 (0.08) (0.09) (0.00335) (0.077) (0.087) (0.04137) 72 -5.151 3.098 0.00485 (0.073) (0.087) (0.00472)

Table A.8 (continued) base-context pair or for diﬀerent base-context-pairs are statistically signiﬁcant. Table A.9 provides a closer look at a small segment of Figure A.15.

Support Measure Measurement Error 16% 24% 32% 40% PFavors-All 0.54 0.91 0.99 0.999 IFavors-All 0.52 0.91 0.99 0.999 Favors-All 0.40 0.87 0.99 0.999 Hedonic-All 0.17 0.71 0.96 0.999 All-All 0.12 0.74 0.98 0.999

Table A.9: A closer look at Figure A.15

A.6.8 The Relation of Support to other Characteristics of Links, Households and Individuals

Link Level Predictors of Support

The ﬁrst collection of statistics, presented in Tables A.10-A.14, builds upon the fact that support is ﬁrstly a property of relationships themselves, rather than a property of agents. We look at the likelihood that a a pair of agents have at least one friend in common conditional on them being similar/disimilar according to a number of individual characteristics: education, age, caste, gender and participation in the APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 180

0.8

0.6

Pfavors-All Ifavors-All 0.4 Favors-All Hedonic-All All-All

0.2

-0.00 0.08 0.16 0.24 0.32 0.4 0.48 0.56 0.64 0.72 0.8 0.04 0.12 0.2 0.28 0.36 0.44 0.52 0.6 0.68 0.76

-0.2

Figure A.15: Fraction of villages with at most the observed support as a function of measurement error microfinance program12. Specifically we break the set of pairs of agents in each village into similarity/dissimilarity classes for each of characteristic and for each class compute the mean across the 75 villages of the fraction of pairs of agents in the class who have at least one friend in common13 in the All network. We present one table for each characteristic. In addition to the mean for each similarity/dissimilarity class, each table shows the number of villages in which the class is the one with the highest fraction of pairs of agents with a friend in common. Note that in each of the 5 individual characteristics considered the presence of a link among dissimilar agents is associated with a significantly greater probability of a friend in common relative to the classes of similar agents, with respect to situations in which the agents are not linked. This is a property which holds for links in the favors networks and also in the hedonic networks. 12As discussed above, the data that we use was collected as part of the deployment of a microfinance program (see Banerjee et al. (2010)). 13A link is supported when the linked agents have at least one friend in common; as above, speaking of having a friend in common lets us refer to linked agents as well as to agents that are not linked. APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 181

Education ij R F avors ij P F avors ij R Hed ij P Hed Mean # Max Mean # Max Mean # Max Mean # Max below median 0.139 26 0.707 25 0.140 25 0.651 23 # pairs in class 600724 9507 600616 9615 ab./bel. med. 0.131 3 0.719 25 0.132 2 0.663 28 # l pairs in class 1003084 11795 1003210 11669 ab. median 0.149 46 0.714 25 0.150 48 0.637 24 # pairs in class 469260 7617 469222 7655

Table A.10: Mean fraction of pairs of agents with at least one friend in common in the All network, by similarity/dissimilarity in formal education, when the agents have a link in the favors network compared to when they do not have a link in the favors network. The column labeled #Max depicts the number of villages (out of 75) in which the fraction of pairs with at least one friend in common is the highest among the 3 education categories. The education scale has 15 diﬀerent levels, ranging from no formal education to graduate degree. The median level of education in the scale is 5.

In addition to the mean for each similarity/dissimilarity class, each table shows the number of villages in which the class is the one with the highest fraction of pairs of agents with a friend in common. Note that in each of the 5 individual characteristics considered the presence of a link among dissimilar agents is associated with a signiﬁcantly greater probability of a friend in common relative to the classes of similar agents, with respect to situations in which the agents are not linked. This is a property which holds for links in the favors networks and also in the hedonic networks.

Household Level Predictors of Support

Here, we examine links at a household level, where we a pair of households to be linked in a given network if there exists a agent in each household who are linked to each other. Table A.16 presents the coeﬃcients associated to two probit regressions relating the likelihood of having a “household friend” in common in the All network to some household characteristics: maximum education, mean age, rooms per person and APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 182

Gender ij R F avors ij P F avors ij R Hed ij P Hed Mean # Max Mean # Max Mean # Max Mean # Max both male 0.203 75 0.707 16 0.202 75 0.637 16 # pairs in class 404266 10343 402374 12235 female/male 0.123 0 0.728 30 0.126 0 0.690 43 # pairs in class 1033918 7572 1038232 3258 both female 0.117 0 0.713 29 0.116 0 0.649 17 # pairs in class 634884 11004 632442 13446

Table A.11: Mean fraction of pairs of agents with at least one friend in common in the All network, by similarity/dissimilarity in gender, when the agents have a link in the favors network compared to when they do not have a link in the favors network. See the caption of Table A.10 for more details.

Age ij R F avors ij P F avors ij R Hed ij P Hed Mean # Max Mean # Max Mean # Max Mean # Max both belong 0.116 3 0.700 23 0.117 3 0.633 17 # pairs in class 552166 7753 551759 8160 belong/not 0.134 0 0.727 30 0.136 0 0.670 36 # pairs in class 1031445 12236 1032157 11524 neither belong 0.167 72 0.705 22 0.168 72 0.644 22 # l pairs in class 489457 8930 489132 9255

Table A.12: Mean fraction of pairs of agents with at least one friend in common in the All network, by similarity/dissimilarity in their age, when the agents have a link in the favors network compared to when they do not have a link in the favors network. See the caption of Table A.10 for more details. APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 183

Caste ij R F avors ij P F avors ij R Hed ij P Hed Mean # Max Mean # Max Mean # Max Mean # Max diﬀerent castes 0.091 0 0.578 3 0.090 0 0.540 7 # pairs in class 1432149 7638 1429094 10693 same caste 0.239 75 0.755 72 0.244 75 0.707 68 # pairs in class 640919 21281 643954 18246

Table A.13: Mean fraction of pairs of agents with at least one friend in common in the All network, by whether they belong to the same caste or not, when the agents have a link in the favors network compared to when they do not have a link in the favors network. See the caption of Table A.10 for more details.

Micro Finance ij R F avors ij P F avors ij R Hed ij P Hed Mean # Max Mean # Max Mean # Max Mean # Max Neither belong 0.103 7 0.696 14 0.102 7 0.629 13 # pairs in class 336948 5018 335703 6263 belong/not 0.100 5 0.691 8 0.099 5 0.637 7 # pairs in class 69383 1087 69092 1378 both belong 0.159 26 0.635 16 0.158 26 0.610 18 # pairs in class 4862 186 4836 212

Table A.14: Mean fraction of pairs of agents with at least one friend in common in the All network, by whether they participate or not in the micro ﬁnance program, when the agents have a link in the favors network compared to when they do not have a link in the favors network. This table was computed only considering pairs agents eligible for participation in the program. That is, women of age 15 or older, living in one of the 38 villages in the treatment group. APPENDIX A. SOCIAL QUILTS: SUPPLEMENTARY RESULTS 184

household size. In the ﬁrst one an observation corresponds to a randomly chosen friend in the favors network of a randomly chosen household in one of the 75 villages in our sample. In the second regression an observation corresponds to a randomly chosen household unrelated to a randomly chosen household in one of the 75 villages.

Linked Not Linked Mean Age 0.244 0.005 Mean Education 0.034 ¡0.010 Rooms per person ¡0.84¦¦ 0.16 Household size ¡0.064 0.041 Intercept 1.54¦ ¡0.56 (**) Signiﬁcant at 5%, (*) Signiﬁcant at 10%

Table A.15: Probit regression of support in the households All network of linked and unlinked pairs in the households favor network. In the regression shown in the ﬁrst column an observation corresponds to a randomly chosen agent linked in the favors network to a randomly chosen household in one of the 75 villages in our sample. In the regression shown in the second column an observation corresponds to a randomly chosen agent unlinked to a randomly chosen household in one of the 75 villages.

Linked Not Linked Mean Age 0.205 0.008 Max Education 0.017 0.007 Rooms per person ¡0.78¦¦ 0.13 Household size ¡0.063 0.029 Intercept 1.70¦¦ ¡0.66¦ (**) Signiﬁcant at 5%, (*) Signiﬁcant at 10%

Table A.16: Probit regression of support in the households all network of linked and unlinked pairs in the households favor network. In the regression shown in the ﬁrst column an observation corresponds to a randomly chosen household linked in the favors network to a randomly chosen household in one of the 75 villages in our sample. In the regression shown in the second column an observation corresponds to a randomly chosen household not linked to a randomly chosen household in one of the 75 villages. Bibliography

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