DOCSLIB.ORG

Asymptotic Stability in the Lovász-Shapley Replicator

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Asymptotic Stability in the Lovász-Shapley Replicator *Manuscript Asymptotic Stability in the Lov´asz-Shapley Replicator Dynamic for Cooperative Games∗ Andr´eCasajusy Michael Krammz Harald Wiesex November 20, 2019 Abstract We derive population dynamics from finite cooperative games with trans- ferable utility, where the players are interpreted as types of individuals. We show that any asymptotically stable population profile is characterized by a coalition: while the types in the coalition have the same positive share, the other types vanish. The average productivity of such a stable coalition must be greater than the average productivity of any proper sub- or super- coalition. In simple monotonic games, this means that exactly the minimal winning coalitions are stable. Possible applications are the analysis of the organizational structure of businesses or the population constitution of eu- social species. ∗We would like to thank Markus Kunze, Pierfrancesco La Mura, Wolfgang Leininger, Lars Metzger, Heinrich Nax, Karl Friedrich Siburg, William Thomson, and J¨orgenWeibull for valu- able comments. This paper also benefited from discussions and remarks at the European Meeting on Game Theory 2015 (SING, St. Petersburg), the 2nd Workshop on Economics and Biology (Toulouse), and the 3rd Workshop on Cooperative Game Theory in Business Practice (Leipzig). We are grateful for the comments of two anonymous referees and an associate editor. Andr´e Casajus: Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Founda- tion) - 388390901. yHHL Leipzig Graduate School of Management, Germany and Dr. Hops Craft Beer Bar, Germany, Email: [email protected] zCorresponding author, Technical University of Dortmund, Vogelpothsweg 87, 44221 Dort- mund, Germany, Email: [email protected] xUniversity of Leipzig, Email: [email protected] Keywords: cooperative game theory, evolutionary game theory, replicator dy- namics, asymptotic stability, Lov´asz-Shapleyvalue, simple monotonic games JEL codes: C71, C73 2 1 Introduction For a long time, non-cooperative game theory has employed the evolution- ary approach to model the behavior of boundedly rational agents and to analyze the relation between the stable outcomes of the dynamical process generated by their repeated interaction and the common static solution concept of Nash equilibrium (see, e.g., Mailath, 1998 or Samuelson, 2002 for discussions). In contrast to non-cooperative game theory, cooperative game theory shifts the focus from modeling strategic interactions and the agents' strategic reasoning to payoffs: it asks questions about which pay- off distribution among the players of a given cooperative game can, for instance, be considered as fair or stable (see, e.g., McLean, 2015). The non-cooperative concept of the Nash equilibrium has taken a cen- tral role in the study of game theory in a rather literary sense. On the one hand, the idea of the Nash program, starting with Nash (1953), is to pro- vide non-cooperative foundations for cooperative solution concepts based on the Nash equilibrium. On the other hand, as mentioned above, Smith and Price (1973) and many other researchers started to think about evolu- tionary foundations of the Nash equilibrium about two decades later. Our modeling approach skips the Nash equilibrium in the chain of reasoning and directly connects cooperative and evolutionary game theory. We in- troduce the evolutionary methodology to cooperative game theory in order to answer questions on the relation between the properties of the underly- ing cooperative game and the stable outcomes of a dynamical evolutionary process. In non-cooperative game theory, evolutionary pressures work against strategies, for instance, dominated strategies may die out (see, e.g., Hof- bauer and Weibull, 1996). In our setup, evolution takes place over types of individuals and not over strategies. In this way, pressures work against 1 certain coalitions, for example, non-minimal winning coalitions (see Corol- lary 2). We can thus analyze the relation between asymptotically stable population profiles and the underlying transferable utility (TU) game. The interpretation of the evolutionary dynamics in our model is il- lustrated with an economic example in Section 2 and can be outlined as follows. We consider a society that consists of infinitely many individuals of finitely many different types. The types correspond to the players of a cooperative game. They are (economic) entities characterized by distinct productive capabilities, such as specific business departments in an organi- zation or specific casts of a eusocial species1. The individuals, e.g., members of a specific business department or cast, are programed to perform cer- tain tasks according to their type and contribute to the organization's or the society's total reproductive success or fitness, i.e., its growth in time. This is modelled by the Lov´aszextension of a finite TU game (see Lov´asz, 1983 and Algaba et al., 2004), which embodies a particular Leontief-type technology for the production of fitness. Eventually, the total fitness has to be distributed among the different types. This is modelled by the Lov´asz- Shapley value (see Casajus and Wiese, 2016) that rewards only the scarce types, which is in line with the Leontief-type technology. Given this dis- tribution, the composition of the society evolves in time according to the replicator dynamic, which is the most commonly used imitative dynamic in the literature (see Sandholm, 2010, Section 5.3) and which is most suitable to capture the idea of individuals reproducing according to their fitness 1A common \criterion for eusociality is the presence of castes, which are groups of individuals that become irreversibly behaviorally distinct at some point prior to repro- ductive maturity. Eusocial societies are characterized by two traits: (1) helping by individuals of the less-reproductive caste, and (2) either behavioral totipotency of only the more reproductive caste (facultative eusociality) or totipotency of neither caste (ob- ligate eusociality)" (Crespi and Yanega, 1995, p. 109). Examples are the European hornet (Vespa crabro) or the naked mole rat (Heterocephalus glaber). 2 derived from the cooperative game. Now, one can study the prolifera- tion of the types and, in particular, stable compositions of the society for the resulting replicator dynamic under the Filippov solution (see Filippov, 1988). To our knowledge, we are the first to derive an evolutionary dynamic from a cooperative game in a similar fashion as it is done in non-cooperative evolutionary game theory without incorporating non-cooperative aspects. This cooperative approach has particular appeal when types represent orga- nizational departments of a collaborative enterprise, such as a large business corporation, or casts of a eusocial species. Usually, it is considered that \[o]n general grounds of methodological individualism, noncooperative games are prior to cooperative games" (see Elster, 1982, p. 466). We do not contest this conventional approach of economic theory (or its value) to analyze aggregate phenomena by exam- ining the actions and incentives of individual entities. Rather, our aim is to provide a complementary view on the \links between aggregate analy- sis and the study of individual behavior" (see Elster, 1982, p. 477). Some researchers have pushed forward that methodological individualism alone cannot explain all aspects of socio-economic interactions and thus that so- cial context and institutions need to be included in the analysis (see New- ton, 2019 and Hodgson, 2007). Furthermore, the debate on the evolution of eusociality2 emphasizes that it might be insightful to provide theories 2The evolution of eusociality is debated controversially among evolutionary biologists. It has challenged the predominant idea of the inclusive fitness theory (in this case, kin selection; see, e.g., Hamilton, 1964), which postulates that evolution takes place at the level of individuals and thus is loosely related to the non-cooperative approach in game theory. In contrast, some researchers advance the idea of group selection to explain eusociality (see Wilson and H¨olldobler,2005 and Nowak et al., 2010), which is an approach similar to cooperative game theory, since it assumes that evolution takes place at the level of groups. In view of this debate, skipping the non-cooperative approach of 3 v(f1g) = 3 v(f1; 2g) = 4 v(N) = 8 v(f2g) = 2 v(f1; 3g) = 5 v(;) = 0 v(f3g) = 1 v(f2; 3g) = 7 Table 1: A three-player cooperative game with transferable utility which take the opposite route, assuming that individual behavior (at least partly) emerges from the properties of a business department, or, in the biological context, of a cast. Our approach adds evolutionary pressures to this perspective, which gives an alternative foundation of the emergence of group properties, such as an evolutionary stable constitution of business units in an entrepreneurial organization. Our goal is to characterize evolutionary stable population profiles in a setting in which the evolutionary process is determined by an underlying transferable utility game. Consider, for instance, the cooperative game with the player set N = f1; 2; 3g defined by the coalition function v as in Table 1, which determines the worth generated by any specific coalition. We derive a simple criterion to characterize stable population profiles even in games as asymmetric and rather complex as the one defined in Table 1. Our results imply that the profiles corresponding to the coalitions f1g and f2; 3g are evolutionary
Load more

Recommended publications
  • Lecture Notes General Equilibrium Theory: Ss205
    LECTURE NOTES GENERAL EQUILIBRIUM THEORY: SS205 FEDERICO ECHENIQUE CALTECH 1 2 Contents 0. Disclaimer 4 1. Preliminary definitions 5 1.1. Binary relations 5 1.2. Preferences in Euclidean space 5 2. Consumer Theory 6 2.1. Digression: upper hemi continuity 7 2.2. Properties of demand 7 3. Economies 8 3.1. Exchange economies 8 3.2. Economies with production 11 4. Welfare Theorems 13 4.1. First Welfare Theorem 13 4.2. Second Welfare Theorem 14 5. Scitovsky Contours and cost-benefit analysis 20 6. Excess demand functions 22 6.1. Notation 22 6.2. Aggregate excess demand in an exchange economy 22 6.3. Aggregate excess demand 25 7. Existence of competitive equilibria 26 7.1. The Negishi approach 28 8. Uniqueness 32 9. Representative Consumer 34 9.1. Samuelsonian Aggregation 37 9.2. Eisenberg's Theorem 39 10. Determinacy 39 GENERAL EQUILIBRIUM THEORY 3 10.1. Digression: Implicit Function Theorem 40 10.2. Regular and Critical Economies 41 10.3. Digression: Measure Zero Sets and Transversality 44 10.4. Genericity of regular economies 45 11. Observable Consequences of Competitive Equilibrium 46 11.1. Digression on Afriat's Theorem 46 11.2. Sonnenschein-Mantel-Debreu Theorem: Anything goes 47 11.3. Brown and Matzkin: Testable Restrictions On Competitve Equilibrium 48 12. The Core 49 12.1. Pareto Optimality, The Core and Walrasian Equiilbria 51 12.2. Debreu-Scarf Core Convergence Theorem 51 13. Partial equilibrium 58 13.1. Aggregate demand and welfare 60 13.2. Production 61 13.3. Public goods 62 13.4. Lindahl equilibrium 63 14.
    [Show full text]
  • Three Essays in Mechanism Design
    THREE ESSAYS IN MECHANISM DESIGN Andrei Rachkov A DISSERTATION PRESENTED TO THE FACULTY OF PRINCETON UNIVERSITY IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY RECOMMENDED FOR ACCEPTANCE BY THE DEPARTMENT OF ECONOMICS Adviser: Stephen Morris January 2013 c Copyright by Andrei Rachkov, 2012. All rights reserved. iii Abstract In the first and second chapters we study whether the current trend of using stronger solution concepts is justified for the optimal mechanism design. In the first chapter, we take a simple auction model and allow for type-dependent outside options. We argue that Bayesian foundation for dominant strategy mechanisms is valid when symmetry conditions are satisfied. This contrasts with monotonicity constraints used before in the literature. In the second chapter we develop the idea further by looking into the practical application of type-dependency of outside options in auctions - namely, a possibility of collusion between agents. We show that in this environment for a certain range of primitives no maxmin foundation for dominant strategy mechanisms will exist. Finally, in the last chapter we study a voting environment and non-transferable utility mechanism design. We argue that strategic voting as opposed to truthful voting may lead to higher total welfare through better realization of preference intensities in the risky environment. We also study optimal mechanisms rules, that are sufficiently close to the first best for the uniform distribution, and argue that strategic voting may be a proxy for information transmission if the opportunities to communicate preference intensities are scarce. iv Acknowledgments To my mother who imbued me with a strong sense of fairness and my father who taught me to always strive for the ideal.
    [Show full text]
  • Personality and Marital Surplus
    IZA DP No. 4945 Personality and Marital Surplus Shelly Lundberg May 2010 DISCUSSION PAPER SERIES Forschungsinstitut zur Zukunft der Arbeit Institute for the Study of Labor Personality and Marital Surplus Shelly Lundberg University of Washington, University of Bergen and IZA Discussion Paper No. 4945 May 2010 IZA P.O. Box 7240 53072 Bonn Germany Phone: +49-228-3894-0 Fax: +49-228-3894-180 E-mail: [email protected] Any opinions expressed here are those of the author(s) and not those of IZA. Research published in this series may include views on policy, but the institute itself takes no institutional policy positions. The Institute for the Study of Labor (IZA) in Bonn is a local and virtual international research center and a place of communication between science, politics and business. IZA is an independent nonprofit organization supported by Deutsche Post Foundation. The center is associated with the University of Bonn and offers a stimulating research environment through its international network, workshops and conferences, data service, project support, research visits and doctoral program. IZA engages in (i) original and internationally competitive research in all fields of labor economics, (ii) development of policy concepts, and (iii) dissemination of research results and concepts to the interested public. IZA Discussion Papers often represent preliminary work and are circulated to encourage discussion. Citation of such a paper should account for its provisional character. A revised version may be available directly from the author. IZA Discussion Paper No. 4945 May 2010 ABSTRACT Personality and Marital Surplus* This paper uses data from the German Socio-economic Panel Study to examine the relationship between psychological traits, in particular personality, and the formation and dissolution of marital and cohabiting partnerships.
    [Show full text]
  • Game Theory for Signal Processing in Networks
    Game Theory for Signal Processing in Networks Giacomo Bacci, Samson Lasaulce, Walid Saad, and Luca Sanguinetti Abstract In this tutorial, the basics of game theory are introduced along with an overview of its most recent and emerging applications in signal processing. One of the main features of this contribution is to gather in a single paper some fundamental game-theoretic notions and tools which, over the past few years, have become widely spread over a large number of papers. In particular, both strategic-form and coalition-form games are described in details while the key connections and differences betweenthemareoutlined.Moreover,aparticularattention is also devoted to clarify the connections between strategic-form games and distributed optimization and learning algorithms. Beyond an introduction to the basic concepts andmainsolutionapproaches,severalcarefullydesigned examples are provided to allow a better understanding of how to apply the described tools. I. INTRODUCTION game theory (GT) is a branch of mathematics that enables the modeling and analysis of the interactions between several decision-makers (called players) who can have conflicting or common objectives. A game is a situation in which the benefit or cost achieved by each player from an interactive situation depends, not only on its own decisions, but also on those taken by the other players. For example, the time a car driver needs to get back home generally depends not only on the route he/she chooses but also on the decisions taken by the other drivers. Therefore, in a game, the actions and objectives of the players are tightly coupled.Untilveryrecently,GThas been used only marginally in signal processing (SP), with notable examples being some applications in robust detection and estimation [1] as well as watermarking [2] (in which the watermarking problem is seen as a game between the data embedder and the attacker).
    [Show full text]
  • Homo Economicus: Why the Economy Cannot Be Divorced from Its Governance
    RESTRAINING THE GENUINE HOMO ECONOMICUS: WHY THE ECONOMY CANNOT BE DIVORCED FROM ITS GOVERNANCE STERGIOS SKAPERDAS CESIFO WORKING PAPER NO. 901 CATEGORY 2: PUBLIC CHOICE APRIL 2003 An electronic version of the paper may be downloaded • from the SSRN website: www.SSRN.com • from the CESifo website: www.CESifo.de CESifo Working Paper No. 901 RESTRAINING THE GENUINE HOMO ECONOMICUS: WHY THE ECONOMY CANNOT BE DIVORCED FROM ITS GOVERNANCE Abstract The Homo economicus of traditional economics is far from being completely self-interested, rational, or as individualistic as he is purported to be; he will haggle to death over price but will not take what he wants by force. Implicitly, he is assumed to behave ruthlessly within a welldefined bubble of sainthood. Based on a simple model, I first examine what occurs when this assumption is relaxed and genuine, amoral Homo economici interact. Productivity can be inversely related to compensation; a longer shadow of the future can intensify conflict; and, more competition among providers of protection reduces welfare. The patently inefficient outcomes that follow call for restraining self-interest, for finding ways to govern markets. I then review some of the different ways of creating restraints, from the traditional social contract, to the hierarchical domination of kings and lords, to modern forms of governance. Checks and balances, wider representation, the bureaucratic form of organization, and other ingredients of modern governance can partly be thought of as providing restraints to the dark side of self-interest. Though highly imperfect, these restraints are better than the alternative, which typically involves autocratic, amateurish, and corrupt rule.
    [Show full text]
  • Games in Coalitional Form 1
    GAMES IN COALITIONAL FORM EHUD KALAI Forthcoming in the New Palgrave Dictionary of Economics, second edition Abstract. How should a coalition of cooperating players allocate payo¤s to its members? This question arises in a broad range of situations and evokes an equally broad range of issues. For example, it raises technical issues in accounting, if the players are divisions of a corporation, but involves issues of social justice when the context is how people behave in society. Despite the breadth of possible applications, coalitional game theory o¤ers a uni…ed framework and solutions for addressing such questions. This brief survey presents some of its major models and proposed solutions. 1. Introduction In their seminal book, von Neumann and Morgenstern (1944) introduced two theories of games: strategic and coalitional. Strategic game theory concentrates on the selection of strategies by payo¤-maximizing players. Coalitional game theory concentrates on coalition formation and the distribution of payo¤s. The next two examples illustrate situations in the domain of the coalitional approach. 1.1. Games with no strategic structure. Example 1. Cost allocation of a shared facility. Three municipalities, E, W, and S, need to construct water puri…cation facilities. Costs of individual and joint facilities are described by the cost function c: c(E) = 20, c(W ) = 30, and c(S) = 50; c(E; W ) = 40, c(E; S) = 60, and c(W; S) = 80; c(E; W; S) = 80. For example, a facility that serves the needs of W and S would cost $80 million. The optimal solution is to build, at the cost of 80, one facility that serves all three municipalities.
    [Show full text]
  • Egalitarianism in Mechanism Design∗
    Egalitarianism in Mechanism Design∗ Geoffroy de Clippely This Version: January 2012 Abstract This paper examines ways to extend egalitarian notions of fairness to mechanism design. It is shown that the classic properties of con- strained efficiency and monotonicity with respect to feasible options become incompatible, even in quasi-linear settings. An interim egali- tarian criterion is defined and axiomatically characterized. It is applied to find \fair outcomes" in classical examples of mechanism design, such as cost sharing and bilateral trade. Combined with ex-ante utilitari- anism, the criterion characterizes Harsanyi and Selten's (1972) interim Nash product. Two alternative egalitarian criteria are proposed to il- lustrate how incomplete information creates room for debate as to what is socially desirable. ∗The paper beneficiated from insightful comments from Kfir Eliaz, Klaus Nehring, David Perez-Castrillo, Kareen Rozen, and David Wettstein. Financial support from the National Science Foundation (grant SES-0851210) is gratefully acknowledged. yDepartment of Economics, Brown University. Email: [email protected] 1. INTRODUCTION Developments in the theory of mechanism design, since its origin in the sev- enties, have greatly improved our understanding of what is feasible in envi- ronments involving agents that hold private information. Yet little effort has been devoted to the discussion of socially desirable selection criteria, and the computation of incentive compatible mechanisms meeting those criteria in ap- plications. In other words, extending the theory of social choice so as to make it applicable in mechanism design remains a challenging topic to be studied. The present paper makes some progress in that direction, with a focus on the egalitarian principle.
    [Show full text]
  • Market Design and Walrasian Equilibrium†
    Market Design and Walrasian Equilibriumy Faruk Gul, Wolfgang Pesendorfer and Mu Zhang Princeton University February 2020 Abstract We establish the existence of Walrasian equilibrium for economies with many dis- crete goods and possibly one divisible good. Our goal is not only to study Walrasian equilibria in new settings but also to facilitate the use of market mechanisms in resource allocation problems such as school choice or course selection. We consider all economies with quasilinear gross substitutes preferences but allow agents to have limited quantities of the divisible good (limited transfers economies). We also consider economies without a divisible good (nontransferable utility economies). We show the existence and efficiency of Walrasian equilibrium in limited transfers economies and the existence and efficiency of strong (Walrasian) equilibrium in nontransferable utility economies. Finally, we show that various constraints on minimum and maximum levels of consumption and aggregate constraints of the kind that are relevant for school choice/course selection problems can be accommodated by either incorporating these constraints into individual preferences or by incorporating a suitable production technology into nontransferable utility economies. y This research was supported by grants from the National Science Foundation. 1. Introduction In this paper, we establish the existence of Walrasian equilibrium in economies with many discrete goods and either with a limited quantity of one divisible good or without any divisible goods. Our goal is not only to study Walrasian equilibria in new settings but also to facilitate the use of market mechanisms in resource allocation problems such as school choice or course selection. To this end, we develop techniques for analyzing al- location problems in economies with or without transfers and for incorporating additional constraints into allocation rules.
    [Show full text]
  • Subgame Perfect Implementation with Almost Perfect Information and the Hold-Up Problem
    NBER WORKING PAPER SERIES SUBGAME PERFECT IMPLEMENTATION WITH ALMOST PERFECT INFORMATION AND THE HOLD-UP PROBLEM Philippe Aghion Drew Fudenberg Richard T. Holden Working Paper 15167 http://www.nber.org/papers/w15167 NATIONAL BUREAU OF ECONOMIC RESEARCH 1050 Massachusetts Avenue Cambridge, MA 02138 July 2009 We owe special thanks to Oliver Hart for numerous discussions, and to Johannes Horner and Andy Skrzypacz for pointing out an error in a previous version. We are also grateful to Mathias Dewatripont, Bob Gibbons, Philippe Jehiel, John Moore, Roger Myerson, Andrew Postlewaite, Olivier Tercieux, Jean Tirole, Ivan Werning and Muhamet Yildiz for helpful discussions and comments. Michael Powell provided excellent research assistance. The views expressed herein are those of the author(s) and do not necessarily reflect the views of the National Bureau of Economic Research. NBER working papers are circulated for discussion and comment purposes. They have not been peer- reviewed or been subject to the review by the NBER Board of Directors that accompanies official NBER publications. © 2009 by Philippe Aghion, Drew Fudenberg, and Richard T. Holden. All rights reserved. Short sections of text, not to exceed two paragraphs, may be quoted without explicit permission provided that full credit, including © notice, is given to the source. Subgame Perfect Implementation with Almost Perfect Information and the Hold-Up Problem Philippe Aghion, Drew Fudenberg, and Richard T. Holden NBER Working Paper No. 15167 July 2009 JEL No. C72,C73,D23,L22 ABSTRACT The foundations of incomplete contracts have been questioned using or extending the subgame perfect implementation approach of Moore and Repullo (1988). We consider the robustness of subgame perfect implementation to the introduction of small amounts of asymmetric information.
    [Show full text]
  • Transferable Utility Cooperative Differential Games with Continuous Updating Using Pontryagin Maximum Principle
    mathematics Article Transferable Utility Cooperative Differential Games with Continuous Updating Using Pontryagin Maximum Principle Jiangjing Zhou 1, Anna Tur 2 , Ovanes Petrosian 3,4 and Hongwei Gao 1,* 1 School of Mathematics and Statistics, Qingdao University, Qingdao 266071, China; [email protected] 2 St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia; [email protected] 3 School of Automation, Qingdao University, Qingdao 266071, China; [email protected] 4 Faculty of Applied Mathematics and Control Processes, St. Petersburg State University, Universitetskiy Prospekt, 35, Petergof, St. Petersburg 198504, Russia * Correspondence: [email protected] Abstract: We consider a class of cooperative differential games with continuous updating making use of the Pontryagin maximum principle. It is assumed that at each moment, players have or use information about the game structure defined in a closed time interval of a fixed duration. Over time, information about the game structure will be updated. The subject of the current paper is to construct players’ cooperative strategies, their cooperative trajectory, the characteristic function, and the cooperative solution for this class of differential games with continuous updating, particularly by using Pontryagin’s maximum principle as the optimality conditions. In order to demonstrate this method’s novelty, we propose to compare cooperative strategies, trajectories, characteristic functions, and corresponding Shapley values for a classic (initial) differential game and a differential game with continuous updating. Our approach provides a means of more profound modeling of conflict controlled processes. In a particular example, we demonstrate that players’ behavior is braver at the beginning of the game with continuous updating because they lack the information for the whole game, and they are “intrinsically time-inconsistent”.
    [Show full text]
  • Evolutionary Game Theory: a Renaissance
    games Review Evolutionary Game Theory: A Renaissance Jonathan Newton ID Institute of Economic Research, Kyoto University, Kyoto 606-8501, Japan; [email protected] Received: 23 April 2018; Accepted: 15 May 2018; Published: 24 May 2018 Abstract: Economic agents are not always rational or farsighted and can make decisions according to simple behavioral rules that vary according to situation and can be studied using the tools of evolutionary game theory. Furthermore, such behavioral rules are themselves subject to evolutionary forces. Paying particular attention to the work of young researchers, this essay surveys the progress made over the last decade towards understanding these phenomena, and discusses open research topics of importance to economics and the broader social sciences. Keywords: evolution; game theory; dynamics; agency; assortativity; culture; distributed systems JEL Classification: C73 Table of contents 1 Introduction p. 2 Shadow of Nash Equilibrium Renaissance Structure of the Survey · · 2 Agency—Who Makes Decisions? p. 3 Methodology Implications Evolution of Collective Agency Links between Individual & · · · Collective Agency 3 Assortativity—With Whom Does Interaction Occur? p. 10 Assortativity & Preferences Evolution of Assortativity Generalized Matching Conditional · · · Dissociation Network Formation · 4 Evolution of Behavior p. 17 Traits Conventions—Culture in Society Culture in Individuals Culture in Individuals · · · & Society 5 Economic Applications p. 29 Macroeconomics, Market Selection & Finance Industrial Organization · 6 The Evolutionary Nash Program p. 30 Recontracting & Nash Demand Games TU Matching NTU Matching Bargaining Solutions & · · · Coordination Games 7 Behavioral Dynamics p. 36 Reinforcement Learning Imitation Best Experienced Payoff Dynamics Best & Better Response · · · Continuous Strategy Sets Completely Uncoupled · · 8 General Methodology p. 44 Perturbed Dynamics Further Stability Results Further Convergence Results Distributed · · · control Software and Simulations · 9 Empirics p.
    [Show full text]
  • Cooperative Games: Core and Shapley Value
    COOPERATIVE GAMES: CORE AND SHAPLEY VALUE Roberto Serrano CEMFI Working Paper No. 0709 July 2007 CEMFI Casado del Alisal 5; 28014 Madrid Tel. (34) 914 290 551. Fax (34) 914 291 056 Internet: www.cemfi.es CEMFI Working Paper 0709 July 2007 COOPERATIVE GAMES: CORE AND SHAPLEY VALUE Abstract This article describes the basic elements of the cooperative approach to game theory, one of the two counterparts of the discipline. After the presentation of some basic definitions, the focus will be on the core and the Shapley value, two of the most central solution concepts in cooperative game theory. JEL Codes: C7. Keywords: Game theory; cooperative game; characteristic function; solution concept; core; Shapley value; axiomatics; equivalence principle; non-cooperative implementation. Roberto Serrano Brown University and IMDEA [email protected] COOPERATIVE GAMES: CORE AND SHAPLEY VALUE by Roberto Serrano Department of Economics, Brown University and IMDEA-Social Sciences July 2007 (to appear in Encyclopedia of Complexity and Systems Science, Springer, Berlin) Outline: 1. Introduction. 2. Cooperative Games. 2.1. Representations of Games. The Characteristic Function. 2.2. Assumptions on the Characteristic Function. 2.3. Solution Concepts. 3. The Core. 3.1. Non-Emptiness. 3.2. The Connections with Competitive Equilibrium. 3.3. Axiomatic Characterizations. 3.4. Non-Cooperative Implementation. 3.5. An Application. 4. The Shapley Value. 4.1. Axiomatics. 4.2. The Connections with Competitive Equilibrium. 4.3. Non-Cooperative Implementation. 4.4. An Application. 1 Glosary of keywords: game theory; cooperative game; characteristic function; solution concept; core; Shapley value; axiomatics; equivalence principle; non-cooperative implementation. Summary: This article describes the basic elements of the cooperative approach to game theory, one of the two counterparts of the discipline.
    [Show full text]