Selection, Learning, and Nomination: Essays on Supermodular Games, Design, and Political Theory Thesis by Laurent Mathevet In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 2008 (Defended May 15th, 2008) ii c 2008 Laurent Mathevet All Rights Reserved iii For my parents, Denise and Ren´eMathevet iv Acknowledgements Many people have helped me along the way. Christelle has been a source of happiness, and without her uncompromising support and faith in me, I would have been miserable. My parents, to whom this dissertation is dedicated, have shown continual and unconditional support despite the distance. They are a constant source of inspiration. I also wish to thank my advisors, Federico Echenique and Matthew Jackson, for their help and encouragement. Federico has spent numerous hours advising me, meeting with me nearly every week since my second year as a graduate student. He has endeavored to extract the most out of me, discarding unpromising thoughts and results, reading my drafts countless times and demanding (countless!) rewrites. Most importantly, he found the right words in those difficult times of a PhD graduate student. Matt has also been a wonderful advisor, available, demanding, and encouraging. I have benefited greatly from his comments, which are the kind that shapes the core of a research project. Matt has funded me for several years, which allowed me to concentrate on research. I also thank him for his invitation to spend a term at Stanford to further my research. I am indebted to Preston McAfee, whose personality, charisma, and sharpness have made meetings with him among the most fruitful and enjoyable. I have learned a great deal from John Ledyard, whose love for mechanism design is contagious. I also thank him for funding my last year at Caltech, which made a difference during the stressful times on the job market. Kim Border patiently met with me any time I had a technical question and excited me with his passion for theory. To this day, I have yet to find a question to which he has no answer. I am grateful to my fellow graduate students, in particular Laura and David for their help and support. I thank my family whose thoughts are with me; and my friends from California and France for their presence and generosity, and in particular, Martina, Rohini, Bertrand, J´er´emy, Manuel, and Philippe. Finally, I cannot forget where I come from and those who made my admission to Caltech v possible, hence this dissertation. First, I wish to thank Philippe Solal, my undergraduate advisor, for raising me as a theoretical economist. He pushed me as few have and instigated the idea of studying abroad. Michel Bellet and many in the Economics department at the University of Saint-Etienne strongly supported my application process to American universities. I could not have been admitted to Caltech without them, and without Alan Kirman who endorsed my application. Of course, I thank the HSS division at Caltech and the members of the recruiting committee that year (such as Federico!) for giving a chance to an unheard of student, from an unheard of university, from an unheard of city. While unusual, I am ending this long acknowledgement section with a quote by Jean Guitton (1961), which I see as an acknowledgement: “Saint-Etienne has nothing to seduce: no rivers, no sites, no past, no monuments. She is only sumptuous for those who love labor and pain, and the familiarity of simple people who work happily, as if they were under the sun, whereas coal mines even cover the face of the land. But one enjoys oneself more in privation. And the miner who goes back up drinks the light.” Une m`ere en sa vall´ee, p. 47–48, translated by Laurent Mathevet. vi Abstract Games with strategic complementarities (GSC) possess nice properties in terms of learning and structure of the equilibria. Two major concerns in the theory of GSC and mechanism design are addressed. Firstly, complementarities often result in multiple equilibria, which requires a theory of equilibrium selection for GSC to have predictive power. Chapter 2 deals with global games, a selection paradigm for GSC. I provide a new proof of equilibrium uniqueness in a wide class of global games. I show that the joint best-response in these games is a contraction. The uniqueness result then follows as a corollary of the contraction principle. Furthermore, the contraction-mapping approach provides an intuition for why uniqueness arises: Complementarities generate multiple equilibria, but the global-games structure dampens complementarities until one equilibrium survives. Secondly, there is a concern in mechanism design about the assumption of equilibrium play. Chapter 3 examines the problem of designing mechanisms that induce supermodular games, thereby guiding agents to play desired equilibrium strategies via learning. In quasilinear environments, I prove that if a scf can be implemented by a mechanism that generates bounded substitutes — as opposed to strategic complementarities — then this mechanism can be converted into a supermodular mechanism that implements the scf. If the scf also satisfies some efficiency criterion, then it admits a supermodular mechanism that balances budget. Then I provide general sufficient conditions for a scf to be implementable with a supermodular mechanism whose equilibria are contained in the smallest interval among all supermodular mechanisms. I also give conditions for the equilibrium to be unique. Finally, a supermodular revelation principle is provided for general preferences. The final chapter is an independent chapter on political economics. It provides three different processes by which two political parties nominate candidates for a general election: Nominations by party leaders, by a vote of party members, and by a spending competition. It is shown that more extreme outcomes can emerge from spending competition and that non-median outcomes can result via any vii process. Under endogenous party membership, median outcomes ensue when nominations are decided by a vote but not with spending competition. viii Contents Acknowledgements iv Abstract vi Contents viii List of Figures xi 1 Introduction 1 2 Global Games 7 2.1 Introduction . 7 2.2 A Motivating Example . 10 2.3 Game and Assumptions . 12 2.3.1 The Payoff Functions . 13 2.3.2 The Beliefs . 16 2.3.3 Increasing Strategies . 17 2.4 The Main Result: Contraction of the Best-Reply . 18 2.4.1 Private Values Case . 19 2.4.1.1 Main Results . 19 2.4.2 Common Values Case . 22 2.5 Examples . 25 2.5.1 Currency Crises . 25 2.5.2 Diamond’s Search Model . 26 2.6 Beliefs and Payoff Uncertainty . 27 2.6.1 Private Values vs. Harsanyi’s Disturbed Payoffs . 27 2.6.2 Private Values vs. Common Values . 28 ix 2.7 Appendix . 28 2.7.1 On the Real Cutoff Points . 28 2.7.2 Proofs . 29 3 Supermodular Implementation 40 3.1 Introduction . 40 3.2 Motivation and Intuition . 46 3.3 Lattice-Theoretic Definitions and Supermodular Games . 51 3.4 Supermodular Implementation: The Framework . 52 3.5 Supermodular Implementation on Quasilinear Domains . 53 3.5.1 Environment and Definitions . 54 3.5.2 General Result and Implementation with Budget Balance . 55 3.5.3 Optimal and Unique Supermodular Implementation . 59 3.5.3.1 Optimal Implementation . 59 3.5.3.2 Unique Implementation and Full Efficiency . 62 3.5.4 Approximate Supermodular Implementation . 67 3.6 Applications . 69 3.6.1 Principal Multi-Agent Problem . 69 3.6.2 Auctions, Bilateral Trading, and Allocation Schemes . 70 3.7 A Revelation Principle . 71 3.8 Discussion . 73 3.8.1 Dominant Strategy vs. Supermodular Implementation . 73 3.8.2 Learning in Bayesian Games . 75 3.9 Conclusion . 76 3.10 Appendix . 77 3.10.1 A Counterexample to the Revelation Principle . 77 3.10.2 Proofs . 80 4 Nomination Processes and Policy Outcomes 96 4.1 Foreword . 96 4.2 Introduction . 96 4.3 The General Model . 99 4.4 Nominations with a Fixed Party Structure . 100 x 4.4.1 Equilibrium Definitions for the Three Nomination Procedures . 101 4.4.2 Nomination by Party Leaders . 103 4.4.3 Nomination by a Vote of Party Members . 105 4.4.4 Nomination by Spending Competition . 107 4.4.4.1 Sufficient Conditions for Existence Under Party Spending with No Overlap in Parties . 109 4.4.4.2 Sufficient Conditions for Existence Under Party Spending: The General Case . 110 4.5 Endogenous Parties . 113 4.5.1 Endogenous Parties and Nomination by Voting . 114 4.5.2 Endogenous Parties and Nomination by Spending Competition . 115 4.6 Concluding Remarks . 117 4.7 Appendix . 119 4.7.1 Justifying the Equilibrium in a Spending Competition via an All-Pay Auction . 119 4.7.2 Proofs of the Propositions . 123 4.7.3 Nomination by Party Leaders and Strong Equilibria . 128 Bibliography 130 xi List of Figures 3.1 Cycling under Cournot Dynamics . 49 4.1 Preferences Generating an Intra-Party Cycle . 109 1 Chapter 1 Introduction The concept of complementarity, whereby two objects are complements if having more of one increases the marginal value of the other, is an old preoccupation of literary and math- ematical economists. It motivated Hicks and Allen [49] and [50] to perform their classical reconsideration of ordinal demand theory in 1934 and Samuelson [95] asserted in 1974 that “the time [was] ripe for a fresh, modern look at the concept of complementarity.” Since then, a large mathematical toolbox to dealing with this notion has been developed, in par- ticular using lattice theory (Topkis [?]). The concept of complementarity has reached game theory with the class of games with strategic complementarities (GSC) (Bulow et al.