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Essays on Bargaining and Repeated Games MASSACHUSETTS Instlfte of TECHNOLOGY by JUN 10 2011 Alexander G Essays on Bargaining and Repeated Games MASSACHUSETTS INSTlfTE OF TECHNOLOGY by JUN 10 2011 Alexander G. Wolitzky LIBRARIES A.B. Economics and Mathematics, Harvard University, 2007 SUBMITTED TO THE DEPARTMENT OF ECONOMICS IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN ECONOMICS ARCHIVES AT THE MASSACHUSETTS INSTITUTE OF TECHNOLOGY JUNE 2011 ©2011 Alexander G. Wolitzky. All rights reserved. The author hereby grants to MIT permission to reproduce and to distribute publicly paper and electronic copies of this thesis document in whole or in part in any medium now known or hereafter created. Signature of Author: Department of Economics May 16, 2011 Certified by: Glenn Ellison Gregory K. Palm Professor of Economics, Thesis Supervisor Daron Acemoglu Elizabeth and James Killian Professor of Economics, Thesis Supervisor f .. uhamet Yildiz P> sorf conomics, Thesis Supervisor Accepted by: Esther Duflo Abdul Latif Jameel Professor of Poverty Alleviation and Development Economics Chair, Department Committee on Graduate Studies Essays on Bargaining and Repeated Games by Alexander G. Wolitzky Submitted to the Department of Economics on May 13, 2011 in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Economics ABSTRACT The thesis consists of four essays on bargaining and repeated games. The first essay studies whether allowing players to sign binding contracts governing future play leads to reputation effects in repeated games with long-run players. Given any prior over behavioral types, a modified prior is constructed with the same total weight on behavioral types and a larger support under which almost all efficient, feasible, and individually rational payoffs are attainable in perfect Bayesian equilibrium. Thus, whether reputation effects emerge in repeated games with contracts depends on details of the prior distribution over behavioral types other than its support. The second essay studies reputational bargaining under the assumption of first-order knowledge of rationality. The share of the surplus that a player can guarantee herself is determined, as is the bargaining posture that she must announce in order to guarantee herself this much. It is shown that this maxmin share of the surplus is large relative to the player's initial reputation, and that the corresponding bargaining posture simply demands this share plus compensation for any delay in reaching agreement. The third essay studies the maximum level of cooperation that can be sustained in sequential equilibrium in repeated games with network monitoring. The foundational result is that the maximum level of cooperation can be sustained in grim trigger strategies. Comparative statics on the maximum level of cooperation are shown to be highly tractable. For the case of fixed monitoring networks, a new notion of network centrality is introduced, which characterizes which players have greater capacities for cooperation and which networks can support more cooperation. The fourth essay studies the price-setting problem of a monopoly that in each time period has the option of failing to deliver its good after receiving payment. Optimal equilibrium pricing and profits are characterized. For durable goods, a lower bound on optimal profit for any discount factor is provided. The bound converges to the optimal static monopoly profit as the discount factor converges to one, in contrast to the Coase conjecture. Thesis Supervisor: Glenn Ellison Thesis Supervisor: Muhamet Yildiz Title: Gregory K. Palm Professor of Economics Title: Professor of Economics Thesis Supervisor: Daron Acemoglu Title: Elizabeth and James Killian Professor of Economics Essays on Bargaining and Repeated Games by Alexander G. Wolitzky Submitted to the Department of Economics on May 13, 2011 in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Economics ACKNOWLEDGEMENTS I would like to begin by thanking my advisors, Glenn Ellison, Daron Acemoglu, and Muhamet Yildiz, for their encouragement, support, and wisdom. I have been very lucky to have had such a dedicated and diverse group of advisors, and I do not look forward to the prospect of writing papers without them looking over my shoulder. I have also been fortunate to receive helpful comments on the essays comprising this thesis from many other people at MIT, including Abhijit Banerjee, Alessandro Bonatti, Gabriel Carroll, Bengt Holmstr6m, Robert Gibbons, Anton Kolotilin, Mihai Manea, Parag Pathak, Michael Powell, Juuso Toikka, and Ivin Werning. For me, the MIT economics department has more than lived up to its reputation as an incredibly stimulating and supportive place to be a graduate student. Finally, I would like to thank my friends and family for their love and support over the past four years, especially, of course, my parents, and Sandy. Contents 1 Indeterminacy of Reputation Effects in Repeated Games with Contracts 1.1 Introduction .. ....... ........ ...... ..... 1.2 Model . 1.3 Example ... 1.4 Indeterminacy of Reputation Effects 1.5 Conclusion .. ............. 2 Reputational Bargaining Under Knowledge of Rationality 2.1 Introduction.. ............... 2.2 Related Literature . ............... 2.3 Model and Key Definitions . ............. 2.3.1 M odel ..................... 2.3.2 Defining the Maxmin Payoff and Posture .. 2.4 Characterization of the Maxmin Payoff and Posture 2.4.1 M ain Result .................. 2.4.2 Offsetting Beliefs and Strategies....... 2.4.3 Maxmin Strategies ............ .. 2.4.4 Proof of Theorem 2 .............. 2.5 Extensions . ............. 2.5.1 Constant Postures .............. 2.5.2 General Convex Bargaining Sets ....... 2.5.3 Heterogeneous Discounting ........... 2.5.4 Rationalizability . ........... 2.6 Two-Sided Commitment . .......... 2.7 Discrete-Time Bargaining with Frequent Offers ... 2.8 Conclusion 2.9 Appendix A: Omitted Proofs for Sections 2.3 and 2.4 . 2.10 Appendix B: Omitted Proofs for Sections 2.5 and 2.7 . 3 Cooperation with Network Monitoring 95 3.1 Introduction................ .... ..... ... .. 95 3.2 Related Literature .. ..... .... ... .... .... .. 98 3.3 Model............. ........ ........ ... 100 3.4 Characterization of Maximum Cooperation ...... ..... .. 103 3.5 Equal M onitoring ... ..... .... ..... .... .... .. 107 3.5.1 Comparative Statics Under Equal Monitoring . .... .. 107 3.5.2 The Effect of Group Size on Global Public Good Provision 109 3.5.3 The Effect of Uncertain Monitoring on Global Public Good Provision 112 3.6 Fixed Monitoring Networks........ .... ..... .. .. 113 3.6.1 Centrality and Maximum Cooperation... ..... .. 114 3.6.2 Comparing Networks. ... ... ..... ..... .. 120 3.6.3 Network Stability .... ....... ....... ..... 121 3.7 Conclusion........ .. .. .... ..... .... ... .. 123 3.8 Appendix A: Omitted Examples .... ..... ...... .... 125 3.8.1 Example Al: Counterexample to Theorem 5 with Non-Sepa rable Utility125 3.8.2 Example A2: Counterexample to Theorem 5 with Unobser ved Moni- toring Network ... ......... ......... ... ...... 126 3.8.3 Example A3: Centrality is not Necessary to Order Maximu m Cooper- ations .. .. .. .. .. .. .. ...... ...... ...... .. 130 3.9 Appendix B: Omitted Proofs, . ....... .... .. .. .. 133 4 Dynamic Monopoly with Relational Incentives 146 4.1 Introduction............ ... .... .... .... .... .. 146 4.2 Relation to the Literature . ... ... ... ... ... ... ... ... 149 4.3 Model .............. .................... ........ 151 4.4 Non-Durable Goods Monopoly ... ... .... ... ... ... ... .. 154 4.5 Durable Goods Monopoly and Rationing......... ..... .. 156 4.5.1 Prelim inaries .... ... .... .... .... .. ... .. 156 4.5.2 Existence of Best Full-Delivery PBE in the Model with Rationing 160 4.5.3 Nonoptimality of Rationing in the Model with Rationing.... .. 165 4.5.4 Equivalence of Best Full-Delivery PBE in the Model with and without Rationing............ ................... .. 166 4.6 Properties of Best Full-Delivery Equilibria................... 167 4.6.1 High Profits and Super-Monopoly Pricing............. .. 167 4.6.2 Declining Prices............... ....... ........ 172 4.7 An Extension: Exogenous Chance of Nondelivery........... .. 176 4.8 Conclusion.................. .............. .. 179 4.9 Appendix A: Omitted Proofs ............ .............. 180 4.10 Appendix B: Non-Full-Delivery Equilibria........... .. 192 1 Indeterminacy of Reputation Effects in Repeated Games with Contracts 1.1 Introduction Does game theory make strong predictions about the outcomes of long-run relationships? It has been known since the seminal papers of Kreps et al (1982), Kreps and Wilson (1982), and Milgrom and Roberts (1982) that reputation effects have important consequences for equilibrium selection in many dynamic games. Fudenberg and Levine (1989) famously showed that a patient long-run player facing a series of short-run opponents receives at least her Stackelberg payoff in any Nash equilibrium, if her "Stackelberg type" has positive prior probability, and similar results hold in two-player repeated games in the limit where one player becomes infinitely more patient than the other.1 However, reputation effects are elusive in two-player games with comparably patient players: 2 indeed, it is not obvious what outcome one would expect reputation effects to select in such games. For this reason, the reputation result of Abreu and Pearce (2007, henceforth AP) is striking: AP show that, in two-player repeated games with common discounting in which players may offer each other binding commitments
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