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University of Pennsylvania ScholarlyCommons Publicly Accessible Penn Dissertations Spring 2010 Incomplete Information and Robustness in Strategic Environments Antonio Penta University of Pennsylvania, [email protected] Follow this and additional works at: https://repository.upenn.edu/edissertations Part of the Economic Theory Commons Recommended Citation Penta, Antonio, "Incomplete Information and Robustness in Strategic Environments" (2010). Publicly Accessible Penn Dissertations. 134. https://repository.upenn.edu/edissertations/134 This paper is posted at ScholarlyCommons. https://repository.upenn.edu/edissertations/134 For more information, please contact [email protected]. Incomplete Information and Robustness in Strategic Environments Abstract Game theoretic modeling involves making assumptions on agents' infinite hierarchies of beliefs. These assumptions are understood to be only approximately satisfied in the actual situation. Thus, the significance of game theoretic predictions depend on robustness properties of the solution concepts adopted. Chapter 1 discusses recent results in this research area and their relations with the results obtained in the subsequent chapters. Chapter 2 explores the impact of misspecification of higher order beliefs in static environments, when arbitrary common knowledge assumptions on payoffs are relaxed. (Existing literature focuses on the extreme case in which all such assumptions are relaxed.) Chapter 3 provides a characterization of the strongest predictions, for dynamic games, that are "robust" to possible misspecifications of agents' higher order beliefs, and shows that such characterization depends on modeling assumptions that have hitherto received little attention in the literature (namely, the distinction between knowledge and certainty), raising novel questions of robustness. Chapter 4 develops a methodology to address classical questions of implementation, when agents' beliefs are unknown to the designer and their private information changes over time. The key idea is the identification of a solution concept that allows a tractable analysis of the full implementation problem: Full "robust" implementation requires that, for all models of agents' beliefs, all the perfect Bayesian equilibria of a mechanism induce outcomes consistent with the social choice function (SCF). It is shown that, for a weaker notion of equilibrium and for a general class of games, the set of all such equilibria can be computed by means of a "backwards procedure" that combines the logic of rationalizability and backward induction reasoning. It is further shown that a SCF is (partially) implementable for all models of beliefs if and only if it is ex-post incentive compatible. In environments with single crossing preferences, strict ex-post incentive compatibility and a "contraction property" are sufficiento t guarantee full robust implementation in direct mechanisms. This property limits the interdependence in agents' valuations. Degree Type Dissertation Degree Name Doctor of Philosophy (PhD) Graduate Group Economics First Advisor George J. Mailath Keywords Robustness; Mechanism Design; Robust Implementation; Dynamic Games; Incomplete Information; Rationalizability; Hierarchies of beliefs Subject Categories Economic Theory This dissertation is available at ScholarlyCommons: https://repository.upenn.edu/edissertations/134 INCOMPLETE INFORMATION AND ROBUSTNESS IN STRATEGIC ENVIRONMENTS Antonio Penta A DISSERTATION in Economics Presented to the Faculties of the University of Pennsylvania in Partial Ful…llment of the Requirements for the Degree of Doctor of Philosophy 2010 Supervisor of Dissertation George J. Mailath, Walter H. Annenberg Professor in the Social Sciences Graduate Group Chairperson Dirk Krueger, Professor of Economics Dissertation Committee: George J. Mailath, Walter H. Annenberg Professor in the Social Sciences Andrew Postlewaite, Harry P. Kamen Professor of Economics Qingmin Liu, Assistant Professor of Economics INCOMPLETE INFORMATION AND ROBUSTNESS IN STRATEGIC ENVIRONMENTS COPYRIGHT 2010 Antonio Penta Per Sara, il suo amore, e il suo sorriso. iii ACKNOWLEDGMENTS I want to thank Prof. George Mailath for his dedication, his support, the valuable insights, and above all for having constantly challenged me. As a true mentor, he has set an example of intellectual honesty and professionalism which will accompany me throughout my own career. I also thank Prof. Qingmin Liu, for the careful and always constructive criticisms, and Prof. Andrew Postlewaite, for the sharpness of his thinking. Special thanks go to Prof. Pierpaolo Battigalli, who more than anyone has shaped my view and understanding of game theory. This very dissertation testi…es the extent of my intellectual debt to him. I owe a lot to the intellectual vivacity of Larbi Alaoui, to the inspiring passion of Dave Cass, and to a long list of colleagues and friends who have made these years at Penn a unique life experience. Everything I achieved though would not have been possible without the invaluable and unconditional support of my parents, the loyalty of my sister, and the friendship of Gil, Larbi, Deniz, Dionissi, Francesca, Leonardo and Michela. Of the many things I should thank them for, there is one that is above the others: their patience. Finally, I thank Sara, without whom everything would have had just a di¤erent ‡avor. iv ABSTRACT INCOMPLETE INFORMATION AND ROBUSTNESS IN STRATEGIC ENVIRONMENTS Antonio Penta George J. Mailath Game theoretic modeling involves making assumptions on agents’in…nite hierar- chies of beliefs. These assumptions are understood to be only approximately satis…ed in the actual situation. Thus, the signi…cance of game theoretic predictions depend on robustness properties of the solution concepts adopted. Chapter 1 discusses re- cent results in this research area and their relations with the results obtained in the subsequent chapters. Chapter 2 explores the impact of misspeci…cation of higher or- der beliefs in static environments, when arbitrary common knowledge assumptions on payo¤s are relaxed. (Existing literature focuses on the extreme case in which all such assumptions are relaxed.) Chapter 3 provides a characterization of the strongest predictions, for dynamic games, that are “robust” to possible misspeci…cations of agents’higher order beliefs, and shows that such characterization depends on model- ing assumptions that have hitherto received little attention in the literature (namely, the distinction between knowledge and certainty), raising novel questions of robust- ness. Chapter 4 develops a methodology to address classical questions of implementa- tion, when agents’beliefs are unknown to the designer and their private information changes over time. The key idea is the identi…cation of a solution concept that allows a tractable analysis of the full implementation problem: Full “robust” implementa- tion requires that, for all models of agents’beliefs, all the perfect Bayesian equilibria of a mechanism induce outcomes consistent with the social choice function (SCF). It v is shown that, for a weaker notion of equilibrium and for a general class of games, the set of all such equilibria can be computed by means of a “backwards procedure”that combines the logic of rationalizability and backward induction reasoning. It is further shown that a SCF is (partially) implementable for all models of beliefs if and only if it is ex-post incentive compatible. In environments with single crossing preferences, strict ex-post incentive compatibility and a “contraction property” are su¢ cient to guarantee full robust implementation in direct mechanisms. This property limits the interdependence in agents’valuations. vi Contents Acknowledgements iv 1 Incomplete Information and Robustness in Strategic Environments 1 1.1 Introduction................................ 2 1.2 Complete Information: Rationalizability and Equilibria . 4 1.2.1 (Static) Games with Complete Information. 5 1.2.2 Common Belief in Rationality and Equilibrium . 7 1.3 (Static) Games with Incomplete Information . 9 1.3.1 Harsanyi’sapproach: Bayesian Games and Equilibrium . 10 1.3.2 Non-Equilibrium Approach . 16 1.4 Interim Robustness in Static Games . 24 1.4.1 Relaxing all CK assumptions on payo¤s . 25 1.4.2 Belief-Free Models and Equilibrium . 32 1.4.3 Discussion............................. 33 1.5 DynamicGames.............................. 33 1.5.1 Relaxing all CK-assumptions on payo¤s (II) . 37 1.5.2 Equilibria in Belief-free environments . 47 2 On the Structure of Rationalizability on Arbitrary Spaces of Un- certainty 62 vii 2.1 Introduction................................ 62 2.2 Game Theoretic Framework . 65 2.2.1 Structure of Rationalizability without Richness . 67 2.3 Discussion................................. 74 3 Higher Order Beliefs in Dynamic Environments 76 3.1 Introduction................................ 77 3.2 Relaxing CK-assumptions and Robustness in Dynamic Games . 82 3.2.1 Preliminaries and Examples . 82 3.2.2 Non-technical Presentation of the Approach and Results . 88 3.3 Game Theoretic Framework . 93 3.4 Interim Sequential Rationalizability . 99 3.4.1 Example: Finitely Repeated Prisoner’sDilemma . 102 3.5 Robustness(-es) . 105 3.5.1 Upper Hemicontinuity . 105 3.5.2 Type Space Invariance . 106 3.5.3 Model Invariance . 106 3.6 The structure of in the Universal Model ............. 107 ISR 3.6.1 Sensitivity of Multiplicity to higher order beliefs . 109 3.6.2 Genericity of Uniqueness . 113 3.7 Related Literature and Concluding
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