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Essays in Economic Theory: Strategic Communication and Information Design Essays in Economic Theory: Strategic Communication and Information Design Andrew Kosenko Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Graduate School of Arts and Sciences COLUMBIA UNIVERSITY 2018 c 2018 Andrew Kosenko All rights reserved ABSTRACT Essays in Economic Theory: Strategic Communication and Information Design Andrew Kosenko This dissertation consists of four essays in economic theory. All of them fall un- der the umbrella of economics of information; we study various models of game- theoretic interaction between players who are communicating with others, and have (or are able to produce) information of some sort. There is a large emphasis on the interplay of information, incentives and beliefs. In the first chapter we study a model of communication and persuasion be- tween a sender who is privately informed and has state independent preferences, and a receiver who has preferences that depend on the unknown state. In a model with two states of the world, over the interesting range of parameters, the equi- libria can be pooling or separating, but a particular novel refinement forces the pooling to be on the most informative information structure in interesting cases. We also study two extensions - a model with more information structures as well as a model where the state of the world is non-dichotomous, and show that analo- gous results emerge. In the second chapter, which is coauthored with Joseph E. Stiglitz and Jungy- oll Yun, we study the Rothschild-Stiglitz model of competitive insurance markets with endogenous information disclosure by both firms and consumers. We show that an equilibrium always exists, (even without the single crossing property), and characterize the unique equilibrium allocation. With two types of consumers the outcome is particularly simple, consisting of a pooling allocation which maximizes the well-being of the low risk individual (along the zero profit pooling line) plus a supplemental (undisclosed and nonexclusive) contract that brings the high risk individual to full insurance (at his own odds). We also show that this outcome is extremely robust and Pareto efficient. In the third chapter we study a game of strategic information design between a sender, who chooses state-dependent information structures, a mediator who can then garble the signals generated from these structures, and a receiver who takes an action after observing the signal generated by the first two players. Among the results is a novel (and complete, in a special case) characterization of the set of posterior beliefs that are achievable given a fixed garbling. We characterize a simple sufficient condition for the unique equilibrium to be uninformative, and provide comparative statics with regard to the mediator’s preferences, the number of mediators, and different informational arrangements. In the fourth chapter we study a novel equilibrium refinement - belief-payoff monotonicity. We introduce a definition, argue that it is reasonable since it captures an attractive intuition, relate the refinement to others in the literature and study some of the properties. Contents List of Figures iii Acknowledgements vi Dedication ix 1 Bayesian Persuasion with Private Information1 1.1 Introduction................................. 1 1.2 Relationship to Existing Literature .................... 6 1.3 Model..................................... 8 1.4 A General Model: Non-dichotomous States................ 43 1.5 Concluding Remarks............................ 52 2 Revisiting Rothschild-Stiglitz 62 2.1 The Model .................................. 66 2.2 Rothschild-Stiglitz with Secret Contracts................. 69 2.3 Pareto Efficiency with Undisclosed Contracts.............. 73 2.4 Definition of Market Equilibrium..................... 77 2.5 Equilibrium Allocations .......................... 83 i 2.6 Equilibrium ................................. 84 2.7 Generality of the Result........................... 89 2.8 Extensions: Non-uniqueness of Equilibrium............... 90 2.9 Extensions to Cases with Many Types .................. 92 2.10 Previous Literature ............................. 94 2.11 The No-disclosure Limited Information Price Equilibria . 97 2.12 Concluding Remarks............................102 2.13 Appendices .................................104 3 Mediated Persuasion: First Steps 115 3.1 Introduction .................................115 3.2 Environment.................................122 3.3 Binary Model ................................136 3.4 Concluding Remarks............................160 3.5 Auxiliary Results ..............................161 4 Things Left Unsaid: The Belief-Payoff Monotonicity Refinement 166 4.1 Introduction .................................166 4.2 Environment.................................170 4.3 Relationship to Other Refinements ....................174 4.4 Concluding Remarks............................184 Bibliography 185 ii List of Figures 1.1 Illustration with pooling on PL, and the deviation to PH.......... 29 2.1 Breaking the RS separating equilibrium in the presence of undisclosed contracts at high-risk odds. .......................... 68 2.2 Sustaining an equilibrium in the presence of a cream-skimming deviant contract D in z................................... 71 2.3 Pareto-efficient allocations ((A∗, C∗), (A 0, C 0)) and the equilibrium allo- cation (A∗, C∗). ................................. 75 2.4 Equilibrium without single-crossing...................... 91 2.5 Equilibrium (A, B, C) with three types, which cannot be broken by D as individuals of higher-risk type supplement it by additional pooling insurance (along the arrow) without being disclosed to the deviant firm. P−L denotes the average probability of accident for the two highest risk types, while Vi indicates an indifference curve for i-risk type (i = H, M, L). 93 2.6 Breaking No-Disclosure-Information Price Equilibrium Pe by a fixed- quantity contract (a 0, b 0), where Pe > P 0 > P. 101 iii 2.7 Nash Equilibrium can be sustained against multiple deviant contracts (A∗B, G) or (A∗B 0, G) offered at different prices as high-risk individu- als also choose G (over A∗B) or as (A∗B 0, G) yields losses for the deviant firm (while inducing self-selection).......................112 3.1 Illustration of the Model.............................124 3.2 Effect of Garbling on Beliefs in a Dichotomy. 125 3.3 An Example....................................131 3.4 Comparing the Feasible Sets of Posteriors...................141 3.5 Increasing Noise Shrinks the Set of Feasible Posteriors. 142 3.6 Tracing the Outer Limit of F(M, p): First Boundary. 144 3.7 Tracing the Outer Limit of F(M, p): Second Boundary. 145 3.8 Tracing the Outer Limit of F(M, p): Third Boundary. 146 3.9 Tracing the Outer Limit of F(M, p): Fourth Boundary. 147 3.10 F(M, p): an Illustration. ............................148 3.11 Key Features of the Feasible Set.........................149 3.12 Blackwell’s Order Implies Set Inclusion for Feasible Sets. 153 3.13 Further Illustration of Set Inclusion. .....................154 3.14 Unranked Feasible Sets. ............................155 3.15 Going Beyond the Dichotomy: Three Signals. 156 3.16 A Simple Non-trivial Example. ........................157 4.1 IC and BPM ...................................176 4.2 D1 and BPM make the same selection.....................177 iv 4.3 NWBR and BPM make the same selection...................178 4.4 D1 and BPM make different selections.....................178 4.5 D1 vs. BPM: which is more convincing?...................180 4.6 D1 does not apply, BPM does..........................180 4.7 A bestiary of refinement concepts. ......................181 v Acknowledgements It is with a profound sense of gratitude and humility that I write these words. I feel that my debt to the people who made the journey possible is greater than that of most other students. There is one person I want to thank before and above all others - my advisor, Navin Kartik. He has been an exceptional role model even before becoming my advisor (in fact, before I even started the program), and will always remain so. It is indeed rare that such a razor-sharp wit should be combined with deep under- standing, and wide knowledge with a warm personality and wisdom. He pushed me to become my best, supported me in so many ways, far beyond any obligation, and believed in me even when I didn’t believe in myself. I would also like to thank and note my profound debt to Joseph Stiglitz. Work- ing with Joe has been a once-in-a-lifetime privilege. He has been incredibly gener- ous with his time, a great mentor and a true joy to work with. He has also effec- tively functioned as an unofficial advisor and I will forever cherish the experience of discussing economic ideas with him as the sun set on the Hudson River. Joe has always been a wellspring of ideas combined with a profound ethical compass, with unparalleled public spirit and an exemplary work ethic. vi I am also grateful to Yeon-Koo Che who was likewise generous with his time, and uncommonly helpful with research, always offering insightful and construc- tive advice. I am deeply indebted to Navin, Joe and Yeon-Koo. I would also like to thank the final two members of my defense committee - Rajiv Sethi and Allison Carnegie - for their participation, help and constructive suggestions, and Allison for the opportunity to work with her. I would be remiss if I didn’t thank Alessandra Casella and Jungyoll Yun, both of whom gave me the opportunity to work with them, Bogachan
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