EssaysinEconomicTheory: StrategicCommunicationandInformationDesign AndrewKosenko Submittedinpartialfulfillmentofthe requirementsforthedegreeof DoctorofPhilosophy intheGraduateSchoolofArtsandSciences COLUMBIAUNIVERSITY 2018 (cid:13)c 2018 AndrewKosenko Allrightsreserved ABSTRACT EssaysinEconomicTheory: StrategicCommunicationandInformationDesign AndrewKosenko Thisdissertationconsistsoffouressaysineconomictheory. Allofthemfallun- 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 ontheinterplayofinformation,incentivesandbeliefs. 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, andareceiverwhohaspreferencesthatdependontheunknownstate. Inamodel 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 asamodelwherethestateoftheworldisnon-dichotomous,andshowthatanalo- gousresultsemerge. 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 thatanequilibriumalwaysexists,(evenwithoutthesinglecrossingproperty),and characterize the unique equilibrium allocation. With two types of consumers the outcomeisparticularlysimple,consistingofapoolingallocationwhichmaximizes 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 extremelyrobustandParetoefficient. Inthethirdchapterwestudyagameofstrategicinformationdesignbetweena 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 providecomparativestaticswithregardtothemediator’spreferences,thenumber ofmediators,anddifferentinformationalarrangements. In the fourth chapter we study a novel equilibrium refinement - belief-payoff monotonicity. Weintroduceadefinition,arguethatitisreasonablesinceitcaptures an attractive intuition, relate the refinement to others in the literature and study someoftheproperties. Contents ListofFigures iii Acknowledgements vi Dedication ix 1 BayesianPersuasionwithPrivateInformation 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 RelationshiptoExistingLiterature . . . . . . . . . . . . . . . . . . . . 6 1.3 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 AGeneralModel: Non-dichotomousStates. . . . . . . . . . . . . . . . 43 1.5 ConcludingRemarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2 RevisitingRothschild-Stiglitz 62 2.1 TheModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 2.2 Rothschild-StiglitzwithSecretContracts . . . . . . . . . . . . . . . . . 69 2.3 ParetoEfficiencywithUndisclosedContracts . . . . . . . . . . . . . . 73 2.4 DefinitionofMarketEquilibrium . . . . . . . . . . . . . . . . . . . . . 77 2.5 EquilibriumAllocations . . . . . . . . . . . . . . . . . . . . . . . . . . 83 i 2.6 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 2.7 GeneralityoftheResult . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 2.8 Extensions: Non-uniquenessofEquilibrium . . . . . . . . . . . . . . . 90 2.9 ExtensionstoCaseswithManyTypes . . . . . . . . . . . . . . . . . . 92 2.10 PreviousLiterature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 2.11 TheNo-disclosureLimitedInformationPriceEquilibria . . . . . . . . 97 2.12 ConcludingRemarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 2.13 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 3 MediatedPersuasion: FirstSteps 115 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 3.2 Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 3.3 BinaryModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 3.4 ConcludingRemarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 3.5 AuxiliaryResults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 4 ThingsLeftUnsaid: TheBelief-PayoffMonotonicityRefinement 166 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 4.2 Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 4.3 RelationshiptoOtherRefinements . . . . . . . . . . . . . . . . . . . . 174 4.4 ConcludingRemarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 Bibliography 185 ii List of Figures Π Π 1.1 Illustrationwithpoolingon ,andthedeviationto . . . . . . . . . . 29 L H 2.1 Breaking the RS separating equilibrium in the presence of undisclosed contractsathigh-riskodds. . . . . . . . . . . . . . . . . . . . . . . . . . . 68 2.2 Sustaininganequilibriuminthepresenceofacream-skimmingdeviant contractDin z. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 2.3 Pareto-efficientallocations((A∗,C∗),(A(cid:48),C(cid:48)))andtheequilibriumallo- cation (A∗,C∗). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 2.4 Equilibriumwithoutsingle-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(alongthearrow)withoutbeingdisclosedtothedeviantfirm. P denotestheaverageprobabilityofaccidentforthetwohighestrisk −L types,whileV indicatesanindifferencecurvefori-risktype(i = H,M,L). 93 i 2.6 Breaking No-Disclosure-Information Price Equilibrium Pe by a fixed- quantitycontract (α(cid:48),β(cid:48)),where Pe > P(cid:48) > P. . . . . . . . . . . . . . . . . 101 iii 2.7 Nash Equilibrium can be sustained against multiple deviant contracts (A∗B,G) or (A∗B(cid:48),G) offered at different prices as high-risk individu- alsalsochooseG(over A∗B)oras(A∗B(cid:48),G)yieldslossesforthedeviant firm(whileinducingself-selection). . . . . . . . . . . . . . . . . . . . . . . 112 3.1 IllustrationoftheModel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 3.2 EffectofGarblingonBeliefsinaDichotomy. . . . . . . . . . . . . . . . . 125 3.3 AnExample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 3.4 ComparingtheFeasibleSetsofPosteriors. . . . . . . . . . . . . . . . . . . 141 3.5 IncreasingNoiseShrinkstheSetofFeasiblePosteriors. . . . . . . . . . . 142 3.6 TracingtheOuterLimitof F(M,π): FirstBoundary. . . . . . . . . . . . . 144 3.7 TracingtheOuterLimitof F(M,π): SecondBoundary. . . . . . . . . . . . 145 3.8 TracingtheOuterLimitof F(M,π): ThirdBoundary. . . . . . . . . . . . 146 3.9 TracingtheOuterLimitof F(M,π): FourthBoundary. . . . . . . . . . . . 147 3.10 F(M,π): anIllustration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 3.11 KeyFeaturesoftheFeasibleSet. . . . . . . . . . . . . . . . . . . . . . . . . 149 3.12 Blackwell’sOrderImpliesSetInclusionforFeasibleSets. . . . . . . . . . 153 3.13 FurtherIllustrationofSetInclusion. . . . . . . . . . . . . . . . . . . . . . 154 3.14 UnrankedFeasibleSets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 3.15 GoingBeyondtheDichotomy: ThreeSignals. . . . . . . . . . . . . . . . 156 3.16 ASimpleNon-trivialExample. . . . . . . . . . . . . . . . . . . . . . . . . 157 4.1 ICandBPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 4.2 D1andBPMmakethesameselection. . . . . . . . . . . . . . . . . . . . . 177 iv 4.3 NWBRandBPMmakethesameselection. . . . . . . . . . . . . . . . . . . 178 4.4 D1andBPMmakedifferentselections. . . . . . . . . . . . . . . . . . . . . 178 4.5 D1vs. BPM:whichismoreconvincing? . . . . . . . . . . . . . . . . . . . 180 4.6 D1doesnotapply,BPMdoes. . . . . . . . . . . . . . . . . . . . . . . . . . 180 4.7 Abestiaryofrefinementconcepts. . . . . . . . . . . . . . . . . . . . . . . 181 v Acknowledgements It is with a profound sense of gratitude and humility that I write these words. I feelthatmydebttothepeoplewhomadethejourneypossibleisgreaterthanthat ofmostotherstudents. 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 metobecomemybest,supportedmeinsomanyways,farbeyondanyobligation, andbelievedinmeevenwhenIdidn’tbelieveinmyself. IwouldalsoliketothankandnotemyprofounddebttoJosephStiglitz. Work- ingwithJoehasbeenaonce-in-a-lifetimeprivilege. Hehasbeenincrediblygener- 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 hasalwaysbeenawellspringofideascombinedwithaprofoundethicalcompass, withunparalleledpublicspiritandanexemplaryworkethic. vi