Information Design, Bayesian Persuasion, and Bayes Correlated Equilibrium†

American Economic Review: Papers & Proceedings 2016, 106(5): 586–591 http://dx.doi.org/10.1257/aer.p20161046

Information Design and Bayesian Persuasion ‡

By Dirk Bergemann and Stephen Morris*

A set of players have preferences over a set However, in each case, there is a revelation of outcomes. Consider the problem of an “infor- principle argument that allows for the analysis mation designer” who can choose an informa- of all information structures or all mechanisms, tion structure for the players to serve his ends, respectively. For the mechanism design prob- but has no ability to change the mechanism lem, we can restrict attention to direct mecha- or force the players to make particular action nisms where the players’ action sets are equal choices( . A mechanism here describes the set to their possible types. Conversely, for the infor- of players,) their available actions, and a map- mation design problem, we can restrict attention ping from action profiles to outcomes. Contrast to information structures where the players’ type this “information design” problem with the sets are equal to their action sets. In this note, “mechanism design” problem, where a “mech- using this observation, we consider information anism designer” can choose a mechanism for design problems when all information structures the players to serve his ends, but has no ability are available to the designer. to choosing the information structure or force When there are many players, but the infor- the players to make particular action choices( .1 mation designer or “mediator” has no informa- In each case, the problem is sometimes studied) tional advantage( over the players,) this problem with a restricted choice set. In the information reduces to the analysis of communication in design problem, we could restrict the designer games Myerson 1991 . If there is only one player to choosing whether the players are given no or “receiver”( but the) information designer or information or complete information about the “sender”( has) an informational advantage over( environment. In the mechanism design prob- the player,) the problem reduces to the “Bayesian lem, we could restrict the designer to choosing persuasion” problem Kamenica and Gentzkow between a first price and a second price auction. 2011 . Some of our recent( work corresponds to the information) design problem when there are both many players and the information designer ‡ Discussants: Drew Fudenberg, Harvard University; has an informational advantage over the players Laura Veldkamp, New York University; Marina Halac, Bergemann and Morris 2013, 2016 . Columbia University; Michael Woodford, Columbia ( ) University. This note explores this unifying perspec- * Bergemann: Department of Economics, Yale University, tive on information design. In the next section, New Haven, CT 06511 e-mail: [email protected] ; we discuss the simplest example of Bayesian Morris: Department of( Economics, Princeton University,) persuasion, with both an uninformed and an Princeton, NJ 08544 e-mail: [email protected] . We ( ) informed receiver. We provide a couple of novel acknowledge financial support from NSF SES 1459899. We perspectives with this treatment. First, we con- would like to thank our discussant, Drew Fudenberg, and Ben Brooks, Jeff Ely, Emir Kamenica, Laurent Mathevet, sider “omniscient persuasion” in the informed and Tymofiy Mylanov for informative comments, and Aron receiver case, where the sender knows the Tobias and Xinyang Wang for valuable research assistance. receiver’s signal, contrasting this with the more † Go to http://dx.doi.org/10.1257/aer.p20161046 to visit usual assumptions that the sender either cannot the article page for additional materials and author disclo- sure statement s . condition on the receiver’s signal or can only 1 We follow( Taneva) 2015 in our use of the term “infor- do so if he can elicit this information from the mation design.” ( ) receiver. Second, we use a two-step procedure 586 VOL. 106 NO. 5 Information Design, Bayesian Persuasion, and Bayes Correlated Equilibrium 587 to solve the problem, by first characterizing the a stochastic action recommendation from an set of outcomes that could be attained by the informed( mediator.) A decision rule is obedient if sender and then analyzing the sender’s choice of the depositor always has an incentive to follow outcome among those that are attainable.2 These the action recommendation. The depositor will novel perspectives are of independent interest then have an incentive to stay if for the Bayesian persuasion literature. But more importantly, they also clarify how the analysis 1 1 2 1 y 1 2 1 0, extends to the many player case. In the final sec- ( ) ( / ) ( − ρG) − ( / ) ( − ρB) ≥ tion, we report the extension to the many player case, discuss the connection to the incomplete and an incentive to run if information correlated equilibrium literature, and survey our own applied and theoretical work 2 0 (1 2) G y ( 1 2) B ( 1). in this area. While the discussion in this note ( ) ≥ / ρ + / ρ − is informal, a companion piece Bergemann Obedience conditions reflect the fact that the and Morris 2016 discusses these (connections agent may have information that we do not formally. ) need to describe explicitly that( leads him to act differently across the two )states, hence and ρG I. A Bayesian Persuasion Example B may differ in value. Since y 1 , 1 is always ρthe binding constraint and we can< rewrite( ) 1 as A bank is solvent in a good state G and ( ) ( ) insolvent in a bad state B . A depositor can 3 B 1 y y G . either run r or stay s ( with) the bank. Each ( ) ρ ≥ − + ρ state of the( world) is equally( ) likely a priori. A Thus in any obedient decision rule, the probabil- regulator can design the depositor’s information ity of running in the bad state has to exceed the in order to influence his behavior. For the depos- probability of running in the good. In particular, itor, the payoff from staying with the bank is 1 staying with the bank for sure can never be an if the bank is insolvent, and y if solvent, with− equilibrium. The regulator’s preferred outcome, 0 y 1; the payoff for running is normalized with the lowest probability of running across < < to zero in either state. The regulator seeks to states, has G 0 and B 1 y. minimize the probability of the depositor run- Now anyρ obedient= decisionρ = −rule corresponds ning. This is the leading example of Kamenica to optimal behavior under some information and Gentzkow 2011 , with the information structure. For the regulator’s preferred outcome, designer being a( regulator) instead of a prose- it suffices to give the depositor a bad signal cutor and the single player( being a depositor with probability 1 y if the state is bad, and instead) of a judge . We rephrase this example otherwise always give− the depositor a good sig- in( order to tie the analysis) with the many player nal. From the point of view of the motivating generalization discussed in Section II. example, this corresponds to a regulator running stress tests to obfuscate the state of a bank in A. The Uninformed Depositor Case order to prevent a run. By pooling good and bad states in this way, the depositor is made indiffer- We briefly review the analysis of Bayesian ent between running and staying. More gener- persuasion with an uninformed depositor ally, any obedient decision rule can always arise receiver . We describe the receiver’s behav- when the depositor is given a signal equal to his ior( by a )decision rule specifying his behavior action recommendation. given the true state of the world, writing for the probability that he will run in eachρ stateθ B. The Informed Depositor Case B, G ; thus a decision rule is a pair θ ∈ { } ( G , B ) . We can think of a decision rule as being Now suppose that the depositor receives ρ ρ information, independent of the regulator. We

2 assume that the depositor receives a signal which We thus do not appeal to a concavification argument is “correct” with probability q 1 2. Formally, Aumann and Maschler 1995 , which is useful to solve > / this( problem at least in specific) settings Kamenica and the depositor observes a signal g or b , with sig- Gentzkow 2011 . ( nals g and b being observed with conditionally ) 588 AEA PAPERS AND PROCEEDINGS MAY 2016 independent probability q when the true state is rule , , , will induce behavior (ρBb ρGb ρBg ρGg) G or B, respectively. ( B, G ) integrating over signals t g, b . The depositor’s information will act as a con- ρ ρ ∈ { } straint on the ability of the regulator to influence Proposition 1 Omniscient Persuasion ( ): the depositor’s decision, since he now has less The probabilities ( B , G ) form an equilibrium control over the depositor’s information. In this outcome for some informationρ ρ structure if enriched setting, a decision rule specifies the probability that the depositor receives a recom- 6 max q 1 y , 1 y y . mendation to run, as a function of both the state ( ) ρB ≥ { ( + ) } − + ρG and the signal. We will write t for the probability of running in state Gρθ, B if the signal The behavior of the equilibrium set as a func- is t g, b . A decision ruleθ ∈ is{ now} described by tion of the precision q of the private information ∈ { } the quadruple ( Bb , Gb , Bg , Gg ) . is illustrated in Figure 1. As the private informa- ρ ρ ρ ρ tion becomes precise and q approaches one, we Omniscient Persuasion.—The analysis of the converge to the complete information outcome informed depositor case will depend on what the with B 1, G 0. The depositor’s private regulator knows about the depositor’s informa- informationρ = thusρ =limits the regulator’s ability to tion. We first consider the case where the regu- influence the depositor’s decision as the private lator knows the information of the depositor as information tightens obedience constraints. well as the state and, in this sense, is omniscient. But if the regulator had to elicit the deposi- This case has not been the focus of the Bayesian tor’s information, the constraints imposed by the persuasion literature with an informed receiver. private information would become even more However, it is a natural case and, in some cases, severe. We briefly contrast omniscient persua- the most natural case to consider. For example, a sion with this case. regulator may know the information available to the depositor in the form of newspapers, public Private Persuasion.—We now consider a reports, etc. .( And while the regulator may be screening problem where the regulator offers a unable to suppress) that information, he may be recommendation, a probability of running as a able to condition the information he releases on function of the reported type and the true state. the depositor’s initial information. Kolotilin et al. 2015 refer to this informational The obedience constraints now reflect the con- environment as( “private) persuasion.”3 We now ditional belief of the agent about the state given have three sets of constraints. First, each type the realization of the signal. The obedience con- has to truthfully report his type; second, each dition for a depositor who observes a good signal type has to be willing to follow the recommen- and an action recommendation to stay is dation, the obedience constraints; and third, double deviations, by means of misreporting 4 q( 1 Gg ) y (1 q) ( 1 Bg ) 0. and disobeying at the same time, are not prof- ( ) − ρ − − − ρ ≥ itable. With these additional constraints, the set The obedience condition for a depositor who of outcomes that can arise in equilibrium with observes a good signal but an action recommen- private persuasion is strictly smaller than under dation to run is omniscient persuasion as can be seen by comparison between panels A and B in Figure 1. 5 0 q Gg y (1 q) Bg . We can verify that the truthtelling constraints ( ) ≥ ρ − − ρ impose restrictions on how the differences in As long as the private information of the agent is the conditional probabilities across signals sufficiently noisy, or q 1 1 y , the binding can vary across states. This imposes additional constraint is 4 , otherwise≤ / (it +is the) inequality restrictions on the ability of the designer when 5 that determines( ) the conditional probabilities. he seeks to attain either very low and very high The( ) obedience conditions for the bad type are running probabilities in both states. derived in an analogous manner. The obedience conditions are defined type by type, and we com- pute the restrictions on the conditional probabil- 3 Bergemann, Bonatti, and Smolin 2015 analyze this ities averaged across types. Now the decision environment with monetary transfers. ( ) VOL. 106 NO. 5 Information Design, Bayesian Persuasion, and Bayes Correlated Equilibrium 589

Panel A Panel B choosing a best response given that he knows 1 1 his initial signal and the action that he is going to play. We call such distributions “Bayes correlated equilibria,” since they correspond to one

B very permissive version of incomplete infor- B 0.5 0.5 ρ ρ mation( correlated) equilibrium. This set clearly reduces to the set of outcomes attainable by omniscient persuasion in the one player case. We 0 0 0 0.5 1 0 0.5 1 have characterized the set of Bayes correlated equilibria in the context of price discrimination ρG ρG and auctions Bergemann, Brooks, and Morris 2015a, b and( symmetric linear best response Figure 1. Bayesian Persuasion with an Informed games with) normal signals Bergemann and Receiver Panel A and Private Persuasion Panel B ( for( y 9 )10 and q 0.575, 0.7, 0.825( ) Morris 2013; Bergemann, Heumann, and Morris = / = 2015; and others . In these applications, we consider both the cases) in which players have initial information as in omniscient persuasion and the case in which( players have no initial infor) - Public Persuasion.—A yet more restrictive mation as in Bayesian persuasion with an unin- model of persuasion with an informed receiver formed (receiver . is to assume that the sender not only does not The second step) of an information design pro- know the receiver’s private information but can- cedure is to identify the information structure not elicit it, either. Kolotilin et al. 2015 call that would be chosen by an interested designer. this scenario “public persuasion.” Such( public) Our results can be and have been used to make persuasion has been the focus of the recent lit- “information design” observations. One novel erature. In our example, one can show that any question that arises in many player informa- outcome that is attainable by private persuasion tion design is whether the information designer is attainable by public persuasion. Kolotilin wants to convey information to the players in the et al. 2015 identify sufficient conditions for an form of private or public signals. If the informa- equivalence.( ) tion designer would like the players’ actions to be correlated, then he will choose to give them II. The Many Player Case public signals, whereas if the designer wants the players’ actions to be uncorrelated, then Omniscient persuasion is the one player ver- he will choose private signals. If he wants the sion of an approach to information design that we actions to be as uncorrelated as possible, then have been pursuing in recent work Bergemann conditional on the amount of payoff relevant and Morris forthcoming . As in the( analysis of information( conveyed to the players , he would omniscient persuasion presented) here, our work like signals to be as uncorrelated as) possible. suggests a two-step procedure to information We have studied the case where an information design. First, what are the set of outcomes that designer acts in the joint interest of the play- could arise for all extra information with which ers. In the online Appendix of Bergemann and the players could be endowed? Second, which of Morris forthcoming , we analyze a two player, those outcomes would be chosen by some inter- two action,( two state) example that is the gen- ested party and what is the information struc- eralization of the one player example above. In ture giving rise( to those outcomes ? this case, public signals are best for the informa- We have analyzed the first step) with many tion designer under strategic complementarities players by giving a general characterization of e.g., with the usual payoffs for a many player the set of outcomes that could arise in some bank( run while private signals are best under equilibrium Bergemann and Morris forthcom- strategic substitutes.) Our analysis of information ing . We consider( a joint distribution of states, sharing in oligopoly in Bergemann and Morris initial) information signals, and action profiles 2013 followed the same logic. Under oligop- that satisfy the relevant obedience constraints: oly( where) actions are strategic substitutes , that is, the constraint that each player is always firms( would like to have accurate information) 590 AEA PAPERS AND PROCEEDINGS MAY 2016 about demand. But they would also like their on any information possessed by the players actions to be as uncorrelated as possible since jointly; private persuasion corresponds to the each firm would like to produce more( when communication equilibria of Forges 1993 , other firms are producing less . There is a con- where the mediator is able only to condition( ) flict between these two objectives.) We show on information that players can be induced to that the optimal information structure trades off report truthfully; and public persuasion corre- these two competing objectives by having firms sponds to the strategic form correlated equi- observe conditionally independent noisy signals librium of Forges 1993 , where a mediator of demand. In the dynamic two player analysis cannot condition on( players’) information at of Ely 2015 , the information designer wants all. Our work on Bayes correlated equilibria to minimize( the) probability of both players run- generalizes omniscient persuasion by allow- ning, and thus wants uncorrelated actions, and ing many players and generalizes the Bayesian thus chooses signals which are as uncorrelated solution by allowing the information designer as possible. to have information that the players do not col- A striking feature of the one player example lectively possess. is that the set of attainable outcomes shrinks as the receiver becomes more informed. This monotonicity arises because i the sender has References the option of giving the receiver( ) any information that he wants, and thus less information Aumann, Robert J., and Michael B. Maschler. does not limit the feasible outcomes for the 1995. Repeated Games with Incomplete Infor- receiver, while ii more information implies mation. Cambridge, MA: MIT Press. tighter obedience( constraints) for the sender. In Bergemann, Dirk, Alessandro Bonatti, and Alex a general omniscient persuasion analysis, there Smolin. 2015. “Selling Experiments: Menu is a tight generalization of this observation: fix- Pricing of Information.” Unpublished. ing a state space, the set of attainable outcomes Bergemann, Dirk, Benjamin Brooks, and Stephen shrinks in all decision problems if and only if Morris. 2015a. “First Price Auctions with Gen- there is more information, in the sense of the eral Information Structures: Implications for Blackwell’s sufficiency order. This result also Bidding and Revenue.” Unpublished. extends to the many player case: more informa- Bergemann, Dirk, Benjamin Brooks, and Stephen tion reduces the set of Bayes correlated equi- Morris. 2015b. “The Limits of Price Discrim- libria in all games, and vice versa. The subtlety ination.” American Economic Review 105 3 : in this statement is formalizing what is meant 921–57. ( ) by “more information” in the many player Bergemann, Dirk, Tibor Heumann, and Stephen case. In Bergemann and Morris forthcoming , Morris. 2015. “Information and Volatility.” we identify the relevant many player( general)- Journal of Economic Theory 158 part B : ization of Blackwell’s order under which this 427–65. ( ) result is tight. Bergemann, Dirk, and Stephen Morris. 2013. We can make a tight connection between the “Robust Predictions in Games with Incom- three versions of persuasion discussed in this plete Information.” Econometrica 81 4 : note and the incomplete information correlated 1251–1308. ( ) equilibrium literature. As discussed in the intro- Bergemann, Dirk, and Stephen Morris. 2016. duction, persuasion problems have an informed “Information Design and Incomplete Informa- information designer choosing information for tion Correlated Equilibrium.” Unpublished. a single player while the communication in Bergemann, Dirk, and Stephen Morris. Forth- games literature has an information designer coming. “Bayes Correlated Equilibrium and with no informational advantage choosing the Comparison of Information Structures in information for many players. The latter sce- Games.” Theoretical Economics. nario corresponds to the classic incomplete Ely, Jeff. 2015. “Beeps.” Unpublished. information correlated equilibrium literature. Forges, Françoise. 1993. “Five Legitimate Defini- Omniscient persuasion corresponds to the tions of Correlated Equilibrium in Games with Bayesian solution of Forges 1993 : the medi- Incomplete Information.” Theory and Decision ator is able to condition his( recommendation) 35 3 : 277–310. ( ) VOL. 106 NO. 5 Information Design, Bayesian Persuasion, and Bayes Correlated Equilibrium 591

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