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Information Design, Bayesian Persuasion, and Bayes Correlated Equilibrium†

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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 ‡ Information Design, Bayesian Persuasion, and Bayes Correlated Equilibrium† 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 each ρstate θ 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 proba- bility 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.
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