Information Design: a Unified Perspective†

Home , Drew Fudenberg, Information design

Journal of Economic Literature 2019, 57(1), 44–95 https://doi.org/10.1257/jel.20181489

Information Design: A Unified Perspective†

Dirk Bergemann and Stephen Morris*

Given a game with uncertain payoffs, information design analyzes the extent to which the provision of information alone can influence the behavior of the players. Information design has a literal interpretation, under which there is a real information designer who can commit to the choice of the best information structure (from her perspective) for a set of participants in a game. We emphasize a metaphorical interpretation, under which the information design problem is used by the analyst to characterize play in the game under many different information structures. We provide an introduction to the basic issues and insights of a rapidly growing literature in information design. We show how the literal and metaphorical interpretations of information design unify a large body of existing work, including that on communication in games (Myerson 1991), Bayesian persuasion (Kamenica and Gentzkow 2011), and some of our own recent work on robust predictions in games of incomplete information. ( JEL C70, D82, D83)

1. Introduction studies how the information designer, through the choice of the information provided, can layers’ payoffs in a game depend on their influence the individually optimal behavior Pactions and also on the realization of a pay- of the players to achieve her objective. She off-relevant state. An “information designer” can achieve this objective even though she can commit how to provide information about has no ability to change outcomes or force the the states to the players. “Information design” lecture at Harvard University, and the 2016 fall Finance * Bergemann: Department of Economics, Yale Univer- Theory group conference at Princeton. Some of the sity. Morris: Department of Economics, Princeton Univer- material in this paper was previewed in Bergemann and sity. We acknowledge financial support from NSF Grants Morris (2016b). We have benefited from discussions and SES 0851200 and 1459899. We are grateful for produc- correspondence about the material in this paper with Ben tive suggestions by the Editor, Steven Durlauf, and five Brooks, Laura Doval, Jeff Ely, Francoise Forges, Drew anonymous referees. This material has been presented in Fudenberg, Olivier Gossner, Tibor Heumann, Atsushi lectures at the 2015 Istanbul meetings of the Society of Kajii, Emir Kamenica, Rohit Lamba, Laurent Mathevet, Economic Design, the 2015 Delhi School of Economics Roger Myerson, Tymofiy Mylanov, Alessandro Pavan, Eran Winter School, the 2016 AEA meetings, the North Amer- Shmaya, Jonathan Weinstein, and Juan Pablo Xandri; and ican Summer Meeting of the Econometric Society, the from valuable research assistance from Ian Ball, Leon Gerzensee Summer Symposium in Economic Theory, Musolff, Denis Shishkin, Áron Tóbiás and Xinyang Wang. the Becker Friedman Institute conference on Frontiers † Go to https://doi.org/10.1257/jel.20181489 to visit the and Economic Theory and Computer Science, the Harris article page and view author disclosure statement(s).

44 Bergemann and Morris: Information Design: A Unified Perspective 45

players to choose particular actions that deter- the information designer has an informa- mine outcomes.1 tional advantage over the players. This case The past decade has seen a rapidly growing has been the focus of some our own work body of literature in information design. An (Bergemann and Morris 2013b, 2016a), influential paper by Kamenica and Gentzkow where we show that the set of outcomes that (2011) phrased the optimal design of infor- can arise in this setting corresponds to a ver- mation as a “Bayesian persuasion” problem sion of incomplete information equilibrium between a sender and single receiver. A (Bayes-correlated equilibrium, or BCE) that large body of work fits this rubric, includ- allows outcomes to be conditioned on states ing important contributions of Brocas and that the players do not know. Carrillo (2007) and Rayo and Segal (2010). A second purpose of the paper is to high- The economic applications of information light a distinction between literal informa- design have been investigated in areas as far tion design and metaphorical information apart as grade disclosure and matching mar- design. The information design problem has kets (Ostrovsky and Schwarz 2010), voter a literal interpretation (given above): there mobilization (Alonso and Câmara 2016), really is an information designer (or media- traffic routing (Das, Kamenica, and Mirka tor, or sender) who can commit to provide 2017), rating systems (Duffie, Dworczak, extra information to players to serve her own and Zhu 2017), and transparency regula- interests. While the commitment assumption (Asquith, Covert, and Pathak 2013) tion may be problematic in many settings, it in financial markets, price discrimination provides a useful benchmark. But the infor- (Bergemann, Brooks, and Morris 2015), and mation design formulation might also be a stress tests in banking regulation (Inostroza metaphor that the analyst uses as a tool. For and Pavan 2017). example, we might be interested in finding an One purpose of the paper is to provide an upper bound (across information structures) overview of information design that unifies on the aggregate variance of output in a given this recent work with a number of litera- economy with idiosyncratic and common tures sometimes treated as distinct. If we shocks to agents’ productivity (Bergemann, assume that there are many players, but the Heumann, and Morris 2015). We can under- information designer (or “mediator”) has no stand this as an information design problem, informational advantage over the players, where the information designer is interested this problem reduces to the analysis of com- in choosing an information structure to max- munication in games (Myerson 1991, section imize aggregate variance in output. But in 6.3) and, more generally, the literature on this case, we do not have in mind that there correlated equilibrium in incomplete infor- is an actual information designer maximiz- mation games (Forges 1993). If there is only ing aggregate variance. We will discuss this one player (or “receiver”) but the informa- application, and other applications where tion designer (or “sender”) has an informa- information design is metaphorical, below. tional advantage over the player, the problem This survey reviews the pure information reduces to the “Bayesian persuasion” prob- design problem where a designer can commit lem of Kamenica and Gentzkow (2011). to a certain information structure for the play- Information design concerns the general ers but has no control over outcomes. This case where there are both many players and problem is a special case of the more general mechanism design problem where a mech- 1 We follow Taneva (2015) in our use of the term “infor- anism designer can control outcomes but mation design” in this context. may also be able to manipulate information 46 Journal of Economic Literature, Vol. LVII (March 2019) in the course of doing so.2 We study the case one-player case. But we will also describe where all information structures are available a general partial order on information to the designer. It is thus possible to appeal structures—generalizing the Blackwell order to the revelation principle from the general for the one-player case—which characterizes mechanism design problem, and without the right definition of “more information” in loss of generality restrict attention to infor- this context (Bergemann and Morris 2016a). mation structures where the signals that the Third, we can ask whether the information information designer sends to a player can designer prefers to give the information to be identified with action recommendations. players in a public or in a private message. This revelation principle/mechanism design Of course, this last question only arises once approach to information design thus contrasts we have multiple players. Public information with work where there is no commitment is optimal if the information designer wants to the information structure or attention is perfect correlation between players’ actions; restricted to a parameterized class of infor- otherwise private information will be optimal. mation structures. While the information designer may have We use a family of two-player, two-action, intrinsic preferences over whether players’ and two-state examples to survey the litera- actions are correlated (or not), the designer ture, and to provide some graphical illustra- may care about correlation for purely instru- tions. We start with the leading example of mental reasons: if there are strategic com- Bayesian persuasion (with a single player/ plementarities between the players’ actions, receiver with no prior information) from the she may want to correlate players’ actions to work of Kamenica and Gentzkow (2011). We relax the obedience constraints on her abil- can use extensions of this example—with ity to attain specific outcomes. The converse many players and prior information—to holds for strategic substitutability. We will illustrate many of the key ideas in the survey. illustrate the case when there are only instru- Three key substantive general insights are mental preferences over correlation. illustrated in these examples. The examples also illustrate a methodolog- First, it is often optimal for the informa- ical point. The information design problem tion designer to selectively obfuscate infor- can be solved in two steps. First, we can iden- mation. This insight is familiar from the case tify the set of outcomes that could be induced of one player without prior information. by the information designer. Second, we can Second, the information designer has less identify which of these outcomes would be ability to manipulate outcomes in his favor if preferred by the information designer. This, players have more prior information: if the too, parallels the mechanism design litera- players are endowed with their own informa- ture: we can first identify which outcomes tion, the designer has less influence over the are implementable and then identify the one information structure that they end up with. most preferred by the designer. As noted This insight can already be illustrated in the above, in the information design problem, the set of implementable outcomes corresponds to the set of Bayes-correlated equi- 2 Myerson (1982, 1991) describes the problem of “com- libria. This approach reduces the problem to munication in games” where the designer cannot control outcomes but can elicit information from players and pass a linear program. it to other players. Thus, what we are defining as the “infor- The information designer is assumed to mation design” problem can be viewed as Myerson’s “com- be able to commit to an information struc- munication in games” problem with the important new feature that the designer may have access to information ture that maps payoff-relevant states of the that is not available to the players. world and prior information of the agents Bergemann and Morris: Information Design: A Unified Perspective 47

(types) into possibly stochastic signals to the information. There may be multiple Bayes– players.3 As the mapping essentially recom- Nash equilibria of the resulting game. In our mends actions to the players, we refer to it treatment of the information design prob- as a decision rule. We initially focus on what lem, we have been implicitly assuming that we sometimes call the omniscient case: here, the designer can pick which equilibrium is the information designer faces no constraints played. Under this maintained assumption, on her ability to condition the signals on the we can appeal to the revelation principle payoff-relevant states of the world and all the and focus attention on information struc- players’ prior information (i.e., their types). tures where the signal space is set equal to But we also consider information design con- the action space, and the signals have the strained by private information, where the interpretation that they are action recom- prior information of the players is not accessi- mendations. In the single-player case, this ble to the information designer, even though maintained equilibrium selection assump- she can condition on the payoff-relevant tion is without loss of generality. But just states. There are two cases to consider here: as the revelation principle breaks down in an information designer may be able to con- mechanism design if the designer does not dition on the reported realizations of the get to pick the best equilibrium (as in Maskin players’ signals even if she does not observe 1999), it similarly breaks down for informa- them (information design with elicitation) tion design.4 We follow Mathevet, Perego, or she may be unable to do so (information and Taneva (2017) in formally describing a design without elicitation). If the informa- notion of maxmin information design, where tion designer cannot condition on the payoff an information designer gets to pick an infor- state at all, and has to rely entirely on the pri- mation structure but the selected equilib- vate information of the players, then these rium is the worst one for the designer. We three scenarios (omniscient, private informa- note how some existing work can be seen tion with elicitation, and private information as an application of maxmin information without elicitation) correspond to versions of design, in particular, an extensive literature incomplete information correlated equilib- on “robustness to incomplete information” rium: in the terminology of Forges (1993), (Kajii and Morris 1997). the Bayesian solution, communication equi- Other assumptions underlying the revela- librium, and strategic form correlated equi- tion principle—and maintained throughout librium, respectively. this paper—are that (i) all information struc- Once the information designer has picked tures are feasible, (ii) there is zero (marginal) the information structure, the players decide cost of using information, (iii) there is a sin- how to play the resulting game of incomplete gle information designer, and (iv) the setting is static. Of course, there are many (static) settings where the impact of different infor- 3 Whether or not the information designer will observe mation structures has been studied, without the payoff relevant state is irrelevant—what is important is whether she can condition the signals she sends on the allowing all information structures. Two clas- realization of the state and the players’ prior signals. For sic examples would be information sharing in example, a prosecutor might never know whether the oligopoly (a literature beginning with Novshek defendant is guilty or innocent, but can nevertheless set up an investigation process that would provide different and Sonnenschein 1982) and the revenue evidence depending on the actual guilt or innocence of comparison across different auction formats the subject and the information of the judge. We thank an anonymous referee who stressed the distinction between conditioning on a state, and actually knowing the realiza- 4 This point has been highlighted by Carroll (2016) and tion of the state. Mathevet, Perego, and Taneva (2017). 48 Journal of Economic Literature, Vol. LVII (March 2019) in auction theory (Milgrom and Weber 1982). two applications emphasize the relevance of In the former, there is a restriction to normally the metaphorical interpretation of informa- distributed signals, and in the latter there is tion design. In section 6, we describe what the restriction to affiliated signals. happens when players’ prior information is Optimal information design in dynamic not known by the information designer; this settings has been studied recently in Ely, discussion allows us to locate the information Frankel, and Kamenica (2015); Passadore design problem within mechanism design and Xandri (2014); Doval and Ely (2016); and within a larger literature on incomplete Ely (2017); Ely and Szydlowski (2017); Ball information correlated equilibrium reviewed (2018); and Makris and Renou (2018). A new by Forges (1993). In section 7, we discuss aspect to information design that appears in the role of equilibrium selection. these dynamic settings is that information Given the synthetic treatment of the lit- can be used as an incentive device to reward erature, there is much terminology that has behavior over time. Horner and Skrzypacz been introduced and used in different con- (2016) survey work on information design in texts (including by us in prior work), and dynamic settings. Gentzkow and Kamenica which is at times inconsistent or redundant. (2014) consider the case of costly informa- To give one example, what we are calling an tion and Gentzkow and Kamenica (2017) “information designer” has in previous work allow for multiple information designers. been called a sender, mediator, principal, This paper provides a conceptual syn- or mechanism designer. We are attempt- thesized guide to the literature; we discuss ing throughout to use a unified and consis- applications when they are relevant for this tent language, but compromising at times purpose, but make no attempt to provide a between the use of a consistent terminology comprehensive survey of the many applica- and precedents set by earlier work. tions of information design. A recent survey by Kamenica (forthcoming) reviews the lit- 2. The Information Design Problem erature of Bayesian persuasion and discusses some leading economic applications. We begin by describing the general setting We describe the basic information design and notation that we maintain throughout problem in section 2. We illustrate the the paper. We will fix a finite set of players main notions and ideas with an investment and a finite set of payoff states of the world. example in section 3. We discuss key ideas There are I players, 1, 2, , I, and we write i … from information design in the investment for a typical player. We write for the set of Θ example in section 4. Here, we discuss pri- payoff states of the world and for a typical θ vate versus public signals, intrinsic versus element of set . Θ instrumental preferences over correlation, A “basic game” G consists of (1) for each the two-step procedure for solving infor- player i, a finite set of actionsA i, and an mation design problems, ordering infor- (ex post) utility function mation, and the use of concavification in information design (instead of pure linear u : A , i × Θ → ℝ programming methods). Section 5 describes, in more detail, two applications of infor- where we write A A A, and = 1 × ⋯ × I mation design with a microeconomic and a a , , a for a typical element of A; = ( 1 … I) macroeconomic perspective, respectively: and (2) a full support prior that ψ ∈ Δ(Θ) limits of price discrimination, and the link is shared by all players and the information I between information and volatility. These designer. Thus G Ai, ui i 1 , , . We = (( ) = Θ ψ) Bergemann and Morris: Information Design: A Unified Perspective 49 define the ex post objective of the informa- (ii) the true state is realized; θ tion designer by: (iii) each agent’s type ti is privately realized; v : A . × Θ → ℝ (iv) the players receive extra messages An “information structure” S consists of (i) according to the information designer’s for each player i, a finite set of types t T; rule; i ∈ i and (ii) a type distribution : T , π Θ → Δ( ) where we write T T1 TI. Thus (v) the players pick their actions based on I = × ⋯ × S Ti i 1 , . their prior information and the mes- = (( ) = π) Together, the “payoff environment” or sages provided by the information “basic game” G and the “belief environment” designer; or “information structure” S define a standard “incomplete information game” (G, S). (vi) payoffs are realized. While we use different notation, this division of an incomplete information game into the We emphasize that the information “basic game” and the “information structure” designer commits to a decision rule before the is a common one in the literature, see, for realization of the state and type profile. This example, Gossner (2000). structure in the timing sets the information We are interested in the problem of an design problem apart from informed prin- information designer who has the ability to cipal, cheap talk, or signaling environments commit to provide the players with addi- where the informed party chooses a message tional information, in order to induce them only after the state has been revealed. to make particular action choices. In this In general, the information designer section, we will consider the leading case could follow any rule for generating mes- where the designer can condition on the sages. However, a “revelation principle” state and on all the players’ types—their argument implies that it is without loss of prior information—if they have any. We will generality to assume that the information sometimes refer to this setting as omniscient designer sends only “action recommen- information design. In section 6, we will con- dations” that are obeyed. The argument is sider the case where prior information of the that any message will give rise to an action players is truly private to them, and hence in equilibrium and we might as well label the information designer cannot condition messages by the actions to which they give on their prior information unless she is able rise. We discuss the revelation principle in to induce them to reveal it.5 more detail below. Given this restriction, If viewed as an extensive form game the information designer is choosing among between the information designer and the decision rules players, the timing is as follows: (1) : T (A) . (i) the information designer picks and σ × Θ → Δ commits to a rule for providing the The information designer can condition players with extra messages; the recommended action profile on the true state and the type vector t T. We θ ∈ Θ ∈ stress that the designer does not need to 5 For this reason, we refer to the types of a player as his prior information, and only in section 6 does this informa- know the true realization of the state or the tion become truly private. type profile—it is sufficient that the decision 50 Journal of Economic Literature, Vol. LVII (March 2019) rule can condition on these. For example, in thus the above inequality is written from a medical test, the information designer, the an interim perspective (conditioning on doctor, may not know the true condition of ti and ai). We can state the above inequal- the patient, but can choose a diagnostic test ity explicitly in terms of the interim beliefs that reveals the condition of the patient with that agent i holds, given his type ti and the desired precision and accuracy. his action recommendation ai, thus using The decision rule encodes the information Bayes’s rule: that the players receive about the realized state of the world, the types and actions of u i( (a i, a i), ) a∑,t, − θ the other players. The conditional depen- i i θ ( (a i, a i)|(t i, t i), ) ( (t i, t i)| ) ( ) dence of the recommended action a on state ______σ − − θ π − θ ψ θ × a , a |t, t , t, t | of the world and type profilet represents a i ,t i ( i i ( i i) ″) i i ″ ( ″) θ ∑ ″ ″ ,θ′′ σ ( −′′ ) −′′ θ π(( −′′ ) θ ) ψ θ the information conveyed to the players. u i a i′ , a i, The key restriction on the decision rule is ≥ (( − ) θ) a∑i,t i, a notion of obedience that we now define. θ ( (a i, a i)|(t i, t i), ) ( (t i, t i)| ) ( ) Obedience is the requirement that the infor- ______σ − − θ π − θ ψ θ . × a , a |t, t , t, t | a i ,t i ( i i ( i i) ″) i i ″ ( ″) mation privately communicated to player ∑ ″ ″ ,θ′′ σ ( −′′ ) −′′ θ π(( −′′ ) θ ) ψ θ i in the form of an action recommendation a according to is such that each player i We observe that the belief of agent i updates i σ would want to follow his recommended independently of whether he is following the action ai rather than choose any other avail- recommendation ai or deviating from it to able action a . a . Moreover, the denominator in Bayes’s rule ′i ′i sums up over all possible profiles a i A i, ′′− ∈ − DEFINITION 1 (Obedience): Decision rule t i T i, ″ , and hence is constant ′′− ∈ − θ ∈ Θ : T (A) is obedient for (G, S) if, across all possible realizations of a i A i, σ × Θ → Δ − ∈ − for each i, ti Ti and ai Ai , we have t i T i, . Hence, we can multiply ∈ ∈ − ∈ − θ ∈ Θ through to obtain the earlier inequality (2), (2) provided that the denominator is strictly positive. ui( ( a i, a i), ) ( ( a i, a i)|( t i, t i), ) ( ( t i, t i)| ) ( ) a∑,t , − θ σ − − θ π − θ ψ θ Bergemann and Morris (2016a) define a i i θ Bayes-correlated equilibrium (BCE) to be u i ( a i′ , a i) , ( a i, a i) |( t i, t i) , ( t i, t i) | ( ), ≥ ∑ ( − θ)σ( − − θ)π( − θ)ψ θ any decision rule satisfying obedience. An a i,t i, σ θ important aspect of this solution concept is for all a A. that the decision rule enters the obedience ′i ∈ i σ constraints, as stated above in (2), in a linear For notational compactness, we only list manner as a probability. Thus, an obedient the elements, but not the sets, over which decision rule can be computed as the solu- we sum under the summation operator, tion to a linear program. thus, e.g., rather than a A . The obedi- ∑ a ∑ ∈ ence inequality requires that each player i, PROPOSITION 1 (Revelation Principle): after receiving his recommendation ai, finds An omniscient information designer can that no other action a could yield him a attain decision rule if and only if it is a ′i σ strictly higher utility. We emphasize that BCE, i.e., if it satisfies obedience. each player, when computing his expected utility, is indeed using the information con- By “can attain decision rule” we mean tained in the action recommendation a i, and that there exists a (perhaps indirect) Bergemann and Morris: Information Design: A Unified Perspective 51

communication rule that gives rise to this The obedience condition (2) thus col- decision rule in Bayes–Nash equilibrium.6 lapses to the best response property (3) in We refer to the resulting (ex ante) joint dis- the absence of payoff uncertainty. Aumann tribution of payoff state and action profile (1987) argued that correlated equilibrium θ a as the outcome of the information design, captured the implications of common knowl- thus integrating out the prior information t: edge of rationality in a complete information game (under the common prior assumption). (a | t, ) (t | ) ( ). An alternative interpretation is that the set of t∑T σ θ π θ ψ θ correlated equilibria is the set of outcomes ∈ attainable by an information designer in the In this (and later) propositions, we omit for- absence of payoff uncertainty. We discuss mal statements and proofs that correspond to these interpretational issues and the litera- revelation principle arguments. Bergemann ture on incomplete information correlated and Morris (2016a) give a formal statement7 equilibrium more broadly in section 6.4. and proof of this proposition as theorem 1. The proof of proposition 1 in Bergemann The argument is a straightforward adapta- and Morris (2016a), like the proof of Aumann tion of standard characterizations of com- (1987), is a “revelation principle” argument, plete and incomplete information correlated establishing that it is without loss of general- equilibrium. ity to focus on a set of signals that equals the If there is no payoff uncertainty—the set set of actions to be taken by the agents—so is a singleton—then the notion of BCE that there is “direct communication”—and Θ exactly coincides with the complete informa- to recommend actions in such way that they tion correlated equilibrium as introduced in will be obeyed—so that “incentive compati- the seminal contributions of Aumann (1974, bility” gives rise to “obedience” conditions. In 1987). In the absence of payoff uncertainty, the case of complete information, Myerson we can simply suppress the dependence of (1991, section 6.2) describes this as the “rev- the payoff function on the state of the world elation principle for strategic form games.” . Thus, the decision rule does not vary Note that while the expression “revelation θ σ with , nor is there any private information principle” is sometimes limited to the case θ t about the state of the world . A decision where agents are sending messages rather θ rule is then simply a joint distribution over than receiving them (e.g., Fudenberg and σ actions, or (A). Now, a distribution Tirole 1991 and Mas-Collel, Whinston, and σ ∈ Δ (A) is defined to be a correlated equi- Green 1995), we follow Myerson in using σ ∈ Δ librium if for each i and a A, we have: the broader meaning throughout the paper. i ∈ i In the basic information design described in (3) ui ai, a i ai, a i this section, the only incentive constraints ( − ) σ( − ) a ∑i A i − ∈ − are obedience conditions, but we discuss the extension to the case where the infor- ui a i , a i ai, a i , a i Ai. ≥ ( ′ − ) σ( − ) ∀ ′ ∈ mation designer must elicit players’ private a ∑i A i − ∈ − information in section 6, where truth-telling incentive constraints also arise. We postpone 6 We do not discuss information design under solution a discussion of how to place information concepts other than Bayes–Nash equilibrium in this paper. design in the broader context of the mecha- Mathevet, Perego, and Taneva (2017) study information nism design literature until then. design under bounded level rationalizability and Inostroza and Pavan (2017) under full rationalizability. Proposition 1 characterizes the set of out- 7 A formal statement also appears in section 7. comes that an information designer could 52 Journal of Economic Literature, Vol. LVII (March 2019) attain, i.e., a feasible set. To complete the 3.1 Single Player without Prior Information description of the basic information design problem, we need an objective. Given the We begin the analysis when the firm has information designer’s ex post utility va, , no prior information about the state (beyond ( θ) her ex ante utility from the decision rule the uniform prior). Together with the above σ for a given game of incomplete information assumptions about the payoff matrix, the (G, S) is: firm would therefore choose to not invest if it had no additional information. We will assume that an information (4) V( ) v( a, ) ( a | t, ) ( t | ) ( ). σ = ∑a,t, θ σ θ π θ ψ θ designer (the “government”) is interested θ in maximizing the probability of investment The (omniscient) information design prob- independent of the state, or lem is then to pick a BCE to maximize σ V . When there is a single player with no (σ) 1 v(invest, ) v(not invest, ) 0, prior information, the information design = θ > θ = problem reduces to the benchmark Bayesian B, G. persuasion problem described by Kamenica θ = and Gentzkow (2011). In this case, the single player is called the “receiver” and the infor- This example is (modulo some changes in mation designer is called the “sender.” labelling) the leading example in Kamenica and Gentzkow (2011). We will describe this example first, but then use variations to illus- 3. An Investment Example trate more general points. The decision rule We now apply this framework to an invest- is now simply: ment game and discuss the main themes of information design through the lens of this : (A), example. σ Θ → Δ We first consider the following benchmark setting. There is a bad state (B) and a good where we can omit the type space T due state ( G). The two states are equally likely: to the absence of prior information. In this binary decision environment, the decision G B _ 1 . rule specifies the probability of invest- ψ( ) = ψ( ) = 2 σ(θ) ment, denoted by p , conditional on the true There is one player (the “firm”). The firm state B, G . Thus,θ a decision rule is a θ ∈ { } can decide to invest or not invest. The payoff pair ( pB , pG ) of investment probabilities. We from not investing is normalized to 0. The can think of a decision rule as a (stochastic) payoff to investing is 1 in the bad state and action recommendation from the govern- − x in the good state, with 0 x 1. These ment. If the recommendations are obeyed, < < payoffs, ua, , are summarized in the fol- the outcome—the ex ante distribution over ( θ) lowing matrix: states and actions—is given by:

u(a, ) bad state B good state G (a ) ( ) bad state B good state G θ σ | θ ψ θ . invest p p . (5) invest 1 x _1 B _1 G − 2 2 not invest (1 p ) (1 p ) not invest 0 0 _1 B _1 G 2 − 2 − Bergemann and Morris: Information Design: A Unified Perspective 53

If the firm receives a recommendation to the BCE, it suffices to give the firm binarya invest, it will update its beliefs about the state information structure S to implement any by Bayes’s rule. The firm’s interim expected BCE decision rule in the binary action envi- utility from following the recommendation ronment. For the outcome that maximizes represents the left-hand side of the inequal- the probability of investment, it suffices to ity below. If the firm were to disobey the rec- generate a no-investment recommendation ommendation and chose not to invest, then with probability 1 x if the state is bad, and − its payoff would be zero. This gives rise to otherwise give the firm an investment rec- the following obedience constraint: ommendation. The resulting outcome—the ex ante distribution over states and actions— _1 _1 is given by: pB pG (6) ______2 ( 1) ______2 x 0. _1 p _1 p − + _1 p _1 p ≥ 2 B + 2 G 2 B + 2 G (a ) ( ) bad state B good state G σ | θ ψ θ invest x . As we discussed following definition 1, we (8) _1 _1 2 2 can simplify the inequality by multiplying not invest (1 x) 0 _1 through with the conditioning probability 2 − _ 1 p _1 p , and thus write the obedience 2 B + 2 G condition equivalently in terms of the interim Thus, a government trying to encourage probabilities: investment will obfuscate the states of the p world in order to maximize investment. By (7) _ 1 p 1 _1 p x 0 p _ B . pooling realizations of the bad and good 2 B(− ) + 2 G ≥ ⇔ G ≥ x states in the recommendation to invest, the There is an analogous obedience constraint firm is made exactly indifferent between corresponding to the recommendation not to investing or not when recommended to invest, namely: invest. The bad state is completely iso- lated in the recommendation not to invest. 0 _ 1 1 p 1 _1 1 p x. Finally, we observe that under complete ≥ 2( − B)(− ) + 2( − G) information the firm would always invest Because the firm would not invest absent in the good state and never invest in the information from the designer—by our bad state. We thus have described three maintained assumption that x 1—the different information structures—zero < binding obedience constraint will be the one information, partial information, and com- corresponding to investment, i.e., inequality plete information—that support the three (7). We see that the highest probability of vertices of the above investment triangle. investment corresponds to the decision rule Thus, the set of all investment probabilities with p 1 and p x. that satisfy the obedience constraints can be G = B = We illustrate the set of BCE decision rules described by a set of linear inequalities that for the case where x 55 100 in figure 1. jointly form a polyhedron of implementable = / Any decision rule (pB , pG ) in the blue shaded outcomes. area can arise as some BCE. We observe that 3.2 Single Player with Prior Information the feasible set of BCE does not depend on the government’s preference va, . We remain with the investment example ( θ) Now every BCE decision rule corresponds where there is still only one firm, but now to optimal behavior under some information the firm has some prior information about structure S. By the revelation principle for the true state that it receives independently 54 Journal of Economic Literature, Vol. LVII (March 2019)

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0 0 0.2 0.4 0.6 0.8 1

Figure 1. Investment Probability with Uninformed Player: x 55 100 = / of the government.8 In particular, the firm Here, signals refer to the prior information has a type (or receives a signal) that is “cor- that firms are endowed with. Conditional on rect” with probability q 1 2. Formally, the type of the firm, the analysis of the obedi- > / the firm observes its typet b, g with ence constraints reduces immediately to the ∈ { } probability q conditional on the true state analysis of the previous section, but where being B or G, respectively: the firm has an updated belief, q or 1 q, − depending on the type. We nonetheless ana- (t ) bad state B good state G lyze this problem because we want to trace π | θ the ex ante implications of a player’s prior bad signal b q 1 q . − information for information design. A deci- good signal g 1 q q sion rule now specifies the probability of − σ investment pt conditional on the true state B, G andθ the type t b, g . Thus a θ ∈ { } ∈ { } decision rule is now a quadruple: 8 Some detailed calculations for this example appear in (9) p p , p , p , p . the appendix. = ( Bb Bg Gb Gg) Bergemann and Morris: Information Design: A Unified Perspective 55

We can solve the problem—conditional on consequence, prior information of the player state and type—as before. For example, the only affects the set of BCE if the prior infor- obedience constraint for the recommenda- mation by itself already generates a differen- tion to invest after receiving a good type g tial response by the player. Second, the slope now becomes: of the boundary is constant across different levels of precision q. This occurs because (10) 1 q p 1 q p x 0. the rate at which the optimal decision of the ( − ) Bg(− ) + Gg ≥ player can be reversed by additional informa- However, we are interested in what we can tion of the designer (and hence indifference say about the joint distribution of states and is attained) is constant across q by Bayes’s law. actions ex ante, integrating out the types, say 3.3 Many Players without Prior Information p qp 1 q p . G = Gg + ( − ) Gb We can now generalize the analysis to two One can show that there is a lower bound on firms and return to the assumption that the investment in the good state given by: firms have no prior information.9 We assume, for now, that the government wants to max- 1 q imize the sum over each individual firm’s (11) p q _− , G ≥ − x probability of investment. If there is no strategic interaction between firms, the previous which approaches 1 as q approaches 1. As analysis can be carried out firm by firm and before, the bound for pG depends on pB will thus be unchanged. and the lowest bound is obtained by taking But we now perturb the problem to make p 0 in that expression. it strategic, assuming that each firm gets an B = The set of BCE is illustrated in figure 2. extra payoff if both invest, where may ε ε More prior information shrinks the set be positive or negative. If is positive, we ε of BCE, since the obedience constraints have a game of strategic complementarities; become tighter. Once q reaches 1 , the firm if 0, we have a game of strategic substi- ε < knows the state and the information designer tutes. We can write firm 1’s state dependent has no ability to influence the outcome. The payoffs for the game as follows (and symmet- firm is simply pursuing the complete infor- rically for firm 2): mation optimal decision, which is to invest in the good state, p 1, and not to invest in G = the bad state, p 0. B = firm 2 The set of BCE across different q has two B invest not invest notable features. First, we notice that the θ = , set of BCE happens to be constant across (12) firm 1 invest 1 1 some information structures near precision − + ε − q 0.5, for example at q 0.5 and q 0.6. not invest 0 0 = = = With low precision in the signals, such as q 0.6, the firm would pursue the same = action for either type realization, absent any additional recommendation by the government. Thus, the weak prior information 9 Other two player, two action, and two state examples by the agent does not constrain the gov- appear in Bergemann and Morris (2013a, 2016a) and ernment in its recommendation policy. In Taneva (2015). 56 Journal of Economic Literature, Vol. LVII (March 2019)

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Figure 2. Investment Probability with Informed Player: x 55 100 = /

firm 2 invest in state G, B ; but we will θ ∈ { } now write r for the probability that both G invest not invest θ θ = invest. Thus a decision rule is a vector . firm 1 invest x x + ε (pB , rB , pG , rG ) . A decision rule can now be represented in a table as not invest 0 0

B invest not invest We can focus on symmetric decision rules, θ = invest r p r , given the symmetry of the basic game, for (13) B B − B any symmetric objective of the information not invest pB rB 1 rB 2pB designer. To see why, note that if we found − + − an asymmetric maximizing decision rule, the decision rule changing the names of the G invest not invest firms would also be optimal, and so would θ = invest r p r . the (symmetric) average of the two decision G G − G rules. Therefore, we will continue to write not invest p r 1 r 2p p for the probability that each firm will G − G + G − G θ Bergemann and Morris: Information Design: A Unified Perspective 57

To ensure that all probabilities are and we can summarize the optimal decision nonnegative, we require that for all rule in the following table: B, G : θ ∈ { } B invest not invest max 0, 2 p 1 r p . θ = { θ − } ≤ θ ≤ θ x , invest + ε 0 ____ 1 − ε The firm has an incentive to invest when told 1 x 2 not invest 0 ____− − ε to do so if 1 − ε

_1 _1 _1 (14) pB pG x (rB rG) 0, G invest not invest −2 + 2 + 2 + ε ≥ θ = . and an incentive to not invest when told to invest 1 0 not do so if not invest 0 0

_1 1 p _1 1 p x −2( − B) + 2( − G) This decision rule entails a public signal: there is common certainty among the _1 p r p r 0. + 2( B − B + G − G) ε ≤ firms that they always observe the same signal. Since x 1, (14) is always the binding If 0, it remains optimal to have both < ε < constraint and—for sufficiently close to firms always invest when the state is good |ε | 0—we can rewrite it by the same reasoning (p r 1). But now we want to mini- G = G = as in section 3.1 as mize rB given ( pB , pG , rG ) . To reduce cases, let us assume that x 1 2 and restrict p > / (15) p _B r r _ . attention to x 1 2. In this case, it G ≥ x − ( B + G) εx |ε | ≤ − / will be optimal to set r 0. Substituting B = Now maximizing the sum of the probabili- these expressions into (15), we have ties of each firm investing corresponds to maximizing p , (or p p , but we will have pB B B + G 1 _ _ ε . p 1 always) subject to (15). For fixed ≥ x − x G = x 1 and 0, it is clearly optimal to < |ε | ≈ have firms always invest when the state is Thus we will now have p x , and we B = + ε good (so p 1 and r 1) and it is not can summarize the optimal decision rule in G = G = possible to get both firms to always invest the following table: when the signal is bad. If 0, (15) implies that it is optimal to ε > B invest not invest choose r as large as possible given p . Thus θ = B B , we will set r p . Substituting these vari- invest 0 x B = B + ε ables into expression (15), we have not invest x 1 2x 2 + ε − − ε p 1 _B p 1 _ , ≥ x − ( B + ) xε G invest not invest and so it is optimal to set θ = invest 1 0 . x p r _ + ε , not invest 0 0 B = B = 1 − ε 58 Journal of Economic Literature, Vol. LVII (March 2019)

Under this decision rule, firms told to invest 4. Issues in Information Design Illustrated know neither whether the state is good or by the Examples bad, nor if the other firm is investing or not. Thus, signals are private to each firm. Given Let us draw out the significance of these that, in the bad state, each firm will invest examples. One basic point that has been with (roughly) probability x and will not with extensively highlighted (e.g., by Kamenica (roughly) probability 1 x, the above infor- and Gentzkow 2011 and the following − mation structure minimizes the uncondi- Bayesian persuasion literature) is that when tional correlation of the signals across firms there is a conflict between the designer (or equivalently minimizes the negative cor- and the player(s), it will, in general, be relation conditional on the bad state.) optimal for the designer to obfuscate, that Strategic complementarities increase the is, hide information from the player(s) in private return from investing if the other order to induce him to make choices that player invests as well. Below, we display are in the designer’s interests. And con- the set of investment probabilities that can ditional on obfuscation being optimal, it be attained by the government while vary- may not be optimal to hide all information, ing the size of the strategic effect . As the but will, in general, be optimal to partially ε strategic effect increases, the boundaries reveal information. This issue already arises ε of the investment probabilities attainable by in the case of one player with no prior the government shift outwards as illustrated information. in figure 3. As the strategic complementar- In this section, we draw out a number of ity increases (or strategic substitutability additional insights about information design decreases), the government can support a that emerged from the examples. First, we larger probability of investment in both states. observe that information will be supplied to The intermediate case of 0 reduces to players publicly or privately depending on ε = the case of a single player, and hence reduces whether the designer would like to induce to the area depicted earlier in figure 1. positive or negative correlation in players’ actions; we also discuss designers’ possible 3.4 Many Players with Prior Information intrinsic or instrumental reasons for want- We analyzed the case of two players and ing positive or negative correlation. Second, prior information in Bergemann and Morris we note that in the case of one player with (2016a). Here, we illustrate this case without prior information, more prior information formally describing it. As in the single-player constrains the ability of the designer to case, an increase in players’ prior informa- control outcomes; we discuss the many- tion limits the ability of the designer to influ- player generalization of this observation. ence the players’ choices. Consequently, Third, we discuss the elegant “concavifi- the impact of prior information on the set cation” approach as an alternative to the of attainable investment probabilities with linear programming representation used many players is similar to the one-player case. above to characterize and provide insights In figure 4 we illustrate the set of attainable into the information designer’s problem. investment probabilities under increasing We also discuss an extension of the con- prior information with strategic complemen- cavification approach to the many-player tarities. The strategic complementarities give case but note limitations of the concavi- rise to a kink in the set of attainable probabil- fication approach, both in the one-player ities (pB , pG ) , unlike in the single-player case case and (even more) in the many-player depicted earlier in figure 2. case. Bergemann and Morris: Information Design: A Unified Perspective 59

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Figure 3. Investment Probability with Negative or Positive Strategic Term ε

4.1 Public versus Private Signals In our analysis of the case of two players and Instrumental versus Intrinsic without prior information, we made the Motivation for Preferences over assumption that the information designer Correlation wanted to maximize the sum of the probabilities that each player invests. Thus, we An information designer will often have assumed that the information designer preferences over whether players’ actions did not care whether players’ actions are correlated with each other or not. The were correlated or not. Put differently, we case of many players without prior informa- assumed that the information designer had tion illustrates the point that if the designer no intrinsic preferences over correlation. wants players’ actions to be correlated, it will Yet, despite this assumption, we observed be optimal to give them public signals, and if that the information designer wants—for he wants players’ actions to be uncorrelated instrumental reasons—to induce correlated he will give them private signals. However, behavior when players’ actions are strate- there are different reasons why the designer gic complements, and to induce negative might want to induce positive or negative correlation when there were strategic sub- correlation in actions. stitutes among the players. This in turn gen- 60 Journal of Economic Literature, Vol. LVII (March 2019)

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0.2 0.5 = 0.6 = 0.7 = 0.8 = 0.9 = 0 0 0.2 0.4 0.6 0.8 1

Figure 4. Investment Probability with Two Players with Prior Information, with Strategic Term 3 10 ε = /

erated the insight that the designer would the action of a second player who has no like to generate public signals when there strategic concerns, i.e., does not care about are strategic complementarities and private the first player’s action. In this case, the signals when there are strategic substitutes. information designer does not have intrin- The reason for this instrumental objective sic preferences over correlation (because is that under strategic complements, the she only cares about the first player’s designer can slacken obedience constraints action), but has an instrumental incentive by correlating play, with the opposite mech- to correlate actions because she can use anism holding under strategic substitutes. information design to influence the action We now describe three environments of the second player and correlate behav- where there will be only instrumental con- ior in order to slacken the first player’s obe- cerns about correlation. First, Mathevet, dience constraint. In the formulation of Perego, and Taneva (2017) consider an Mathevet, Perego, and Taneva (2017), the environment with one-sided strategic com- information designer is a manager, the first plementarities. The designer cares about player is a worker, and the second player is the action of a first player who cares about a supervisor. Bergemann and Morris: Information Design: A Unified Perspective 61

Second, Bergemann and Morris (2016a) player to take an action that is optimal given consider an environment with two-sided their shared preferences. In the many-player strategic complementarities but where a case, however, the players themselves may nonstrategic payoff externality removes not act in their joint interest for the usual intrinsic preferences over correlation. To (noncooperative strategic) reasons. In this illustrate this, suppose that we take our many case—as in the above example—a benev- player with no prior information example olent information designer might want to from section 3.3, but now suppose that—in obfuscate information. addition to the existing payoffs—each firm For a last case with only instrumen- would like the other firm to invest, and thus tal concerns over correlation, Bergemann there are spillovers. In the following payoff and Morris (2013b) considered quantity table, we are assuming that each firm gets an (Cournot) competition in a market, where extra payoff of z 0 if the other firm invests: the information designer wants to maximize > the sum of the firms’ payoffs, i.e., the indus- 10 B invest not invest try profits. A continuum of firms choose θ = output where there is uncertainty about the invest 1 z 1 , − + ε + − intercept of the demand curve, i.e., the level not invest z 0 of demand. In this case, the information designer would like the firms’ total output to be correlated with the level of demand, but G invest not invest total profits do not depend on the correla- θ = tion of firms’ output conditional on the level invest x z x . + ε + of aggregate output. However, firms would not invest z 0 like their actions to be negatively correlated (because the game is one of strategic substitutes); but they would also like output to be Observe that this change in payoffs has no correlated with the state. The information impact on the firms’ best responses: neither designer can induce players to make total firm can influence whether the other firm output choices that are closer to the opti- invests. But now suppose that the govern- mal level, but allow them to negatively cor- ment is interested in maximizing the sum of relate their output. In the optimal outcome the firms’ payoffs. Consider the case that z is (for some parameters), firms observe con- very large. As z becomes larger and larger, ditionally independent private signals about the government’s objective will approach the state of demand, trading off these two maximizing the sum of the probabilities objectives.11 that each firm invests. In this sense, the Having considered the case where the government’s instrumental preference for information designer cares about correlation correlation is micro-founded in the benevo- for instrumental but not intrinsic reasons, we lent government’s desire to make each firm can also consider the opposite case where the invest in the interests of the other firm. This information designer cares about correlation example illustrates a distinctive point about strategic information design. Recall that 10 This corresponds to a large literature on information in the one-player case where the designer sharing in oligopoly following Novshek and Sonnenschein and the player have common interests, it (1982). 11 In this setting, the information designer would like to is always optimal for the designer to fully induce firms to lower output on average, but cannot do so. reveal all information in order to allow the The designer can only influence correlation. 62 Journal of Economic Literature, Vol. LVII (March 2019) for intrinsic but not instrumental reasons. players have binary actions and the infor- We can illustrate this case with the example mation designer has an intrinsic motive for of section 3.3, also. Suppose that the payoffs correlation, but there is no strategic interac- remain the same, but now the government tion—and thus no instrumental motive for would like to maximize the probability that caring about correlation. With supermodular at least one firm invests, so that the govern- payoffs, public signals are optimal, whereas ment has intrinsic preferences over correla- with submodular payoffs, private signals tion. But in this case—under our maintained are optimal and it is optimal to minimize assumption that x 1 2—it is possible to correlation.12 > / ensure that one firm always invests. Consider 4.2 Tightening Obedience Constraints and the following decision rule: BCE Outcomes

B invest not invest There is never any reason for an informa- θ = tion and/or mechanism designer to provide 1 , invest _ 0 2 players with more information than they will not invest _1 0 use in making their choices. Giving more 2 information will impose more incentive constraints on players’ choices, and thus reduce G invest not invest the ability of an information designer to θ = attain outcomes that are desirable for him. In . invest 1 0 dynamic mechanism design, giving players not invest 0 0 information about others’ past reports will tighten truth-telling constraints. Myerson (1986) emphasizes that a similar observation If were equal to 0, this decision rule would ε is true in dynamic problems of communica- be obedient, with all constraints holding tion in games; in these games the extra infor- strictly: a firm told to not invest would have a mation imposes more obedience constraints strict incentive to obey, since it would know as well. Recall that in our language, commu- that the state was bad; a firm told to invest nication in games corresponds to informa- would have a strict incentive to obey, since tion design when the information designer its expected payoff will be _ 2 x _1 0. 3( − 2) > has no information of her own. Because the obedience constraints hold Our examples have illustrated this general strictly, this decision rule will continue to be observation: giving players more informa- obedient, for positive or negative , as long as tion will impose more obedience constraints ε | | is sufficiently small. Note that the govern- and thus reduce the set of (BCE) outcomes ε ment’s objective of maximizing the probabil- that can occur. However, the examples illus- ity that at least one firm invests necessitates trate a more subtle point that is the focus of private signals. Bergemann and Morris (2016a): it is not only In a dynamic setting, Ely (2017) shows true that sending additional signals reduces how an information designer with intrin- the set of outcomes that can occur; it is sic preferences for negative correlation will optimally use private signals to induce it (he also shows that this is consistent with players’ 12 In a recent survey on the algorithmic aspects of infor- strategic objectives). Arieli and Babichenko mation design, Dughmi (2017) emphasizes how the structure of the payoff environment impacts the algorithms to (2016) and Meyer (2017) provide a general compute the optimal information structure and their com- analysis of optimal information design when putational complexity. Bergemann and Morris: Information Design: A Unified Perspective 63 also possible to construct a partial order on incentive constrained than another if the set arbitrary information structures that exactly of BCE outcomes under the former experi- characterizes the notion of “more informa- ment is smaller (reflecting the tighter obedi- tive,” and one that corresponds exactly to ence constraints) in every decision problem adding more obedience constraints. (or one-player basic game). This was illustrated in our one-player Taken together, there is now an elegant set example with prior information. In that of connections between Blackwell’s theorem example, the set of implementable BCE out- and the information design problem. One comes shrunk in size as the accuracy q of the can show that there is a three-way equiva- prior information increased (as illustrated in lence between (i) the “more informative” figure 2). As q increases, we are intuitively ordering; (ii) the “sufficiency” ordering; and giving the player more information. The (iii) the “incentive-constrained” ordering. additional information is given to the player Blackwell’s theorem shows an equivalence not in the form of more signals, but rather between (i) and (ii). Thus, if an experiment is more precise signals. “more informative” in the sense of Blackwell, We will now informally describe how this then—in any decision problem—the set of observation can be generalized in many BCE outcomes for a given experiment is directions. In the one-player case, an infor- smaller under the more informative experi- mation structure reduces to an experiment ment. There is naturally also a converse. If in the sense of Blackwell (1951, 1953). He an experiment is not more informative than defines the “more informative” ordering in another, then one can find a decision prob- terms of a feasibility ordering. An experiment lem and an outcome that is a BCE for the is said to be more informative than another first experiment but not for the second. experiment if the set of outcomes (joint dis- This equivalence result holds in the one- tributions over actions and states) that can be player case for general games, i.e., deci- induced by decision rules mapping signals sion problems with finitely many states and into actions is larger—in any decision prob- actions. This result is the one-player special lem—under the first experiment.13 case of the main result (for many-player Blackwell (1951, 1953) offered an alter- information structures) from Bergemann native, entirely statistical ordering on and Morris (2016a). experiments, without reference to actions The definition of the incentive ordering, as or payoffs: one experiment is sufficient for well as the definition of the feasibility order- another if the latter can be attained by adding, generalizes naturally to the many-player ing noise to the former. case. Bergemann and Morris (2016a) offer a We have described an incentive ordering new statistical ordering, termed individual on experiments: one experiment is more sufficiency, for many players, and show that is equivalent to the incentive ordering in the many-player case. Individual sufficiency is 13 Blackwell (1951, 1953) defines “more informative” in terms of “risk vectors” rather than joint distributions defined as follows. Fix two information struc- between states and action. These two feasibility conditions tures. A combined information structure is are equivalent. Blackwell defines as risk vector the vector one where players observe a pair of signals, of expected payoffs that can be sustained by a decision rule measurable with respect to the information structure corresponding to the two information struc- alone. The resulting payoff vector is, however, computed tures, with the marginal on signal profiles of conditional (on the vector) of the true state. It then follows each information structure corresponding easily that a larger set of risk vectors is sustained if and only if a larger set of joint distribution of actions and states is to the original information structures. Thus, sustained. there are many combinations of any two 64 Journal of Economic Literature, Vol. LVII (March 2019) information structures corresponding to dif- to the “more valuable” ordering: see, for ferent ways of correlating signals across the example, Hirshleifer (1971); Neyman (1991); two information structures. One information Gossner (2000); and Bassan et al. (2003). structure is now individually sufficient for The above discussion provides a novel per- another if there is a combined information spective. Intuitively, there are two effects of structure such that each player’s signal in the giving players more information in a strategic former information structure is a sufficient setting. First, it allows players to condition statistic for his beliefs about the state of the on more informative signals, and thus—in world and others’ signals in the latter infor- the absence of incentive constraints—attain mation structure. A subtle feature of this more outcomes. Second, more information ordering is that one information structure can reduce the set of attainable outcomes by being individually sufficient for another nei- imposing more incentive constraints on play- ther implies nor is implied by the property ers’ behavior. The value of information in that players’ joint information in the former strategic situations is ambiguous in general case is sufficient (in the statistical sense) for because both effects are at work. Following their joint information in the latter case. We Lehrer, Rosenberg, and Shmaya (2010), should add that in the special case of a single we can abstract from the second (incen- player individual sufficiency and sufficiency tive) effect by focusing on common-interest naturally coincide. The ordering of indi- games. Here, more information in the sense vidual sufficiency has a number of natural of individual sufficiency translates into more properties. Two information structures are attainable outcomes. But looking at BCE individually sufficient for each other if and abstracts from the first (feasibility) effect by only if they correspond to the same beliefs allowing the information designer to sup- and higher-order beliefs about states, and ply any information to the players. Now, differ only in the redundancies of the type more information in the sense of individ- identified in Mertens and Zamir (1985). One ual sufficiency translates into less attainable information structure is individually suffi- outcomes. cient for another only if we can get from the 4.3 BCE Outcomes and Information Design latter to the former by providing additional without Concavification information and removing redundancies. In the one-player setting, there is an alter- We have described a “two-step” approach native but equivalent ordering to the fea- to solving information design problems. sibility ordering. It is phrased in terms of First, provide a linear algebraic characteri- optimality and appears more frequently in zation of implementable outcomes, meaning the economics, rather than statistic, litera- the set of joint distributions over actions ture. For example, Laffont (1989) defines and states that can be induced by some one experiment to be “more valuable” than information structure that the information another if it leads to (weakly) higher maximal designer might choose to give the play- expected utility for every decision problem. ers. The set of implementable outcomes is In a single-player problem, a larger set of exactly the set of BCE. Second, we select feasible joint distributions clearly implies a among the BCE the one that is optimal for larger maximal expected utility. Less obvious the information designer. This second step perhaps, the converse also holds, as the rank- implicitly identifies the optimal information ing has to hold across all decision problems. structure. The first problem is solved by find- A common observation is that in strategic ing the set of outcomes that satisfy a set of situations, there is no many-player analogue linear (obedience) constraints. The second Bergemann and Morris: Information Design: A Unified Perspective 65 problem corresponds to maximizing a lin- analysis. Mathevet, Perego, and Taneva ear objective subject to linear constraints. (2017) describe this generalization of con- Both steps of this problem are well behaved. cavification for the many-player case. There is a separate reason why we might Concavification and its many-player pursue this two-step procedure: for many analogue are important for two reasons. questions of interest, it is critical to first First, they offer structural insights into the understand the set of BCE outcomes. The information design problem. Second, they next two subsections describe two contexts provide a method for solving information where the structure of the BCE outcomes is design problems. As a solution method, the the focus of the analysis. concavification approach and its generaliza- However, there is a different approach tion do not always help without some spe- to information design: concavification. The cial structure. Our own work (Bergemann, “concavification” approach is based on a Brooks, and Morris 2015) on (one player) geometric analysis of the function mapping price discrimination, discussed in the next receiver posterior beliefs to sender payoffs. section, relies heavily on linear program- Concavification focuses more on the “exper- ming; but while the solution must corre- iments” or the distribution of posteriors that spond to the concavification of an objective are induced for the receivers, rather than function, it is very difficult to visualize the on the joint distribution between actions concavification or provide a proof using and states. In the one-person problem, we it. However, linear programming meth- can identify the payoff that the information ods do not always help, either: in our work designer receives for any given probabil- on (many player) auctions (Bergemann, ity distribution over states, subject to the Brooks, and Morris 2017a), neither gener- fact that the player will make an optimal alized concavification nor linear program- choice. But the information designer has ming methods are used in stating or proving the ability to split the player’s beliefs about our results (although linear programming the state, i.e., supply the player with infor- played an important role in supplying con- mation that will induce any distribution jectures for the results). over posteriors with the constraint that the prior equals the expected posterior. This 5. Metaphorical Information Design and implies that the set of attainable payoffs for Applications the information designer, as a function of prior distributions of states, is the concavi- Mechanism design sometimes has a lit- fication of the set of payoffs of the designer eral interpretation. For example—in some in the absence of information design. This settings—a seller may be able to commit concavification argument (building on to an auction for selling an object. In other Aumann and Maschler 1995) is the focus of settings, the mechanism design problem is both Kamenica and Gentzkow (2011) and studied even though there does not exist a the large and important literature inspired mechanism designer able to commit. For by their work. The many-players case is example, suppose that we are interested in significantly harder than the single-player a buyer and seller bargaining over an object. case, as it is no longer the set of probabil- There may be no rules for how the play- ity distributions over states that matters, but ers bargain and no one who could enforce rather the set of (common prior) subsets such rules. Nonetheless, Myerson and of the universal type space of Mertens and Satterthwaite (1983) studied what would be Zamir (1985) that are relevant for strategic the optimal mechanism for realizing gains 66 Journal of Economic Literature, Vol. LVII (March 2019) from trade because it bounds what could and a linear interaction game with a focus happen under any bargaining protocol that on aggregate variance, and macroeconomic ends up being used. In this sense, there is implications (Bergemann, Heumann, and not a literal mechanism designer, but we Morris 2015). are rather using the language of mechanism Caplin and Martin (2015) adopt a simi- design for another purpose. lar, metaphorical, approach to the recovery Similarly for information design, the most of preference orderings and utility from literal interpretation of the information choice data. They allow for the possibility design problem is that there is an actual that the decision maker has imperfect infor- information designer who can commit to mation while satisfying Bayes’s law and iter- choosing the players’ information structure ated expectation. They ask what they can in order to achieve a particular objective. In learn from the observed choice data about many contexts, this commitment assumption the underlying preference profile without may not be plausible.14 Yet, the information making strong assumptions on the informa- design perspective can be used to address tion available to the decision maker at the many important questions even where there moment of choice. In related work, Caplin is not a literal information designer. In par- and Dean (2015) develop a revealed pref- ticular, understanding the set of outcomes erence test giving conditions under which that an information designer can induce apparent choice “mistakes” can be explained corresponds to identifying the set of all out- in terms of optimal costly information acqui- comes that could arise from some informa- sition by the player in the presence of imper- tion structure. fect information. In our own applications of information 5.1 The Limits of Price Discrimination design, we have mostly been interested in metaphorical interpretations. The set of A classic issue in the economic analysis BCE is precisely the set of outcomes that can of monopoly is the impact of discriminatory arise with additional information for a given pricing on consumer and producer surplus. basic game and prior information struc- A monopolist engages in third-degree price ture. If there are properties that hold for all discrimination if he uses additional informa- BCE, we have identified predictions that are tion about consumer characteristics to offer robust to the exact information structure. different prices to different segments of the Identifying the best or worst outcome that aggregate market. Bergemann, Brooks, and can arise under some information structure Morris (2015) characterize what could hap- according to some objective function as cri- pen to consumer and producer surplus for all terion is the same as solving an information possible segmentations of the market. design problem where the designer is maxi- One can provide some elementary bounds mizing or minimizing that criterion. In this on consumer and producer surplus in any section, we will review two such economic market segmentation. First, consumer sur- applications of information design. We will plus must be nonnegative as a consequence highlight the implications of this approach in of the participation constraint: a consumer the context of third-degree price discrimina- will not buy the good at a price above his tion (Bergemann, Brooks, and Morris 2015) valuation. Second, the producer must get at least the surplus that he could get if there was no segmentation and he had no addi- 14 Forges and Koessler (2005) observe that conditioning on players’ exogenous information makes sense if players’ tional information beyond the prior distri- types are ex post verifiable. bution. In this case, an optimal policy is Bergemann and Morris: Information Design: A Unified Perspective 67 always to offer the product with probabil- triangle is the solution to some maximization ity one at a given price to all buyers. We problem of the information designer defined therefore refer to it as the uniform monop- by some preferences over producer and con- oly price, and correspondingly, uniform sumer surplus. monopoly profit. Third, the sum of con- The information design problem has a sumer and producer surplus cannot exceed very clear literal interpretation in the case the total social value that is generated where the monopolist knows the consumer’s by the good, which is willingness-to-pay valuation. She can then achieve perfect price minus unit cost of production. The shaded discrimination at point B. However, giving a right-angled triangle in figure 5 illustrates literal information design interpretation of these three bounds. point C is more subtle. We would need to The main result in Bergemann, Brooks, identify an information designer who knew and Morris (2015) is that every welfare out- consumers’ valuations and committed to come satisfying these constraints is attain- give partial information to the monopolist able by some market segmentation. This is in order to maximize the sum of consum- the entire shaded triangle in figure 5. If the ers’ welfare. Importantly, even though the monopolist has no information beyond the disclosure rule is optimal for consumers as prior distribution of valuations, there will a group, individual consumers would not be no segmentation. The producer charges have an incentive to truthfully report their the optimal monopoly price and gets the valuations to the information designer, given associated monopoly profit, and consumers the designer’s disclosure rule, since they receive a positive surplus; this is marked by would want to report themselves to have low point A in figure 5. If the monopolist has values. complete information, then he can charge As discussed in the previous section, it each buyer his true valuation, i.e., engage in seems hard to explain the main result using perfect or first-degree price discrimination; concavification, but there is an elemen- this is marked by point B. The point marked tary geometric argument. One can show C is where consumer surplus is maximized; that any point where the monopolist is the outcome is efficient and the consumer held down to his uniform monopoly prof- gets all the surplus gains over the uniform its with no information beyond the prior monopoly profit. At the point marked D, distribution—including outcomes A, C, social surplus is minimized by holding pro- and D in figure 5—can be achieved with ducer surplus down to uniform monopoly the same segmentation. In this segmenta- profits and holding consumer surplus down tion, consumer surplus varies because the to zero. monopolist is indifferent between charging The main result states that we can make different prices. only very weak predictions about producer We can use a simple example to illus- and consumer surplus. It can be understood trate these results. There are three valu- as the outcome of a set of metaphorical ations, v V 1, 2, 3 , which arise in ∈ = { } information design problems. If an infor- equal proportions, and there is zero mar- mation designer wanted to maximize con- ginal cost of production. The feasible social sumer surplus, she would choose point C. If surplus is w ∗ (1 3) 1 2 3 2. = / ( + + ) = she wanted to minimize consumer surplus, The uniform monopoly price is v ∗ 2. = or producer surplus, or any weighted com- Under the uniform monopoly price, profit is bination of the two, she could choose point ∗ (2 3) 2 4 3 and consumer surplus π = / × = / D. Any other point on the boundary of the is u ∗ (1 3) 3 2 (1 3) 2 2 1 3. = / ( − ) + / ( − ) = / 68 Journal of Economic Literature, Vol. LVII (March 2019)

B Producer surplus ( π ) D A C

Comsumer surplus (u)

Figure 5. The Bounds on Profits and Consumer Surplus in Third-Degree Price Discrimination

Segment x x x (x) A segment x is a vector of probabilities of 1 2 3 σ {1, 2, 3} 1 1 1 2 each valuation, thus x x1, x2, x3 , and by x _ _ _ _ = ( ) 2 6 3 3 (x) we denote the total mass of a segment σ x {2, 3} 0 _1 _2 _1 x. A segmentation of the market is therefore (16) 3 3 6 . a collection of segments x X and and a x {1} 0 1 0 _ 1 ∈ 6 probability distribution over the seg- (⋅ ) 1 1 1 σ x ⁎ _ _ _ 1 ments. We give an example of a segmenta- 3 3 3 tion below. In the example, there are three segments and each segment is identified by its support on the valuations indicated by the The particular segmentation has a num- set in the superscript. The frequency of ber of interesting properties. First, in each {· } each segment x is given by x : segment, the seller is indifferent between σ( ) Bergemann and Morris: Information Design: A Unified Perspective 69 charging as price any valuation that is in the economics is how aggregate and individual support of the segment. Second, the uniform productivity shocks translate into variation in monopoly price, p ∗ 2, is in the support GDP. Another classical question is when and = of every segment. Thus, this particular seg- how asymmetric information can influence mentation preserves the uniform monopoly this mapping, and in particular exacerbate profit. If the monopolist charges the uni- aggregate volatility. form monopoly price on each segment, we These questions are studied in a set- get point A. If he charges the lowest value in ting with a continuum of agents whose best the support of each segment (which is also an responses are linear in the (expectation of optimal price, by construction), we get point the) average action of others and in the idio- C; and if he charges the highest value in the syncratic as well as aggregate shocks. Shocks, support, we get point D. actions, and signals are symmetrically nor- Roesler and Szentes (2017) consider a mally distributed across agents, maintaining related information design problem in which symmetry and normality of the information a single buyer can design her own informa- structure. The maximal aggregate volatility is tion about her value before she is facing a attained in an information structure in which monopolist seller. While the analysis of the the agents confound idiosyncratic and aggre- third-degree price discrimination proceeds gate shocks, and display excess response to as a one-player application, the arguments the aggregate shocks, as in Lucas (1972) and extend to many-player settings. Bergemann, more recently in Hellwig and Venkateswaran Brooks, and Morris (2017a) pursue the ques- (2009), Venkateswaran (2013), and Angeletos tion of how private information may impact and La’O (2013). Our contribution is to high- the pricing behavior in a many-buyer envi- light that, in this setting with idiosyncratic and ronment. There, we derive results about aggregate shocks, a class of noise-free con- equilibrium behavior in the first-price auc- founding information structures are extremal tion that hold across all common-prior infor- and provide global bounds on how much vola- mation structures. The results that we obtain tility can arise. In particular, for any given vari- can be used for a variety of applications, e.g., ance of aggregate shocks, the upper bound on to partially identify the value distribution in aggregate volatility is linearly increasing in the settings where the information structure is variance of the idiosyncratic shocks. unknown and to make informationally robust In this application, we do not think there is comparisons of mechanisms. any economic agent who is able to or wants to maximize aggregate volatility. But because we 5.2 Information and Volatility are interested in bounds on aggregate volatil- Bergemann, Heumann, and Morris ity across equilibria of different information (2015) revisit a classic issue in macroeco- structures, the problem is naturally repre- nomics. Consider an economy of interacting sented as an information design problem. agents—each of whom picks an action— The basic setting is that the payoff shock where the agents are subject to idiosyncratic of individual i is given by the sum of an θi and aggregate shocks. A fundamental eco- aggregate shock and an idiosyncratic shock θ nomic question in this environment is to ask : εi how aggregate and idiosyncratic shocks map into “aggregate volatility”—the variance of . θi ≜ θ + εi the average action. Versions of this question arise in many different economic contexts. The aggregate shock is common to all agents θ In particular, a central question in macro- and the idiosyncratic shock is independent εi 70 Journal of Economic Literature, Vol. LVII (March 2019) and identically distributed across agents, as The information structure is noise free in λ well as independent of the aggregate shock. the sense that every signal si is a linear com- Each component of the payoff shock is bination of the idiosyncratic and the aggre- θi normally distributed: gate shock, and , and no extraneous noise εi θ or error term enters the signal of each agent. 2 0 Nonetheless, since the signal s combines the (17) θ μθ , σθ . i ( ) ∼  ( 0 ) 2 idiosyncratic and the aggregate shock with ε i ( ( 0 )) σε weights and 1 , each signal s leaves λ − λ i The variance of the individual payoff shock agent i with residual uncertainty about can be expressed in terms of the variance his true individual payoff shock , unless θi θi of the sum of the idiosyncratic and the aggre- 1 1 2. λ = − λ = / gate shock: 2 2 . The correlation (coeffi- In the decision-theoretic case, r 0, the σθ + σε = cient) between the payoff shocks i and j best response of each agent simply reflects ρθ θ θ of any two agents i and j is: a statistical prediction problem, namely to predict the payoff shock i given the signal si: 2 θ _ (18) σθ . ______(1 ) (1 ) ρ θ ≜ 2 2 (20) ai [ i si] − λ ρθ + λ − ρθ si . = 피θ | = (1 ) 2 2 (1 ) σθ + σε − λ ρθ + λ − ρθ The best response of agent i is given by a The individual prediction problem is more linear function responsive to the signal si, that is, assigns a larger weight to si if and only if the signal a 1 r r A , contains more information about the individ- i = ( − ) 피i[θi] + 피i[ ] ual payoff shock . The noise-free informa- θi where A is the average action. Now, the tion structure 1 2 allows each agent to λ = / results described above regarding the struc- perfectly infer the individual payoff shock . θi ture of the extremal information structure It follows that the responsiveness, and hence hold independently of whether the weight the variance of the individual action 2 , is σa on the average action, r, is negative (the stra- maximized at 1 2: λ = / tegic substitutes case), zero (the purely deci- 2 2 2 sion theoretic case), or positive (the strategic a . σ = σθ + σε complements case). A striking property is that the set of feasible correlations between Now, to the extent that the individual payoff individual and average actions and individual shocks, and , are correlated, we find that θi θj and aggregate shocks is independent of r and even though each agent i only solves an indi- determined only by statistical constraints. vidual prediction problem, his actions are Here we will thus convey the flavor of the correlated by means of the underlying cor- result in a setting where the decision of the relation of the individual payoff shocks, and agent is independent of any strategic consid- the resulting aggregate volatility is: erations, thus the decision-theoretic case. 2 2 2 It suffices to consider the following A a a . σ = ρ σ = σθ one-dimensional class of signals: We can ask whether the aggregate volatility (19) s (1 ) , can reach higher levels under information i ≜ λεi + − λ θ structures different from 1 2. As the λ = / where the linear composition of the signal information structure departs from 1 2, λ = / s is determined by the parameter 0, 1 . we necessarily introduce a bias in the signal i λ ∈ [ ] Bergemann and Morris: Information Design: A Unified Perspective 71

si toward one of the two components of the maintain a large aggregate volatility even in payoff shock . Clearly, the signal s is losing the presence of vanishing aggregate payoff θi i its informational quality with respect to the shocks by confounding the payoff-relevant individual payoff shock as moves away information about the idiosyncratic shock θi λ from 1 2 in either direction. Thus, the indi- with the (in the limit) payoff-irrelevant infor- / vidual prediction problem (20) is becoming mation about the aggregate shock. noisier and in consequence, the response of the individual agent to the signal s is atten- i 6. Information Design with Private uated. But a larger weight, 1 , on the − λ Information aggregate shock , may support correlation θ in the actions across agents, and thus support We have thus far considered the scenario larger aggregate volatility. As the response where the designer knows not only the true of the agent is likely to be attenuated, a state but also the players’ prior informa- θ , trade-off appears between bias and loss of tion about the state. We now consider what information. We show that the maximal happens when the information designer does aggregate volatility: not have access to players’ prior information 2 (but still can condition on the state). Here, _2 2 ______( ) we consider two alternative assumptions (21) max { var(A)} σθ + √σ θ + σε λ = 4 about the designer’s ability to condition recommendations on players’ prior information. is achieved by the information structure ∗: If the designer can elicit the private infor- λ mation, then we have information design with elicitation. If the designer cannot elicit (22) ∗ arg max varA λ ≜ { ( )} the private information, we have informa- λ tion design without elicitation. We present ______ _1 a self-contained discussion of these exten- σθ_ , = 2 2 2 < 2 sions and then discuss how—with these σθ + √σθ + σε extensions—information design fits into the which biases the signal toward the aggregate mechanism design and incomplete-informa- shock. We can express the information struc- tion-correlated equilibrium literatures more ture that maximizes the aggregate volatility broadly in sections 6.3 and 6.4. in terms of the correlation coefficient : ρθ 6.1 Players’ Prior Information is not Known to the Designer _ arg max varA _√ρθ , { ( )} = 1 2 _ When the information designer cannot λ + √ρθ observe the players’ prior information, she and the maximal volatility can be expressed may or may not be able to ask the players as: about it. In the case of information design with elicitation, she will be able to condition max var(A) _1 (1 _ ) 2 2 2 . her recommendations on the reported types. { } = 4 + √ρ θ (σθ + σε) λ In the case of information design without Surprisingly, as we approach an environment elicitation, she can only send a list of recom- with purely idiosyncratic shocks, the maxi- mendations, namely one recommendation mal aggregate volatility does not converge to for each possible type of the player. zero; rather, it is bounded away from zero, In the case of information design with elic- and given by 2 4. Thus, the economy can itation, the revelation principle still implies σε/ 72 Journal of Economic Literature, Vol. LVII (March 2019) that we can restrict attention to the case designer by choosing a rather than a. δi( i) i where the information sent by the informa- Thus, incentive compatibility implies, but is tion designer consists of action recommen- not implied by, separately requiring truth dations. However, we will now require an telling and obedience. incentive compatibility condition that entails truth telling as well as obedience, so that the PROPOSITION 2: An information designer information designer can only condition on with elicitation can attain a decision rule if a player’s signal if the player can be given an and only if it is incentive compatible. incentive to report it truthfully. Following Myerson (1991, section 6.3), we can think In the case of information design without of the information designer choosing a elicitation, the designer cannot condition decision rule : T (A), but each the action recommendation on the reported σ × Θ → Δ type of each player can choose a deviation type, but has to offer a contingent recom- : A A with the interpretation that mendation, a vector of action recommen- δi i → i a is the action chosen by player i if the dations, where each individual entry is an δi( i) information designer recommended action action recommendation for a specific type of a. The decision rule is incentive compati- the player, hence contingent on the realized i σ ble if each player does not have an incentive type. The set of feasible recommendations to to deviate. player i is therefore given by B A Ti. The i = i set of player i’s contingent recommendations DEFINITION 2 (Incentive Compatible): A therefore has a typical element decision rule : T (A) is incentive σ × Θ → Δ compatible for (G, S) if for each i 1, , I = … b : T A . and t T , i i → i i ∈ i I ui((a i, a i), ) ((a i, a i) | (t i, t i), ) ((t i, t i) | ) ( ) We define B ∏ i 1 B i and let a generic a∑,t , − θ σ − − θ π − θ ψ θ = = i θ element be given by b (b , , b) B, = 1 … I ∈ u i(( i( a i), a i), ) ((a i, a i) | (t i , t i), ) ((t i, t i ) | ) ( ) ≥ ∑ δ − θ σ − ′ − θ π − θ ψ θ and so b(t) (b1(t1), , bI(tI)), a,t i, = … θ We are now interested in contingent for all t T and : A A. action recommendations : (B) ′i ∈ i δi i → i ϕ Θ → Δ rather than action recommendations.

The displayed inequality will be referred DEFINITION 3 (Public Feasibility): A to as player i’s type-t incentive constraint. It decision rule : T (A) is publicly i σ × Θ → Δ ensures that player i, after observing signal feasible if there exists a contingent recom- t, finds it optimal to report his signal truth- mendation : (B) such that for each i ϕ Θ → Δ fully and then, after observing and updating a A, t T, and with (t | ) 0, ∈ ∈ θ ∈ Θ π θ > on the information contained in the resulting action recommendation a i, finds it optimal to follow this recommendation. Thus it builds (a | t, ) (b | ). σ θ = b B∑:b(t) a ϕ θ in both truth telling and obedience. In addi- { ∈ = } tion, the notion of incentive compatibility requires that the decision rule is immune In this case, we say that is induced by . σ ϕ to “double deviations” in which the player misreports his type to be t (rather than Public feasibility is the restriction that a ′i ti) and disobeys the recommendation of the given player’s contingent recommendation Bergemann and Morris: Information Design: A Unified Perspective 73

cannot depend on the type of the player u i((b i(t i), b i(t i)), ) ((b i, b i) | ) ((t i, t i) | ) ( ) b∑,t , − − θ ϕ − θ π − θ ψ θ himself nor on the types of the other players. i i θ We refer to the above requirement as public u i((a i , b i(t i)), ) ((b i, b i) | ) ((t i, t i) | ) ( ) ′ − − − − feasibility, since the recommendation vector ≥ b∑,t , θ ϕ θ π θ ψ θ i i θ cannot be tailored to the private information of the players and it has to be feasible in for all a A. ′i ∈ i the sense that it induces the decision rule . σ We emphasize that each contingent recom- The displayed inequality will be referred mendation bi is still communicated to each to as player i’s ( t i, b i)-publicly feasible obedi- agent i separately and privately, and thus is ence constraint. It ensures that player i, after not public in the sense of a public announce- observing signal ti and receiving and updating ment to all players. on the recommendation b i, finds it optimal to When I 1, public feasibility is a vac- take the action b (t ) prescribed by the vec- = i i uous restriction. Every decision rule is tor b for his type t . Note that in so far as a σ i i induced by the contingent recommendation contingent recommendation bi reveals more given by information about the state and thus possi- ϕ θ bly and indirectly about the type profile t i − (b | ) ∏ (b(t) | t, ). than just an action recommendation, publicly ϕ θ = t T σ θ ∈ feasible obedience will be a more demanding concept than mere obedience as it allows the Under this choice of , the components of agents to contemplate deviations based on ϕ the contingent recommendation vector b(t) more accurate information. After all, type ti is are drawn independently. When I 1, how- able to observe the recommendation tailored > ever, public feasibility is a substantive restric- toward him as well as the recommendation tion. By recommending to a particular player offered to all other types t t. To the ′i ≠ i a strategy rather than an action, the designer extent that the recommendation across types can condition that player’s action on his type. is not perfectly correlated, it then follows that By judiciously choosing a distribution over B , the type ti will receive additional information the designer can even correlate the players’ through the contingent recommendation strategies. But she cannot correlate one play- rather than the action recommendation. er’s strategy on another player’s type. We are not interested in all contingent PROPOSITION 3: An information designer recommendations, but rather those that are without elicitation can attain a decision rule obedient in the sense defined earlier, in defi- if and only if it is publicly feasible obedient. nition 1. Below, we adapt the definition to account for the larger space of contingent 6.2 The Investment Example Revisited recommendations, b , rather than action recommendations, a. We now illustrate information design with private information using the investment DEFINITION 4 (Publicly Feasible Obe- example introduced earlier in section 3. The dience): A decision rule : T (A) issues that arise in the one-player case with σ × Θ → Δ is publicly feasible obedient if there exists a private information are already interesting and contingent recommendation : (B) subtle. Here we do not present any examples ϕ Θ → Δ such that (i) induces , and (ii) satisfies of elicitation with many players. We thus do ϕ σ ϕ obedience in the sense that for each i, t T , not consider any additional issues of public fea- i ∈ i and b B , sibility that only arise in the many-player case. i ∈ i 74 Journal of Economic Literature, Vol. LVII (March 2019)

We therefore now allow the information of described below for a good type t g. The = the firm to be private. The government does truth-telling constraint for the good type not know the realization of the signal that the t g is: = firm observes but can or cannot elicit it. qp x 1 q p qp x 1 q p , (23) Gg − ( − ) Bg ≥ Gb − ( − ) Bb 6.2.1 Information Design with Elicitation and correspondingly for the bad type t b: = In the case of elicitation, we have a screen- 1 q p x qp 1 q p x qp . ing problem where the designer offers a rec- (24) ( − ) Gb − Bb ≥ ( − ) Gg − Bg ommendation that induces a probability of investing as a function of the reported sig- By misreporting and then following the nal and the true state. As noted above, we resulting recommendation afterwards, each have three sets of constraints that need to type can change the probability of investing. be satisfied. First, each type has to truth- We can write the above two truth-telling con- fully report his signal; second, each type has straints in terms of a bracketing inequality: to be willing to follow the recommendation, the obedience constraints; and third, double 1 q (25) _ − x p p p p deviations, by means of misreporting and q ( Gg − Gb) ≤ Bg − Bb disobeying at the same time, must not be q profitable. Kolotilin et al. (2017) refer to this _ x p p . ≤ 1 q ( Gg − Gb) informational environment as “private per- − suasion.”15 Kolotilin (2018) pursues a linear These inequalities highlight how the dif- programming approach to identify the opti- ferential between type t b and t g in = = mal information disclosure policy under a the recommendation p Bt in the bad state are single-crossing assumption. bounded, below and above, by the differen- A decision rule now specifies the proba- tial in the recommendation pGt in the good bility of investment pt conditional on the state. Notice also that since true state B, G andθ the reported type θ ∈ { } 1 q q t b, g . Thus, as before in sec- _ − _1 _ x, ∈ { } q x < 1 q tion 3.2, a decision rule is now a vector − p p , p , p , p . The information the above bracketing inequality requires that = ( Bb Bg Gb Gg) designer offers a recommendation (stochas- tically) as a function of the true state and the p p 0, p p 0, Gg − Gb ≥ Bg − Bb ≥ reported type. The obedience conditions are as in section 3.2, where information was not thus, the conditional probability of investing private. A truthful reporting constraint is has to be larger for the good type than the bad type in either state. An implication specific to the binary action, binary state envi- 15 Bergemann, Bonatti, and Smolin (2018) consider a model of private persuasion with quasilinear utility. Their ronment is the fact that double deviations do main objective is to analyze the revenue maximizing solu- not impose any additional restriction on the tion to the information design problem subject to the elic- behavior of the player. We state and prove itation constraints. Daskalakis, Papadimitriou, and Tzamos (2016) also consider an information design with quasilinear this result formally in the appendix as prop- utility. The novel aspect of their analysis is that the object osition 6. In particular, this means that the for sale has many attributes, and the seller chooses opti- obedience and truth-telling constraints dis- mally how much to disclose about each individual attri- bute. Their analysis reveals a close relationship to the cussed above imply the incentive compatibil- classic bundling problem of a multi-item monopolist. ity condition of definition 2. Bergemann and Morris: Information Design: A Unified Perspective 75

0.8

0.6

0.4

0.2

Omniscient with prior info Elicitation

0 0 0.2 0.4 0.6 0.8 1

Figure 6. Investment Probability with Private Information

With these additional constraints, the set of to the outcomes that can arise under infor- outcomes that can arise in equilibrium under mation design with elicitation; adding in information design with elicitation is weakly, the pink region, we get back to the triangle and typically strictly, smaller than under an that corresponds to omniscient information omniscient designer, i.e., the case in the design where the designer knows players’ previous section where the designer can con- prior information. dition his recommendation directly on the 6.2.2 Information Design without players’ prior information. The truth-telling Elicitation constraints impose restrictions on how the differences in the conditional probabilities We could also consider a government, across types can vary across states. These who does not know the signal of the firm and impose additional restrictions on the ability cannot even elicit it. Kolotilin et al. (2017) of the government to attain either very low call this scenario “public persuasion.” Such or very high investment probabilities in both information design without elicitation has states, as highlighted by equation (25). been the focus of the recent literature. Figure 6 illustrates the case where x 0.9 Clearly, the designer can replicate any = and q 0.7; the dark red region corresponds decision rule without elicitation with a = 76 Journal of Economic Literature, Vol. LVII (March 2019)

decision rule with elicitation. This inclusion set of feasible actions of the player. The holds without any restrictions on the state decision to invest small comes with a higher space, the number of players, or the players’ rate of return but smaller total return than actions. In the specific investment example the (regular) investment decision. The above, with a single player, two states, and payoff from a small investment is 1 2 in − / two actions, the converse happens to be true the bad state and y (x 2, x) in the good ∈ / as well. That is, any decision rule attainable state: with elicitation can also be attained without elicitation. In other words, in the binary bad state B good state G setting and with a single player, there is no invest 1 x need for elicitation. The designer can induce − any incentive-compatible decision rule 1 invest small _ y . by recommendations alone. We state and −2 prove these two results in the appendix as not invest 0 0 propositions 5 and 6.16 The equivalence breaks down immediately if either of the binary assumptions regarding For simplicity we restrict attention to decision action and state are relaxed or we consider rules that put zero probability on the small more than one player. We illustrate this fail- investment in equilibrium. We note that the ure of the equivalence result with a minor small investment decision still plays a role in generalization of the investment example. In the characterization of incentive compatible particular, in a single-player environment, decision rules as it is a feasible action to the we allow the player to either consider a small player. It will hence generate additional obe- or a large investment. For completeness, we dience constraints that the designer has to present, in the appendix, examples where respect as the player has now two possible one of the other two hypotheses fails. This deviations from the recommended action, modified example allows us to find a strict one of which is to invest at a small scale. The nesting of the set of outcomes without prior decision rules—restricted to invest and not information, with prior information and an invest—can still be represented by a vector omniscient designer, with elicitation, and p ( p , p , p , p ) that records the = Bb Bg Gb Gg finally without elicitation. For the purpose of probability of investing. this example, it will be sufficient to focus on As a benchmark, first suppose the player the case of a single player. has no prior information. Then a decision rule that never recommends the small 6.2.3 Beyond the Binary Setting investment can be represented as a pair Consider the basic investment example ( p , p ) [0, 1] 2 that specifies the proba- B G ∈ with I 1, {B, G}, uniform prior, and bility of the large investment in each state. = Θ = symmetric types that are correct with prob- When the firm has no prior information, ability q 1 2. We now add an additional there are two binding obedience constraints, > / investment decision, to invest small, to the one for the big investment against the small investment:

16 Kolotilin et al. (2017) showed such an equivalence (26) p x p p y _1 p , under a different set of assumptions. In their model, the G − B ≥ G − 2 B designer and the player are privately informed about distinct payoff states rather than having distinct information about the same state as in the present setting. Bergemann and Morris: Information Design: A Unified Perspective 77

0.8

0.6

0.4

0.2 Omniscient without prior info Omniscient with prior info Elicitation Without elicitation 0 0 0.2 0.4 0.6 0.8 1

Figure 7. Investment Probability under Different Information Design Scenarios

and one for no investment against the small levels analyzed earlier, there is now a kink investment:17 in the area of attainable decision rules that reflects a change in the binding obedience 0 (1 p ) y _1 1 p . constraint, from zero investment to small ≥ − G − 2( − B) investment. The equilibrium regions are depicted in fig- If we consider the case in which the firm ure 7. If the firm has no prior information, has prior information, then we have three the government faces only the above two different communication protocols for the constraints. The set of attainable decision government. An omniscient designer faces rules is described by the light red area. In the obedience constraints that we analyzed contrast to the setting with two investment earlier in section 3.2. By contrast, if the firm holds private information, then the information designer is no longer omniscient. Now 17 The other two possible incentive constraints, namely the firm has two possible ways to disobey. for the big investment against no investment and for no investment against the big investment, are supplanted by If the government does not observe the sig- the above two. nal but can elicit the information from the 78 Journal of Economic Literature, Vol. LVII (March 2019) firm, then we have truth-telling constraints between elicitation and no elicitation in the as described by (23) and (24) in addition to binary action and state environment.18 the obedience constraints. Importantly, in this We conclude with a few observations setting with more than two actions, the possi- about the comparative statics with respect to bility of double deviations—misreporting and the information structure. As the precision disobeying—generates additional constraints of the information q decreases toward 1 2, / on the incentive compatible decision rules. the inner three regions expand outward and Finally, a designer without elicitation faces converge to the no-prior-information equi- additional obedience constraints that rule librium set. By contrast, as the precision q out deviations conditional on a particular increases towards 1, the three inner regions vector recommendation. The corresponding contract and converge to the single coordi- areas in figure 7 illustrate that the sequence nate vector (0, 1). of additional constraints from omniscient to 6.3 Information Design within Mechanism elicitation to no elicitation impose increas- Design ingly more restrictions on the government and hence generate a sequence of strictly In the information design problem, the nested sets. “information designer” can commit to pro- We already discussed how these three viding information to the players to serve regimes offer an increasing number of con- his ends, but has no ability to choose out- straints. It remains to discuss the specific comes (or force the players to take particu- impact of being able to elicit (or not) the lar actions). The set of available actions and private information. With elicitation, the a mapping from action profiles to outcomes player only learns the designer’s recommen- and thus payoffs is fixed. How does this relate dation for one type, namely the type that to “mechanism design”? he reports. But a designer who cannot elicit Myerson (1982) describes a class of Bayes must reveal her action recommendations for incentive problems, which constitutes a all types, hence the contingent recommen- leading definition of mechanism design dation. This enables a player to contemplate (see also Myerson 1987, 1994). In this set- additional contingencies and hence deviating, players may have control over some tions. With three possible actions, as in this actions affecting outcomes but the mecha- example, there are two additional deviations nism designer may be able to commit to pick that take advantage of this finer informa- other outcomes as a function of the play- tion. In particular, the high type can disobey ers’ reports. For example, in many classical the recommendation to invest by deviating mechanism design problems with individual to invest small only when the designer also rationality constraints, players do have con- recommends not to invest to the low type. trol over some actions: participation versus Likewise, the low type can disobey the rec- nonparticipation. And even if the mecha- ommendation not to invest by deviating to nism designer may not have any information invest small only when the designer also that is unavailable to the players, he can— recommends investing to the high type. The via the mechanism—implicitly control the additional options for the player induce further constraints on the information designer. 18 We mentioned earlier that double deviations were Naturally, these additional deviations were not relevant in the binary environment in the sense that not available in the binary action environ- they do not add additional restrictions. This changes in the richer environment here, where the communicating ment. And in fact, the absence of this large designer indeed faces additional restrictions coming from set of deviations accounts for the equivalence the possibility of double deviations. Bergemann and Morris: Information Design: A Unified Perspective 79

information that players have about each rationale for thinking about correlated equi- other. Myerson (1991) then labels the case librium when the information designer has where the mechanism designer has no direct information of her own.19 In this section, we control over outcomes “Bayesian games with review the existing literature on incomplete communication” (section 6.3) and the setting information correlated equilibrium to relate where the designer has complete control it to the version of incomplete information over outcomes “Bayesian collective choice correlated equilibrium—BCE—that is rele- problems” (section 6.4). Thus, what we are vant for information design. calling information design corresponds to While Aumann (1987) provides an Myerson’s Bayesian games with communica- information design foundation for com- tion with the proviso that the mediator brings plete information correlated equilibrium, his own information to the table, rather than he offers a broader interpretation of the merely redistributing others’ information. characterization, arguing that correlated There is also an important literature on an equilibrium captures the implications of informed player (referred to as informed prin- common knowledge of rationality in a com- cipal) who can commit to choose outcomes as plete information game, under the common a function of messages (see Myerson 1983). prior assumption.20 A large literature on the But in this setting, the information designer epistemic foundations of game theory has (principal) is typically assumed to be able to developed since then (Dekel and Siniscalchi commit to a mechanism only after receiving 2015), elaborating on the formal language his private information and the principal is and questions suggested by Aumann’s work, not choosing action herself; see Mylovanov although focused on the case of complete and Troeger (2012, 2014) and Perez-Richet information without the common prior (2014) for recent contributions. By contrast, assumption. Formal treatment of the impli- in the information design setting there is a cations of common knowledge of rationality principal who cannot pick a contract mech- and the common prior assumption under / anism but can commit to a disclosure rule incomplete information ties in with many of prior to observing her information. the issues discussed in this survey; we will discuss one issue that arises in this case in 6.4 Correlated Equilibrium and Incomplete the next subsection. Information To understand the literature on incom- Aumann (1974, 1987) introduced cor- plete information correlated equilibrium, it related equilibrium as a solution concept is useful to identify two kinds of constraints for games with complete information (about in the literature on incomplete information the payoff matrix). He showed that the set correlated equilibrium: feasibility conditions of correlated equilibria equals the set of distributions over actions that could arise in a Bayes–Nash equilibrium if players 19 Bergemann and Morris (2017) consider foundations for other solution concepts based on informational robust- observed some additional payoff-irrelevant ness and information design considerations, when the signals (consistent with the common prior). common prior assumption is not maintained. Equivalently, the set of correlated equilibria 20 Hillas, Kohlberg, and Pratt (2007) propose a related alternative foundation for correlated equilibrium. Consider corresponds to the set of outcomes that could an external observer who observes an infinite sequence of be induced by an (uninformed) information plays of a complete information game, has exchangeable designer. What we are calling “information beliefs about them, but does not believe he can offer ben- eficial advice to players on how to improve their payoffs. design” thus corresponds to an incomplete This observer must believe that play corresponds to a cor- information elaboration of this original related equilibrium. 80 Journal of Economic Literature, Vol. LVII (March 2019)

(constraints on what kind of information tions of common certainty of rationality and decision rules can condition on) and incen- players’ beliefs and higher-order beliefs. The tive compatibility conditions (what decision solution concept by construction imposes rules are consistent with optimal behavior). belief invariance. Liu (2015) observes that In this paper so far, we have introduced the set of interim correlated rationalizable one feasibility condition—public feasibil- actions corresponds to the set of actions that ity (definition 3)—and three incentive con- can be played in a correlated equilibrium straints—obedience (definition 1), incentive with incomplete information and subjective compatibility (definition 2) and publicly fea- priors. If, then, the common prior assump- sible obedience (definition 4). Recall that tion is imposed, this corresponds to the set of BCE—our characterization of outcomes that belief-invariant Bayes-correlated equilibria. can be induced by an omniscient informa- Thus, the solution concept of belief-invariant tion designer—imposed only obedience. We BCE is the “right” one for understanding the will discuss two further feasibility conditions implications of common knowledge assump- to provide an overview of correlated equilib- tions under the common prior assumption. rium with incomplete information. Second, Mathevet, Perego, and Taneva (2017) consider a situation where the infor- 6.4.1 Belief Invariance mation designer can convey information Consider the requirement that the infor- only about beliefs and higher-order beliefs, mation designer can correlate players’ but is not able to send additional information actions, but without changing players’ beliefs about correlation. Now the set of belief-in- and higher-order beliefs about the state of variant BCE, once some higher-order belief the world. This is formalized as follows. information has been sent, is equal to the set of BCE. Bergemann and Morris (2016a) DEFINITION 5 (Belief Invariant): Decision describe how an arbitrary information struc- rule : T (A) is belief invariant ture can be decomposed into information σ × Θ → Δ for (G, S) if i(ai | (ti , t i), ) is independent of about beliefs and higher-order beliefs, and σ − θ t i for each i, ai, ti , and , where additional belief-invariant signals. − θ 6.4.2 Join Feasibility i a i | t i, t i, i a i, a i| t i, t i, . σ ( ( − ) θ) ≜ σ (( − ) ( − ) θ) a ∑i A i − ∈ − Twenty five years ago, Forges (1993) (see also Forges 2006) gave an overview of We then say that a decision rule is a incomplete information correlated equilib- belief-invariant BCE if it satisfies belief rium. A maintained assumption in that liter- invariance and obedience. It is not obvious ature was that the information designer (or how this feasibility condition arises under “mediator”) did not bring any information of an information design interpretation: if the her own to the table, but simply rearranged designer can condition his information on information, telling players privately about , why not allow him to change beliefs and others’ information. This can be formalized θ higher-order beliefs? as follows. There are two conceptual reasons why one might nonetheless be interested in belief-in- DEFINITION 6 (Join Feasibility): Decision variant BCE. First, Dekel, Fudenberg, and rule : T A is join feasible for σ × Θ → Morris (2007) introduced the solution con- (G, S) if (a | t, ) is independent of , i.e., σ θ θ cept of interim correlated rationalizability. (a | t, ) (a | t, ) for each t T, a A, σ θ = σ θ′ ∈ ∈ They show that it characterizes the implica- and , . θ θ′ ∈ Θ Bergemann and Morris: Information Design: A Unified Perspective 81

Thus, join feasibility allows the designer (v) A strategic form correlated equilibrium to use the join of the private information of is a decision rule satisfying join feasi- all the players, the information contained in bility and publicly feasible obedience the entire type profile t. At the same time, (which implies belief invariance, pub- it requires that the information designer can lic feasibility, obedience, and incentive send information only about the type profile compatibility). of the players and thus can only condition on the type profile, not on the state of the world Thus, the Bayesian solution, communi- or . Join feasibility is imposed implicitly cation equilibrium, and strategic form cor- θ θ′ in some work on incomplete information related equilibrium correspond to omniscient correlated equilibrium—Forges (1993) inte- information design, information design with grates out uncertainty other than the players’ elicitation, and information design without types—but explicitly in others, e.g., Lehrer, elicitation, respectively. The belief invariant Rosenberg, and Shmaya (2010). Bayesian solution and the agent normal form As noted in the introduction, information correlated equilibrium do not have natural design adds to the old incomplete informa- information design interpretations. tion correlated literature the twist that the Forges (1993) noted inclusions implied by designer brings information of her own to these definitions. In particular, if we write the table. In turn, this allows the designer to (n) for the set of incomplete information cor- choose the optimal design and provision of related equilibria of type n above, we have the information to the players. Forges’s 1993 paper was titled “Five (5) (3) (2) (1) and ⊆ ⊆ ⊆ Legitimate Definitions of Correlated Equilibrium in Games with Incomplete (5) (4) (1). Information.” The feasibility and incentive ⊆ ⊆ conditions described so far allow us to com- Forges (1993) reports examples showing pletely describe the five solution concepts that these inclusions are the only ones that she discusses: can be shown, i.e., there exist decision rules that (i) are Bayesian solutions but not belief (i) A Bayesian solution is a decision rule invariant BCE or communication equilib- satisfying join feasibility and obedience. ria; (ii) are belief invariant Bayes solutions but not communication equilibria or agent (ii) A belief invariant Bayesian solution is a normal form correlated equilibria; (iii) are decision rule satisfying join feasibility, communication equilibria but not belief belief invariance, and obedience. invariant Bayesian solutions; (iv) are belief invariant Bayesian solutions and commu- (iii) An agent normal form correlated nication equilibria but not agent normal equilibrium is a decision rule satisfy- form equilibria; (v) are agent normal form ing join feasibility, public feasibility, correlated equilibria but not communica- (which implies belief invariance) and tion equilibria; (vi) are agent normal form obedience. correlated equilibria and communication equilibria. (iv) A communication equilibrium is a deci- Forges (1993) discusses one more solution sion rule satisfying join feasibility and concept: the universal Bayesian solution. The incentive compatibility (which implies universal Bayesian solution corresponds—in obedience). our language—to the set of Bayes-correlated 82 Journal of Economic Literature, Vol. LVII (March 2019) equilibria that would arise under join feasi- strategy for player i in this game is a mapping bility if players had no information. b : T M A . A communication i i × i → Δ( i) rule C and strategy profileb will now induce a decision rule 7. Information Design with Adversarial Equilibrium and Mechanism Selection ( a | t, ) ( m | t, ) b i(a i | t i , m i) . σ θ = m∑M ϕ θ (∏i ) We have so far examined settings where ∈ the revelation principle holds: we can, with- We will write E(C) for the set of Bayes–Nash out loss of generality, assume that the set of equilibria of the game with communication signals, or types, is equal to the set of actions. rule C. We can now give a more formal state- We now consider two natural extensions of ment of proposition 1. information design where the revelation principle breaks down. PROPOSITION 4: Decision rule is a BCE σ of (G, S) if and only if there exists a commu- 7.1 Adversarial Equilibrium Selection nication rule C and a Bayes–Nash equilib- In section 2, it was implicitly assumed rium b E(C) that induce . ∈ σ that the information designer could, having designed the information structure, also This is a revelation principle argument select the equilibrium to be played. With one that was formally stated as theorem 1 in player, the equilibrium selection problem Bergemann and Morris (2016a). reduces to breaking ties and is not of sub- Recall that in section 2, we defined the stantive interest. However, Carroll (2016) information designer’s utility from BCE : σ and Mathevet, Perego, and Taneva (2017) highlighted that this issue is of first-order V( ) v(a, ) (a | t, ) (t | ) ( ). σ = a∑,t, θ σ θ π θ ψ θ importance in the many-player case, and that θ the revelation principle argument breaks We can also define the information design- down and alternative arguments must be er’s utility from communication rule C and used; Mathevet, Perego, and Taneva (2017) strategy profile b: formalize and analyze the case where play- ∗ V ( C, b) v(a, ) b i(a i | ti, m i) (m | t, ) (t | ) ( ). ers with no prior information will choose the = ∑, t, a θ (∏i m∑M ϕ θ )π θ ψ θ worst equilibrium for the designer. θ ∈ For our representation, we define a Let us consider the problem of an informa- “communication rule” for the information tion designer who can pick both the commu- designer. Players have the prior informa- nication rule and the equilibrium and is thus tion encoded in the information structure solving the problem I S Ti i 1 , . The information designer = (( ) = π) sends each player i an extra message max max V ∗( C, b). m M, according to rule : T M , C b E(C) i ∈ i ϕ × Θ → Δ( ) ∈ where M M1 MI. A communi- = × ⋯ × I cation rule is then C Mi i 1 , . Now Proposition 4 established that = (( ) = ϕ) the basic game G , the prior information structure S, and the communication rule max max V ∗( C, b) max V( ). C b EC = BCE σ C describe a Bayesian game (G, S, C).21 A ∈ ( ) σ∈ But one could also consider the problem of 21 Bergemann and Morris (2016a) call the pair (S , C) an an information designer who can pick the “expanded information structure.” communication rule, but wants to maximize Bergemann and Morris: Information Design: A Unified Perspective 83 his utility in the worst equilibrium, and is of inefficient outcomes by creating common thus solving the problem knowledge of payoffs and picking the good equilibrium. To make the problem interest- max min V ∗ C, b .22 ing, they then study the maxmin problem. C b EC ( ) ∈ ( ) Finally, a literature on robustness to We now discuss three applications where incomplete information (Kajii and Morris maxmin information design problems have 1997) can be understood as an information been motivated and studied; in each appli- design problem with adversarial equilibrium cation, players have prior information. First, selection. We will give an example to illus- Carroll (2016) considers the problem of trate this connection.23 We will consider a bilateral trade where he wants to know the slightly adapted version of the incomplete worst possible gains from trade for a given information investment game discussed ear- distribution over the known private values lier with payoffs: of a buyer and a seller. If we picked the worst equilibrium, we could always support B invest not invest no trade with probability one, so instead he θ = invest x, x 1, 0 , considers the best equilibrium. This is equiv- − alent to having an information designer pick not invest 0, 1 0, 0 an information structure to minimize the − efficiency of trade anticipating that the buyer and seller will play an equilibrium that max- G invest not invest imizes efficiency (i.e., maximizes the gains θ = , from trade). invest x, x x, 0 Second, Inostroza and Pavan (2017) con- not invest 0, 1 0, 0 sider global game models of regime change − and the problem of an information designer trying to minimize the probability of regime for some 0 < x < 1, and the probability change (they are motivated by the design of of state G is , and is small. Assume that ε ε stress tests to minimize the probability of a the prior information is that player 1 knows run on a bank). What information should the the state and player 2 knows nothing. Thus, information designer send—as a function of player 1 has a dominant strategy to invest in the state and players’ initial information—to state G, while there are multiple equilibria minimize the probability of a run? They call in the complete information game corre- this scenario “discriminatory” because the sponding to state B. In this setting, we can information designer can condition on play- study the standard information design (with ers’ prior information. As in Carroll’s bilateral prior information) described above. Suppose trade problem, the problem is not interest- that the information designer wants to max- ing if the designer is able to pick the equi- imize the probability that at least one player librium as well as the information structure: invests. Maintaining the assumption that in this case, he can prevent the possibility the designer can pick the equilibrium, the answer is trivial: the information designer can simply give the players no additional 22 Of course, the minmax problem min max V ∗( C, b) is C b EC ∈ ( ) a reinterpretation of a maxmin problem where the objective is replaced by V ∗ C, b . The minmin problem is sim- − ( ) ilarly a reinterpretation of the maxmax problem. 23 See also Hoshino (2017) on this connection. 84 Journal of Economic Literature, Vol. LVII (March 2019) information and there will be an equilibrium to invest. Now player 2, with information where players always invest. set {1, 2}, must have a best response to invest, But what if the information designer since he attaches probability q to player 1 anticipated that the worst equilibrium would investing. Now suppose that we have estab- be played? This is information design with lished that both players are investing at adversarial equilibrium selection. What information sets of the form , 1 if {ω ω + } information structure would the informa- k. Now consider the player with infor- ω ≤ tion designer choose and what would be the mation set k 1, k 2 . He attaches { + + } induced probability that at least one player probability q to the other player being at invests? It is convenient and more transpar- information set k, k 1 and therefore { + } ent to the describe information structures investing. So the player with information set using the language of partitions. k 1, k 2 will invest. This argument { + + } Consider the information structure defined establishes that it is possible to ensure that— by a set {1, 2, , } where player 1 if is sufficiently small—both players invest Ω = … ∞ ε q observes an element of the partition: with probability _____ . Since this is true for 2q 1 − ε any q ____1 , it implies that it is possible to 1 x ( { 1}, { 2, 3}, { 4, 5}, , { }) , > + … ∞ get both players to invest with probability and player 2 observes an element of the arbitrarily close to partition: 1 (1 x) ______/ + _1 . 2 (1 x) 1 ε = 1 x ε ( { 1, 2}, { 3, 4}, , { }) . / + − − … ∞ The information structure we used to get Thus, an element of the partition now con- arbitrarily close to this bound was (count- stitutes a signal realization. Let payoffs be ably) infinite, but we can also get arbitrarily given as defined by the payoff state G close using finite information structures θ = if 1 is realized and by the payoff state as shown in Kajii and Morris (1997). Now, ∈ σ arguments from Kajii and Morris (1997) B everywhere else. For some q _1 , 1 , θ = ∈ (2 ) imply that this information structure is (arbi- let the probability of state be ω ≠ ∞ trarily close to) optimal for the information 1 q ω ____− and so the probability of state designer in this problem. To get a flavor of ε( q ) q the argument, say that a player p-believes is 1 _____ (if is sufficiently small). 2q 1 an event if he attaches probability at least ∞ − − ε ε This information structure could arise from p to the event occurring, and that there is the prior information described above (only common p-belief of that event if each player player 1 can distinguish between states p-believes it, each player p-believes that B and G) and communicating additional both p-believe it, and so on. One can information. Now suppose that q ____ 1 > 1 x show that not invest is rationalizable only ; this condition implies that a player assign+ - if there is common ____ x - belief that pay- 1 x ing probability q to the other player invest- offs correspond to state+ B. But since x 1, it follows that ____ x _1 and one ing will always have a strict incentive to < 1 x > 2 invest. Following the induction argument can show that if the event+ that payoffs of Rubinstein (1989), invest is the unique are given by state B has probability at rationalizable action for both players at all least 1 , then—for sufficiently small − ε states . To see this, observe that at —the ex ante probability that there is ω ≠ ∞ ε common ____ x -belief that the state is B is state 1, player 1 has a dominant strategy 1 x + Bergemann and Morris: Information Design: A Unified Perspective 85 at least 1 ____1 . This establishes that the basic game, or mechanism, was going to 1 x − − ε bound is tight. If x 1, similar arguments be? In particular, suppose that the choice of > can be used to show that the information mechanism was adversarial. Again, we will designer can ensure that both players invest lose the revelation principle. Once the infor- with probability 1. mation designer has picked the information Arguments from Kajii and Morris (1997) structure (and thus the set of signals), the and the follow-up literature (Ui 2001 and adversarial mechanism designer could pick a Morris and Ui 2005) can be used to analyze mechanism with a different set of messages. maxmin payoffs more generally when—as in Bergemann, Brooks, and Morris (2016) the above example—the incomplete informa- consider the problem of an information tion game has each player either knowing that designer picking an information structure payoffs are given by a fixed complete informa- for a set of players with a common value of tion game or having a dominant strategy. an object to minimize revenue, anticipat- It is worth emphasizing that the above ing that an adversarial mechanism designer definition of the maxmin problem, and all will then pick a mechanism to maximize three applications, correspond to the omni- revenue (a minmax problem). This gives an scient case where the information designer upper bound on the revenue of the seller can condition on players’ prior information as of a single object who is picking a mecha- well as on the state. An alternative case that nism anticipating that the worst information has been studied is when the information structure will be chosen (a maxmin prob- designer can only send public signals and only lem). Du (2018) constructs elegant bounds condition on the state (and not players’ prior for the latter problem and shows that these information). Goldstein and Huang (2016) bounds are sharp in the limit as the number and Inostroza and Pavan (2017) have studied of bidders increases. The former establishes this problem in global game models of regime the equivalence between minmax and max- change. (Inostroza and Pavan 2017 call this min exactly when there are two bidders and the nondiscriminatory case, to contrast with when the support of the value is binary, and the discriminatory case described previously.) the latter solves the auction design problem This case can be illustrated by our example in the limit when the number of bidders goes above. An information designer interested in to infinity. In a recent paper, Brooks and Du maximizing the probability of both investing (2018) provide a general solution to the com- would send a public signal to invest always if mon value auction problem with a general common prior and a finite number of bid- the state was good and with probability ____ ε x 1 − ε ders. Both problems are studied without the if the state was bad. This would make player 2 common prior assumption by Chung and Ely indifferent between investing and not invest- (2007). ing if he got the “invest” signal. 8. Conclusion 7.2 Adversarial Mechanism Selection We have provided a unified perspec- We considered an information designer tive for a rapidly expanding literature on who was choosing additional information Bayesian persuasion and information design. for the players, holding fixed the basic game In contrast with the recent literature on and players’ prior information. But what Bayesian persuasion that is concerned with if the information designer had to pick the a single player (receiver), we emphasized information structure not knowing what the the implications of information design for 86 Journal of Economic Literature, Vol. LVII (March 2019)

many-player strategic environments. We studied in conjunction is the area of mar- presented a two-step approach to informa- kets with resale. Here, the information that tion design: first identify the set of outcomes is disclosed in the first stage fundamentally that are attainable under some information affects the interaction in the resale mar- structure, then identify the optimal informa- ket, see for example Calzolari and Pavan tion structure. We have described the close (2006); Dworczak (2017); Carroll and Segal connection between the information design (forthcoming); and Bergemann, Brooks, and problem and the earlier literature on cor- Morris (2017b). related equilibrium with incomplete infor- The information design problem— mation; but whereas players are receiving whether literal or metaphorical—identifies real payoff-relevant information in the infor- a mapping from the economic environment mation design problem, in the older cor- to possible outcomes, allowing for different related equilibrium literature, the designer choices of information structures. There is (mediator) was merely providing correlating a second, reverse use of information design devices. for robust identification, identifying a map- We have focused on a pure version of the ping from outcomes to possible parameters static information design problem where the of the economic environment, allowing that designer has no ability to control outcomes. different information structures might have But—as argued in Myerson (1982) and generated the data.24 For example, in an auc- Myerson (1987) and discussed in section 6.3— tion setting, one might consider a sample of there are settings where a designer can con- bids from a sequence (or cross-section) of trol some outcomes (as a function of players’ independent auctions. We can then ask what messages), but cannot control others and then we can infer about the underlying distribu- can only use information to influence the out- tion of valuations under weak assumptions comes outside her control. In other settings, on the information structure, that is without the principal may be able to jointly choose the assuming a specific information structure. mechanism and the information structures. Syrgkanis, Tamer, and Ziani (2017) pursue For example, Bergemann and Pesendorfer such an approach for inference and identi- (2007) consider the optimal design of infor- fication in an auction setting. Magnolfi and mation structure and auction format in an Roncoroni (2017) adopt a similar perspective independent private value environment. in the analysis of discrete games, in particu- More recently, Daskalakis, Papadimitriou, lar entry and exit games. and Tzamos (2016) solve for the optimal Many interesting avenues remain open in auction and information structure when the information design. There are many open seller and the bidders each have some private methodological questions. The concavifica- information about the valuation of the object. tion approach has been very influential in the Their analysis is motivated by online advertis- single-player (receiver) setting, so it is natural ing auctions, where the two-way information to ask if it can be as useful in the many-player asymmetry among seller and bidder is a cen- setting. In both the linear programming and tral feature of the environment. the concavification approach, the optimal As one moves into dynamic settings, information structure is identified by a global an overlap between the tools of informa- optimization problem. It might be insight- tion design and mechanism design more ful to find a more local approach that could generally become more central. A specific setting where the tools of mechanism design 24 This reverse use is exposited in Bergemann and and information design have recently been Morris (2013b). Bergemann and Morris: Information Design: A Unified Perspective 87 identify the direction of valuable information Amazon, or service platforms such as Uber provision. We briefly mentioned a number or OpenTable. This suggests that information of applications of information design in the design will naturally be part of the solution introduction. Digital information is becoming of a large class of allocation problems. To the widely used in the allocation and distribution extent that the relevant information is arriv- of services and commodities, as in traffic nav- ing sequentially and improving over time, the igation apps such as Waze or Google Maps, resulting models will likely incorporate and recommender systems used by Netflix and address dynamic aspects.

Appendix

Additional Computation for Section 3.2

We observed in section 3.1 that, absent any additional information beyond the common prior, the firm does not invest. For any additional private information of the firm to change the “default” behavior, it has to be that the firm is investing after receiving the good signal, or that

1 q (28) qx 1 q 1 0 x _− q _1 . + ( − )(− ) ≥ ⇔ ≥ q ⇔ ≥ 1 x + In other words, the information has to be sufficiently precise—thus q sufficiently large—to induce a change in the behavior. Conditional on being type g , the firm will have an incentive to invest (when told to invest) under p p , p , p , p if = ( Bb Bg Gb Gg)

1 q pBg (29) _1 1 q p _1 qp x 0 p _− _ , −2( − ) Bg + 2 Gg ≥ ⇔ Gg ≥ q x and an incentive to not invest (when told to not invest) if

1 q pBg 1 q (30) 0 _1 1 q 1 p _1 q 1 p x p _ − _ 1 _− _1 . ≥ −2( − )( − Bg) + 2 ( − Gg) ⇔ Gg ≥ q x + − q x

A similar pair of incentive constraints apply to the recommendations conditional on being type b. As long as the private information of the firm is sufficiently noisy, orq 1 1 x , the ≤ /( + ) binding constraint is (29) as in the uninformed case; otherwise it is the inequality (30) that determines the conditional probabilities. The obedience conditions for the firm observing a bad type b are derived in an analogous manner. The obedience conditions are defined type by type, and we compute the restrictions on the conditional probabilities averaged across types. Now the decision rule (pBb , pBg , pGb , pGg ) will induce behavior (pB , pG ) integrating over types t b, g . ∈ { } 88 Journal of Economic Literature, Vol. LVII (March 2019)

Additional Results for Section 6.2

For a given Bayesian game (G, S), let E(G, S), respectively, NE(G, S), denote the set of decision rules that can be attained with and without elicitation, respectively.

PROPOSITION 5: For each (G, S), we have

NE(G, S) E(G, S). ⊆ PROOF: Let NE(G, S), and let be an obedient contingent recommendation that induces . To σ ∈ ϕ σ show that E(G, S), we will verify that is incentive compatible. σ ∈ σ Fix player i, types t, t T, and a function : A A. For each strategy b B, take i ′i ∈ i δi i → i i ∈ i a (b (t )) in player i’s (t, b ) publicly feasible obedience constraint. Then sum the result- ′i = δi i ′i i i ing inequalities over b B. After regrouping the summation, we have i ∈ i

ui (bi (ti), b i (t i )), ((bi, b i) | ) (ti, t i | ) ( ) ( − − θ) ϕ − θ π − θ ψ θ t i T∑ i, (bi, ∑ b i) B − ∈ − θ∈Θ( − ∈ )

ui ( i (bi (t ′i )), b i (t i)), ((bi, b i) | ) (ti, t i | ) ( ). ≥ ( δ − − θ) ϕ − θ π − θ ψ θ t i T∑ i, (bi, ∑ b i) B − ∈ − θ∈Θ( − ∈ ) We focus on the term in parentheses on each line. In the first line, group the summation according to the value of (bi (ti ), b i (t i )) and use the fact that induces to obtain − − ϕ σ

ui ((ai, a i ), ) ((ai, a i) | (ti, t i), ). − θ σ − − θ (ai, ∑ a i) A − ∈

In the second line, group the summation according to the value of (bi (t i ), b i (t i )) and use the ′ − − fact that induces to obtain ϕ σ

ui (( i (ai), a i), ) ((ai, a i) | (t i , t i ), ). δ − θ σ − ′ − θ (ai, ∑ a i) A − ∈

Substituting these expressions into the inequality gives player i’s type-ti incentive constraint with deviation t , . Since i, t, t , are all arbitrary, the proof is complete. ′i δi i ′i δi ∎ PROPOSITION 6: Let (G, S) be a Bayesian game with I 1. If |A | 2 and | | 2, then = = Θ = NE(G, S) (G, S). = E PROOF: By proposition 5, it suffices to proveNE (G, S ) E(G, S). First, we simplify the nota- ⊇ tion. Label the states and actions so that {G, B} and A {0, 1}. If either action is weakly Θ = = dominant, the desired result can be verified by directly computingNE (G, S) and E(G, S). Bergemann and Morris: Information Design: A Unified Perspective 89

Therefore, we assume u(1, G ) u(0, G) and u(1, B ) u(0, B) are each nonzero and − − have opposite signs. Then without loss, we may assume the payoffs take the form u(0, G) u(0, B) 0, u(1, B) 1, and u (1, G) x 0.25 Action 1 can be interpreted = = = − = > as investment. We will represent a decision rule by a vector p ( pt )( , t) T, where σ = θ θ ∈Θ× pt (1 | , t). For each signal t T, let θ = σ θ ∈ (G ) (t | G) q(t ) ______ψ π , = (t) π where (t ) (G ) (t | G ) (B ) (t | B ) 0 by assumption. π = ψ π + ψ π > Let p ( pt ) E(G, S). To show that p NE(G, S), we will explicitly construct an = θ ∈ ∈ obedient contingent recommendation that induces p . Let t, t T and set q q(t) and ϕ ′ ∈ = q q(t ) . Since p satisfies the truth-telling constraint, ′ = ′ qp x (1 q) p qp x (1 q) p , Gt − − Bt ≥ Gt′ − − Bt′ q p x (1 q ) p q p x (1 q ) p . ′ Gt′ − − ′ Bt′ ≥ ′ Gt − − ′ Bt Taking (1 q , 1 q) and (q , q) linear combinations of these two inequalities respectively − ′ − ′ yields

(q q ) ( p p ) x 0 and (q q ) ( p p ) 0. − ′ Gt − Gt′ ≥ − ′ Bt − Bt′ ≥

So q(t ) q(t′ ) implies pt pt for {G, B}. In the case q q′ , both inequalities must < θ ≤ θ ′ θ ∈ = hold with equality so

q( p p )x (1 q ) ( p p ), Gt − Gt′ = − Bt − Bt′ and hence p p if and only if p p . Therefore, we can label the signals t, , t Gt ≥ Gt′ Bt ≥ Bt′ 1 … n so that

(31) q( t1 ) q( tn ) and pt pt for B, G. ≤ ⋯ ≤ θ 1 ≤ ⋯ ≤ θ n θ =

To simplify notation, defineq l q( tl ) for each l 1, , n; set pt0 0 and ptn 1 1 for = = … θ = θ + = all . For each k 1, , n 1, define the cutoff strategy b k by θ = … +

k 1 if l k b ( tl ) ≥ = {0 otherwise.

25 First, swap the labels G and B if needed to obtain u (1, G ) u(0, G ) 0. Then rescale the utility function so that − > u(1, G ) u(0, G ) 1. Finally, translate the functions u( , G) and u ( , B) separately so that u (0, G ) u(0, B ) 0. The − = ⋅ ⋅ = = separate translations may change the agent’s preferences over states but not over actions. 90 Journal of Economic Literature, Vol. LVII (March 2019)

1 n 1 In particular, b is unconditional investment and b + is unconditional non-investment. Define the stochastic contingent recommendation : (B) by ϕ Θ → Δ k ptk ptk 1 if b b for some k 1, , n 1 (b | ) θ − θ − = = … + ϕ θ = {0 otherwise.

By (31), ptk ptk 1 0, so ( ⋅ | ) is a probability distribution for each {G, B}. It is easy θ − θ − ≥ ϕ θ θ ∈ to check that induces the decision rule p. ϕ To complete the proof, we verify that is obedient. For each l 1, , n and ϕ = … k 1, , n 1, type t’s expected utility from investing if and only if being recommended = … + l b k is:

Ul|k ql ( pG tk pGtk 1 ) x (1 ql )( pB tk pBtk 1 ) . = − − − − − −

Since both expressions in parentheses are nonnegative, Ul|k is weakly increasing in l. Therefore, for types t with l k, l ≥

Ul|k Uk|k qk pG tk x (1 qk ) pB tk qk pG tk 1 x (1 qk ) pB tk 1 0, ≥ = ( − − ) − ( − − − − ) ≥ where the last inequality holds by truth telling for k 1 and obedience for k 1. Similarly, > = for types t with l k, l <

Ul|k Uk 1|k qk 1 pG tk x (1 qk 1 ) pB tk qk 1 pG tk 1 x (1 qk 1 ) pB tk 1 0. ≤ − = ( − − − − ) − ( − − − − − − ) ≤

The last two inequalities establish the obedience of , so the proof is complete. ϕ ∎

Now we return to the main problem of finding( pBb , pBg , pGb , pGg ). To compare these decision rules to the benchmark, we will ultimately integrate over the signals to compute the probability of investment in each state. Formally,

( p , p , p , p ) (1 q) p qp , qp (1 q) p . Bb Bg Gb Gg → ( − Bg + Bb Gg + − Gb)

With an informed receiver, the omniscient designer faces four obedience constraints:

1 q (32a) 1 qp x (1 q) p q p y _− p 0, Δg ≜ Gg − − Bg − ( Gg − 2 Bg) ≥ q (32b) 1 (1 q) p x qp (1 q) p y _ p 0, Δb ≜ − Gb − Bb − ( − Gb − 2 Bb) ≥ 1 q (32c) 0 q(1 p ) y _− (1 p ) 0, Δg ≜ − Gg − 2 − Bg ≤ q (32d) 0 (1 q )(1 p ) y _ (1 p ) 0. Δb ≜ − − Gb − 2 − Bb ≤ Bergemann and Morris: Information Design: A Unified Perspective 91

More precisely, 1 denotes the difference in type t’s utility between investing large and small Δt when told to invest large and 0 denotes the difference in type t’s utility between investing Δt small and not investing when told not to invest. We thus have only ruled out profitable truthful deviations to the small investment, but it can be shown that this implies that there are no profitable truthful deviations to not invest or to invest large. An information designer with elicitation faces four additional constraints ruling out non-truthful deviations:

(33a) qp x (1 q) p qp x (1 q) p , Gg − − Bg ≥ Gb − − Bb 1 p (33b) qp x (1 q) p q( p x (1 p ) y ) (1 q) _+ Bb , Gg − − Bg ≥ Gb + − Gb − − 2 (33c) (1 q) p x qp (1 q) p x qp , − Gb − Bb ≥ − Gg − Bg q (33d) (1 q) p x qp (1 q) p y _ p . − Gb − Bb ≥ − Gg − 2 Bg

Again, it is sufficient to consider a smaller class of deviations because the high type finds investment more attractive than the low type does. Formally, E(G, S) is the set of p [0, 1] 4 ∈ satisfying (32a)–(33d). Now we determine the additional constraints faced by an information designer without elicitation. Since there are only two signals, we may represent each strategy b : T A as an → ordered pair (b(g ) , b(b )) A 2. (In the second component, the letter b is used in two differ- ∈ ent ways, to denote a strategy and a signal.) The strategy b (not invest, invest) can never = be obedient for both types, so for any p NE(G, S), there is only one candidate , namely ∈ ϕ

((0, 0 ) | ) 1 pg , ϕ θ = − θ ((0, 1 ) | ) 0, ϕ θ =

((1, 0 ) | ) pg pb , ϕ θ = θ − θ

((1, 1 ) | ) pb , ϕ θ = θ for each {B, G}. A designer without elicitation faces two additional obedience constraints, θ ∈ which prevent deviations following the recommendation (invest, not invest):

1 q (34a) q( p p ) x (1 q ) ( p p ) q( p p ) y _− ( p p ) , Gg − Gb − − Bg − Bb ≥ Gg − Gb − 2 Bg − Bb q (34b) (1 q ) ( p p ) y _ ( p p ) 0. − Gg − Gb − 2 Bg − Bb ≤

Formally, NE(G, S) is the set of decision rules in E(G, S) satisfying (34a) and (34b). After some algebra, we can see that (34a) is equivalent to

(35) 1 1 p (2q 1 )(x y ) p (2q 1 ) 2. Δg ≥ Δb + Gb − − + Bb − / 92 Journal of Economic Literature, Vol. LVII (March 2019)

When p puts positive probability on investing after a bad signal, (35) eliminates decision rules for which (32a) has little slack. Similarly, (34b) is equivalent to

(36) 0 0 (1 p ) (2q 1 ) y (1 p )(2q 1 ) 2. Δb ≤ Δg − − Gg − − − Bg − / When p puts positive probability on not investing after a good signal, (36) eliminates decision rules for which (32d) does not have too much slack.

EXAMPLE 1 (Two Agents): Suppose I 2, {B, G}, A A {invest, not invest}, = Θ = 1 = 2 = and ui (ai , a i , ) u( ai , ) with u as in the opening example given by (5). Each player i − θ = θ receives a conditionally independent signal t {g, b} that is correct with probability q 1 2. i ∈ i > / Suppose q q , so that player 1 receives a more accurate signal. Consider the following deci- 1 > 2 sion rule: both players invest if player 1’s signal is good and neither agent invests if player 1’s signal is bad. For x sufficiently near one, this decision rule is incentive compatible. However, it is not even publicly feasible because following any contingent recommendation, player 2’s choice of action will depend on her own signal, not on player 1’s.

EXAMPLE 2 (Three States): Consider the single player, single investment setting of the opening example given by (5), but now split the bad state into two bad states B1 and B2 , each with prior probability 1 4 and the same payoffs as in state B of the original example. Suppose the / agents receive a completely uninformative binary signal t taking values t1 and t2 with equal probability. Consider the following decision rule: type ti invests precisely in states G and Bi . For x (1 2, 1), this decision rule is incentive compatible. It is uniquely induced by recom- ∈ / mending (b( t ) , b( t )) (invest, invest) in state G; (b(t ), b(t )) (invest, not invest) in 1 2 = 1 2 = state B; and (b(t ), b(t )) (not invest, invest) in state B . However, this contingent recom- 1 1 2 = 2 mendation perfectly reveals the state of the world, so the agent can profitably deviate to his first-best strategy of investing if and only if the state is G. Therefore, the decision rule is not publicly feasible obedient.

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