Bayes correlated equilibrium and the comparison of information structures in games

Bergemann, Dirk; Morris, Stephen Article Bayes correlated equilibrium and the comparison of information structures in games Theoretical Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Bergemann, Dirk; Morris, Stephen (2016) : Bayes correlated equilibrium and the comparison of information structures in games, Theoretical Economics, ISSN 1555-7561, The Econometric Society, New Haven, CT, Vol. 11, Iss. 2, pp. 487-522, https://doi.org/10.3982/TE1808 This Version is available at: https://hdl.handle.net/10419/150284 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/3.0/ Theoretical Economics 11 (2016), 487–522 1555-7561/20160487 Bayes correlated equilibrium and the comparison of information structures in games Dirk Bergemann Department of Economics, Yale University Stephen Morris Department of Economics, Princeton University A game of incomplete information can be decomposed into a basic game and an information structure. The basic game defines the set of actions, the set of payoff states, the payoff functions, and the common prior over the payoff states. The information structure refers to the signals that the players receive in the game. We characterize the set of outcomes that can arise in Bayes Nash equilibrium if players observe the given information structure but may also observe additional signals. The characterization corresponds to the set of (a version of) incomplete information correlated equilibria, which we dub Bayes correlated equilibria. We identify a partial order on many-player information structures (individual sufficiency) under which more information shrinks the set of Bayes correlated equilibria. This order captures the role of information in imposing (incentive) constraints on behavior. Keywords. Correlated equilibrium, incomplete information, Bayes Nash equilibrium, Bayes correlated equilibrium, robust predictions, information structure, sufficiency, Blackwell ordering. JEL classification. C72, D82, D83. 1. Introduction 1.1 Motivation and results We investigate behavior in a given game of incomplete information, where the latter is described by a “basic game” and by an “information structure.” The basic game refers Dirk Bergemann: [email protected] Stephen Morris: [email protected] We acknowledge financial support through NSF Grants SES 0851200 and ICES 1215808. Earlier versions (Bergemann and Morris 2011) were circulated under the title “Correlated Equilibrium in Games of Incomplete Information.” We are grateful for discussions with Ben Brooks, with whom we are working on related projects, and to Hanming Fang and Ilya Segal, with whom we discussed related ideas for possible joint work in the past. We have benefited from comments from seminar participants and also from Josh Cherry, Mark Dean, Francoise Forges, Matthew Gentzkow, Olivier Gossner, Atsushi Kajii, Emir Kamenica, Qingmin Liu, Daniel Martin, Lones Smith, Eran Shmaya, Bruno Strulovici, Jonathan Weinstein, Alex Wolitsky, and especially Marcin Peski. We acknowledge valuable research assistance from Constantinos Kalfarentzos, Alex Smolin, and Áron Tóbiás. Copyright ©2016 Dirk Bergemann and Stephen Morris. Licensed under the Creative Commons Attribution- NonCommercial License 3.0. Available at http://econtheory.org. DOI: 10.3982/TE1808 488 Bergemann and Morris Theoretical Economics 11 (2016) to the set of actions, the set of payoff states, the utility functions of the players, and the common prior over the payoff states. The information structure refers to the type space of the game, which is generated by a mapping from the payoff states to a probability distribution over types or signals. We ask what might happen in equilibrium if players may have access to additional signals beyond the given information structure. We show in Theorem 1 that behavior corresponds to a Bayes Nash equilibrium for some extra information that the players might observe if and only if it is an incomplete information version of correlated equilibrium that we dub Bayes correlated equilibrium. Adecision rule specifies a distribution over actions for each type profile and payoff state. A decision rule is a Bayes correlated equilibrium if it satisfies an obedience condition: a player does not have an incentive to deviate from the action recommended by the decision rule if he knows only his type and the action recommendation. There are a number of reasons why the notion of Bayes correlated equilibrium and its characterization result are of interest. First, it allows the analyst to identify properties of equilibrium outcomes that are going to hold independently of features of the information structure that the analyst does not know; in this sense, properties that hold in all Bayes correlated equilibria of a given incomplete information game constitute robust predictions. Second, it provides a way to partially identify parameters of the underlying economic environment independently of knowledge of the information structure. Third, it provides an indirect method of identifying socially or privately optimal information structures without explicitly working with a space of all information structures. In Bergemann and Morris (2013b), we illustrate these uses of the characterization result in a particular class of continuum player—linear best response games—focussing on normal distributions of types and actions, and symmetric information structures and outcomes. In this paper, we focus on game theoretic foundations.1 The separation between the basic game and the information structure enables us to ask how changes in the information structure affect the equilibrium set for a fixed basic game. A second contribution of this paper is that (i) we introduce a statistical partial order on information structures—called individual sufficiency—that captures intuitively when one information structure contains more information than another, and (ii) we show that the set of Bayes correlated equilibria shrinks in all games if and only if the informativeness of the information structure increases. Thus, if the information structure of the players contains more information, then a smaller set of outcomes is incentive compatible. To describe the order on information structures, it is useful to note that a one-player version of an information structure is an “experiment” in the sense studied by Blackwell (1951, 1953). An experiment consists of a set of signals and a mapping from states to probability distributions over signals. Suppose that we are interested in comparing a pair of experiments. A combination of the two experiments is a new experiment where a pair of signals—one from each experiment—is observed and the marginal probability over signals from each of the original experiments corresponds to the original distribution over signals for that experiment. One way to characterize the classic sufficiency 1We report an example in the Appendix that illustrates both Bayes correlated equilibrium and these applications in the context of a finite game and thus the setting of this paper. Theoretical Economics 11 (2016) Bayes correlated equilibrium 489 condition of Blackwell (1951) is the following: one experiment is sufficient for another if it is possible to construct a combined experiment such that the former experiment is a sufficient statistic for the combined experiment. Our partial order on (many-player) information structures is a player-by-player generalization of sufficiency. One information structure is individually sufficient for another if there exists a combined information structure where each player’s signal from the former information structure is a sufficient statistic for the state and other players’ signals in the latter information structure. This partial order has a couple of key properties, each generalizing well known properties in the one-player case, that suggest that it is the “right” ordering on (many-player) information structures. First, two information structures are individually sufficient for each other if and only if they have the same canonical representation, where signals are identified with higher-order beliefs about states. Second, one information structure is individually sufficient for another if and only if it is possible to start with the latter information structure and then have each player observe an extra signal, so that the expanded information structure has the same canonical representation as the former information structure. We analyze an “incentive ordering” on information structures: an information structure is more incentive constrained than another if it gives rise to a smaller set of Bayes correlated equilibria. Our main result, Theorem 2, is that one information structure is more incentive constrained than another if and only if the former is individually sufficient for the latter. Thus we show the equivalence between a statistical ordering and an incentive ordering. Blackwell’s theorem showed that if one experiment was sufficient for another, then making decisions based on the former experiment allows a decision maker to attain a richer set of outcomes, and thus higher ex ante utility. Thus Blackwell’s theorem showed the equivalence of a “statistical ordering” on experiments (sufficiency) and a “feasibility ordering” (more valuable than). Our main result, restricted to the one-player case, has a natural interpretation and shows an equivalence between a statistical ordering and an incentive ordering, and thus can be seen as an extension of Blackwell’s theorem. To further understand the connection to Blackwell’s theorem, we also describe a feasibility ordering on many-player information structures and establish in Theorem 3 that it is equivalent to individual sufficiency and is “more incentive constrained than.” Taken together, our main result and discussion of the relation to Blackwell’s theorem highlight the dual role of information. By making more outcomes feasible, more information allows more outcomes that can occur. By adding incentive constraints, more information restricts the set of equilibrium outcomes that do occur. The same partial order–individual sufficiency, reducing to sufficiency in the one-player case, captures both roles of information simultaneously. 1.2 Related literature Hirshleifer (1971) showed how information might be damaging in a many-player context because it removed options to insure ex ante. In mechanism design, it is well understood how more information may reduce the set of attainable outcomes by adding incentive 490 Bergemann and Morris Theoretical Economics 11 (2016) constraints. Our result on the incentive constrained ordering can be seen as a formalization of the idea behind the observation of Hirshleifer (1971): we give a general statement of how more information creates more incentive constraints and thus reduces the set of incentive compatible outcomes. Aumann (1974, 1987) introduced the notion of correlated equilibrium in games with complete information and a number of definitions of correlated equilibrium in games with incomplete information have been suggested, notably in Forges (1993, 2006). A maintained assumption in that literature, which we dub join feasibility, is that play can only depend on the combined information of all the players. This restriction is natural under the maintained assumption that correlated equilibrium is intended to capture the role of correlation of the players’ actions but not unexplained correlation with the state of nature. Our different motivation leads us to allow such unexplained correlation. Liu (2015) also relaxes the join feasibility assumption, but imposes a belief invariance assumption (introduced and studied in combination with join feasibility in Forges 1993, 2006), requiring that, from each player’s point of view, the action recommendation that he receives from the mediator not change his beliefs about others’ types and the state. Intuitively, the belief invariant Bayes correlated equilibria of Liu (2015) capture the implications of common certainty of rationality and a fixed information structure, while our Bayes correlated equilibria capture the implications of common certainty of rationality and the fact that the players have observed at least the signals in the information structure, and possibly additional information. Gossner (2000) introduced a partial order on information structures, expressed in statistical terms, that characterized when the set of outcomes that can arise in Bayes Nash equilibrium shrinks going from one information structure to another. We perform the analogous exercise for Bayes correlated equilibrium. The analysis of the Bayes Nash equilibrium inevitably conflated issues of incentives—more information imposes more incentive constraints—and feasibility—more information allows more things to happen. As a result, the statistical partial order of Gossner (2000) never ranks information structures corresponding to different beliefs and higher-order beliefs about the state; it simply characterizes when one information structure permits more correlation than another.2In contrast, the notion of Bayes correlated equilibrium abstracts from feasibility considerations by construction. In fact, we show that mere correlation possibilities are irrelevant in our partial order and information structures are ranked based only on beliefs and higher-order beliefs about the state. Nonetheless, our arguments are closest to those of Gossner (2000), as our main result can be seen as removing feasibility considerations from his main argument. Lehrer et al. (2013) study solution concepts that are intermediate between Bayes correlated equilibrium and Bayes Nash equilibrium, and provide partial characterizations of how the set of equilibrium outcomes vary with the information structure. 2The main result in Gossner (2000) is about complete information games, but our discussion of Gossner (2000) here and in the rest of the paper refers to Section 6 and Theorem 17, which briefly reports the extension to incomplete information. See Cherry and Smith (2012) for an alternative approach to Gossner’s question in the complete information case. Theoretical Economics 11 (2016) Bayes correlated equilibrium 491 Our characterization result also has an important one-player analogue. Consider a decision maker who has access to an experiment, but may have access to more information. The joint distribution of actions, signals, and states that might result in a given decision problem is equal to the set of one-player Bayes correlated equilibria. Such one-player Bayes correlated equilibria have already arisen in a variety of contexts. Kamenica and Gentzkow (2011) consider the problem of cheap talk with commitment (“Bayesian persuasion”). So as to understand the behavior that a sender/speaker can induce a receiver/decision maker to choose, one must first characterize all outcomes that can arise for some committed cheap talk (independent of the objectives of the speaker). This, in our language, is the set of one-player Bayes correlated equilibria in the case of a null experiment (where the information structure of the receiver is simply the common prior over the payoff states and hence contains null additional information). In this sense, our work provides an approach for studying a many receiver version of Kamenica and Gentzkow (2011) where receivers have prior information. Kamenica and Gentzkow (2011) is based on a concavification argument introduced in the study of repeated games by Aumann et al. (1995).3Thus our work can be seen as an extension of Aumann (1987) to environments with incomplete information by extending the analysis of Aumann et al. (1995) to many players and general, many-player, information structures. Our main result concerns an ordering on information structures based on the idea that more information reduces the set of outcomes by imposing more incentive constraints, i.e., an incentive ordering. By contrast, for the one-player case, Blackwell (1951) characterized an order on information structures based on the idea that more information increases the set of feasible outcomes, and thus increases the set of attainable payoffs, i.e., a feasibility ordering.Lehrer et al. (2010) propose a natural way to study feasibility orderings in the many-player case: see what can happen in equilibria in common interest games under different solution concepts. If we look for the best (common) payoff under feasible strategy profiles (under a given solution concept), then more information, by making more outcomes feasible, will lead to a higher maximum common payoff. They characterize the ordering on information structures that increases the maximum payoff in all common interest games, for different solution concepts. The relevant ordering on information structures varies with the feasibility constraints built into the solution concept. It is an easy corollary of the results of Lehrer et al. (2010) that an information structure is individually sufficient for another if and only if, in any common interest game, the maximum payoff attainable in belief invariant Bayes correlated equilibrium (as defined above) is weakly higher under the former information structure than under the latter information structure. Thus our feasibility result, Theorem 3, follows Lehrer et al. (2010, 2013) in showing that the same ordering on infor- 3Aumann et al. (1995) showed that in infinitely repeated zero sum games with one sided uncertainty and without discounting, the outcome of the repeated game is as if the informed player can commit to reveal only certain information about the state in the corresponding static game. They then showed that a concavification of the complete information payoff function yields the complete characterization of the set of feasible payoffs in the one-player game of private information. 492 Bergemann and Morris Theoretical Economics 11 (2016) mation structures that is relevant for incentive orderings is also relevant for feasibility orderings.4 The structure of the remainder of the paper is as follows. In Section 2,wedefinethe notion of Bayes correlatedequilibrium for a general finite game and establish the first result, Theorem 1, showing the relationship betweenBayes correlated equilibria and Bayes Nash equilibria of games with more information. In Section 3,wedescribeamanyplayer generalization of the sufficiency ordering of information structures, dubbed individual sufficiency. We also relate individual sufficiency to beliefs and higher-order beliefs, and illustrate the different notions with binary information structures. In Section 4, we present the second result, Theorem 2, which establishes an equivalence between the incentive based ordering and the statistical ordering. In Section 5,weplaceBayescorrelated equilibrium in the context of the literature on incomplete information correlated equilibrium, discuss the relation to alternative orderings on information structures, including feasibility orderings and Blackwell’s theorem, and show how our results can be used to give a many-player approach to Bayesian persuasion. 2. Bayes correlated equilibrium 2.1 Definition There are Iplayers, 12I, and we write ifor a typical player. There is a finite set of states, , and we write θfor a typical state. A basic game Gconsists of (i) for each player i, a finite set of actions Ai,wherewewriteA=A1×···×AI, and a utility function ui: A×→R, and (ii) a full support common prior ψ∈++().ThusG=((Aiui)I i=1ψ). An information structure Sconsists of (i) for each player i, a finite set of signals (or types) Ti,wherewewriteT=T1×···×TI, and (ii) a signal distribution π:→(T).Thus S=((Ti)I i=1π). A possible (and natural) information structure is the null information structure in which each player’s set of signals Tiis a singleton, Ti={ti}. This corresponds to the situation in which each player has no information over and above the common prior ψ. Together, the basic game Gand the information structure Sdefine a standard incomplete information game. This division of an incomplete information game into the basic game and the information structure has now been widely used (see, for example, Gossner 2000). Adecision rule in the incomplete information game (GS) is a mapping σ: σ:T×→(A) (1) One way to mechanically understand the notion of the decision rule is to view σas the strategy of an omniscient mediator who first observes the realization of θ∈chosen according to ψand the realization of t∈Tchosen according to π(·|θ),andthenpicksan action profile a∈Aand privately announces to each player ithe draw of ai.Forplayers to have an incentive to follow the mediator’s recommendation in this scenario, it would have to be the case that the recommended action aiwas always preferred to any other 4Gossner (2010) also highlights the dual role of information in a different analytic setting. Theoretical Economics 11 (2016) Bayes correlated equilibrium 493 action a iconditional on the signal tithat player ihad received and his knowledge of the recommended action ai. This is reflected in the following “obedience” condition. Definition 1 (Obedience). Decision rule σis obedient for (GS) if, for each i=1I, ti∈Ti,andai∈Ai,wehave  a−it−iθ ψ(θ)π((tit−i)|θ)σ((aia−i)|(tit−i)θ)ui((aia−i)θ) ≥ a−it−iθ ψ(θ)π((tit−i)|θ)σ((aia−i)|(tit−i)θ)ui((a ia−i)θ) for all a i∈Ai. Our definition of Bayes correlated equilibrium requires obedience and nothing else. Definition 2 (Bayes correlated equilibrium). A decision rule σis a Bayes correlated equilibrium (BCE) of (GS) if it is obedient for (GS). If there is complete information, i.e., if is a singleton, then this definition reduces to the Aumann (1987) definition of correlated equilibrium for a complete information game. If Sis the null information structure, then this is essentially the “universal Bayesian solution” of Forges (1993). If, in addition, there is only one player, then this definition reduces to behavior in the concavification problem of Aumann et al. (1995) and the Bayesian persuasion of Kamenica and Gentzkow (2011). We postpone until Section 5 a discussion of these connections and how this definition relates to (and is in general weaker than) other definitions in the literature on incomplete information correlated equilibrium. We provide our motivation for studying this particular definition next. Consider an analyst who had the following knowledge: 1. The basic game Gdescribes actions, payoff functions depending on states, and a prior distribution on states. 2. The players observe at least information structure S, but may observe more. 3. The players’ actions constitute a Bayes Nash equilibrium given the actual information structure. What joint distributions of actions, signals (in the original information structure, S), and states can arise in such an equilibrium? We will formalize this question and show that the answer is the set of Bayes correlated equilibria of (GS). We first note the standard definition of Bayes Nash equilibrium in this setting. A (behavioral) strategy for player iin the incomplete information game (GS) is βi:Ti→ (Ai). 494 Bergemann and Morris Theoretical Economics 11 (2016) Definition 3 (Bayes Nash equilibrium). A strategy profile βis a Bayes Nash equilibrium (BNE) of (G S) if for each i=1I,ti∈Ti,andai∈Aiwith βi(ai|ti)>0,wehave  a−it−iθ ψ(θ)π((tit−i)|θ) j=i βj(aj|tj)ui((aia−i)θ) ≥ a−it−iθ ψ(θ)π((tit−i)|θ) j=i βj(aj|tj)ui((a ia−i)θ) for each a i∈Ai. 2.2 Foundations We want to discuss situations where players observe more information than that contained in a given information structure. To formalize this, we use the concept of combinations of information structures. If we have two information structures S1=(T1π1) and S2=(T2π2), we will say that information structure S∗=(T ∗π∗)is a combination of information structures S1and S2if the combined information structure S∗=(T∗π∗) is obtained by forming a product space of the signals, T∗ i=T1 i×T2 ifor each i,anda signal distribution π∗:→(T1×T2)that preserves the marginal distribution of its constituent information structures. Definition 4 (Combination). The information structure S∗=(T∗π∗)is a combination of information structures S1=(T1π1)and S2=(T2π2)if T∗ i=T1 i×T2 ifor each i and  t2∈T2 π∗(t1t2|θ) =π1(t1|θ) for each t1∈T1and θ∈  t1∈T1 π∗(t1t2|θ) =π2(t2|θ) for each t2∈T2and θ∈ Note that the above definition places no restrictions on whether signals t1∈T1and t2∈T2are independent or correlated, conditional on θ,underπ∗. Thus any pair of information structures S1and S2will have many combined information structures. Definition 5 (Expansion). An information structure S∗is an expansion of S1if S∗is a combination of S1and some other information structure S2. Suppose strategy profile βwas played in (GS∗),whereS∗is a combination of two information structures S1and S2. Now, if the analyst did not observe the signals of the combined information structure S∗, but only the signals of S1, then the behavior under Theoretical Economics 11 (2016) Bayes correlated equilibrium 501 the latter given the former: θ0t0t 0θ0t1t 0θ0t1t 1θ1t1t 0θ1t1t 1 t0t 010000 t1t 0001 3 2 30 t1t 101 6 1 602 3  Now from the above table, we can see that combined types t1t 0and t1t 1both assign probability 1 3to state θ0(and 2 3to state θ1), and thus cannot be distinguished on the basis of their first order beliefs. But we also see that combined types t1t 0and t1t 1both assign probability 1 3to the event that θ=θ0and the other player assigning probability 1 3to state 0. Thus combined types t1t 0and t1t 1cannot be distinguished on the basis of their second order beliefs, and so on. ♦ 4. Comparing information structures Giving players more information will generate more obedience constraints and thus reduce in size the set of Bayes correlated equilibria. If “giving players more information” is interpreted to mean that we expand their information structures, allowing them to keep their previous signals and observe more, then this claim follows trivially from the definition and characterization of Bayes correlated equilibria in Section 2.Inthissection, we strengthen this observation by showing that it is also true if by “giving player more information,” we mean that we replace their information structure with one that is individually sufficient for it. And we prove a converse, showing that if an information structure, S, is not individually sufficient for another, S, then there exists a basic game G such that (GS) has a Bayes correlated equilibrium that generates outcomes that could not arise under a Bayes correlated equilibrium of (GS). So as to compare outcomes across information structures, we will be interested in what can be said about actions and states if signals are not observed. We will call a mapping ν:→(A) an outcome and say νis induced by decision rule σif it is the marginal of σon A,sothat ν(a|θ)  = t∈T σ(a|tθ)π(t|θ) for each a∈Aand θ∈.Outcomeνis a Bayes correlated equilibrium outcome of (GS) if it is induced by a Bayes correlated equilibrium decision rule σof (GS). We now define a partial order on information structures that corresponds to shrinking the set of BCE outcomes in all basic games. Writing BCE(GS) for the set of BCE outcomes of (GS), we can state the following definition. Definition 7 (Incentive constrained). Information structure Sis more incentive constrained than information structure Sif, for all basic games G, BCE(GS) ⊆BCE(GS) 502 Bergemann and Morris Theoretical Economics 11 (2016) We call this partial order more incentive constrained than because, given our definition of Bayes correlated equilibrium, it captures exactly the role of information in imposing more incentive constraints. Thus an information structure giving rise to a smaller set of Bayes correlated equilibria in all games corresponds to a more informed information structure. By contrast, if we replaced Bayes correlated equilibrium in the above definition with Bayes Nash equilibrium—which corresponds to the problem studied by Gossner (2000)—a smaller set of Bayes Nash equilibria corresponds to a less informed information structure as the notion of Bayes Nash equilibrium imposes strong feasibility constraints on the outcome mapping. Theorem 2. Information structure Sis individually sufficient for information structure Sif and only if Sis more incentive constrained than S. We report an example illustrating the theorem in the Appendix. To prove the result, we first show constructively that if Sis individually sufficient for Sand νis a BCE outcome of (GS), then we can use the BCE decision rule inducing ν and the combined information structure establishing individual sufficiency to construct a decision rule of (GS)that induces ν. The incentive constraints under Sare averages of the incentive constraints under S, and therefore the incentive compatibility of the original decision rule for (GS) is preserved for (GS).Versionsofthisargumenthave been used by Gossner (2000),Lehrer et al. (2013),andLiu (2015) to prove similar claims working with different solution concepts and orderings on information structures. To prove the converse, we consider, for any information structure S,aparticularbasic game Gand a particular BCE outcome νof (GS).IfSis more incentive constrained than S, that particular νmust also be a BCE outcome of (GS). We then show that our choice of Gand νimply that if νis a BCE outcome of (GS), there must exist a combined information structure establishing that Sis individually sufficient for S. To show this, we use the basic game Gwhere each player iis asked to report either atypeinTi(which is associated under Swith a belief over T−i×)or an arbitrary belief over T−i×(which does not in general correspond to an element of Ti). Players are then given an incentive to truthfully report their beliefs over T−i×(which may or may not correspond to a type in Ti) using a quadratic scoring rule. There is a BCE of (GS) where players “truthfully” report their types in S. This BCE thus induces the outcome π:→(T ). Now consider any decision rule σfor (GS) that induces the same outcome π. Combining π:→(T)and σ:T×→(T) gives a combined information structure for Sand Swith π∗(t t|θ) =π(t|θ)σ(t|tθ). Obedience of σin the game (GS)now implies that, under the combined information structure, the beliefs of type t iabout (t−iθ)when recommended to take action timust equal the beliefs of tiabout (t−iθ) under information structure Salone. But now we have a combined information structure establishing individual sufficiency. This heuristic argument uses an infinite action basic game, and we are restricted to finite games. In the formal proof, we use finite approximations of this infinite action game and a continuity argument to establish our result. Theoretical Economics 11 (2016) Bayes correlated equilibrium 503 This step has parallels in an similar argument in Gossner (2000).9There are two differences. First, and less substantively, Gossner (2000) allows general games (not just finite games), which changes technical aspects of the argument. More importantly, because Gossner (2000) works with the solution concept of Bayes Nash equilibrium, feasibility constraints matter in the argument and so, in addition to establishing that there is a combined experiment establishing individual sufficiency, the combined experiment must satisfy additional properties reflecting feasibility restrictions not present in our analysis, giving rise to a very different statistical ordering that we will discuss in Section 5.2. Proof of Theorem 2. Suppose that Sis individually sufficient for S.Takeanybasic game Gand any BCE σof (GS).Wewillconstructσ:T×→(A), which is a BCE of (G S)that gives rise to the same outcome as σ.WriteVi(aia iti)for the expected utility for player iunder decision rule σif he is type tiand receives recommendation ai but chooses action a i,sothat Vi(aia iti) = a−i∈A−it−i∈T−iθ∈ ψ(θ)π((tit−i)|θ)σ((aia−i)|(tit−i) θ)ui((a ia−i)θ) Now, by Definition 1, for each i=1I,ti∈Ti,andai∈Ai,wehave Vi(aiaiti)≥Vi(aia iti)(7) for each a i∈Ai. Since Sis individually sufficient for S, there exists a combined information structure satisfying (4). Define σ:T×→(A) by σ(a|tθ)=t∈Tπ∗(tt|θ)σ(a|tθ) π(t|θ) (8) for all (atθ)∈A×T×whenever π(t|θ) > 0(and if π(t|θ) =0, we are free to choose an arbitrary probability distribution σ(a|tθ)). By construction, decision rules σ(a|tθ) and σ(a|tθ) induce the same outcome function ν:→(A).WriteV i(aia it i)for the expected utility for player iunder decision rule σif he is type t iand receives recommendation aibut chooses action a i,sothat V i(aia it i) = a−i∈A−it −i∈T −iθ∈ ψ(θ)π((t it −i)|θ)σ((aia−i)|(t it −i)θ)ui((a ia−i)θ) Now σsatisfies the obedience condition (Definition 1) to be a correlated equilibrium of (G S)if for each i=1I,t i∈T i,andai∈Ai, V i(aiait i)≥V i(aia it i) 9We are grateful to Marcin Peski for clarifying the connection to Gossner (2000), which also suggested a simplification of the proof of Theorem 2. In private communication, Peski has suggested how our proof could be unified with (a finite version of) Gossner (2000). 504 Bergemann and Morris Theoretical Economics 11 (2016) for all a i∈Ai. Condition (4) in the definition of individual sufficiency implies the existence of φi:Ti→(T i)such that φi(t i|ti)π((tit−i)|θ) = t −i π∗((tit−i)(t it −i)|θ) (9) for each t i,ti,t−i,andθ.Now V i(aia it i) = a−i∈A−it −i∈T −iθ∈ ψ(θ)π((t it −i)|θ)σ((aia−i)|(t it −i)θ)ui((a ia−i)θ) = a−i∈A−it −i∈T −iθ∈t∈T ψ(θ)π∗(t t|θ)σ((aia−i)|tθ)ui((a ia−i)θ) [by the definition of σ;see(8)] = a−i∈A−iθ∈t∈T ψ(θ)σ((aia−i)|tθ)ui((a ia−i)θ)  t −i∈T −i π∗(t(t it −i)|θ) (10) = a−i∈A−iθ∈t∈T ψ(θ)σ((aia−i)|tθ)ui((a ia−i)θ)π((tit−i)|θ)φi(t i|ti)[by (9)] = ti∈Ti φi(t i|ti) × a−i∈A−iθ∈t−i∈T−i ψ(θ)π((tit−i)|θ)σ((aia−i)|(tit−i) θ)ui((a ia−i)θ) = ti∈Ti φi(t i|ti)Vi(aia iti) Now for each i=1I,t i∈T i,andai∈Ai, V i(aiait i)= ti∈Ti φi(t i|ti)Vi(aiaiti)[by (10)] ≥ ti∈Ti φi(t i|ti)Vi(aia iti)[by (7) for each ti∈Ti] =V i(aia it i)[by (10)] for each a i∈Ai.Thusσis a BCE of (GS).Byconstructionσand σinduce the outcome ν:→(A). Since this argument started with an arbitrary BCE outcome νof (G S) and an arbitrary G,wehaveBCE(GS) ⊆BCE(GS)for all basic games G. We now show the converse. We first introduce a notion of approximate individual sufficiency. Fix a full support prior ψ∈++().Letλi:Ti→(T−i×) denote the induced belief of type tiabout (t−iθ)under S: λi(t−iθ|ti) =ψ(θ)π((tit−i)|θ)  t−i θψ( θ)π((ti t−i)| θ) Theoretical Economics 11 (2016) Bayes correlated equilibrium 505 For any combined information structure S∗=(T ×Tπ∗)for Sand S,write λπ∗ i(t−iθ|tit i)for the induced beliefs of player iabout (t−iθ) given a combined type (tit i), λπ∗ i(t−iθ|tit i) =t −iψ(θ)π∗((tit−i)(t it −i)|θ)  t−i θ t−iψ( θ)π∗((ti t−i)(t i t −i)| θ)(11) provided that the denominator does not vanish. We say that Sis ε-individually sufficient for Sif there exists a combined information structure S∗=(T ×Tπ∗)with λπ∗ i(t−iθ|tit i)−λi(t−iθ|ti)≤ε for all such t i,ti,t−i,andθthat (11) is well defined. We will now construct a finite basic game such that Gε=((Aiui)I i=1ψ)and an outcome ν∗:→(A) such that (i) ν∗∈BCE(GS) and (ii) ν∗∈BCE(GS)imply that Sis ε-individually sufficient for S.Letibe any ε-grid of (T−i×), i.e., a finite subset of (T−i×) satisfying the property that, for all ξi∈(T−i×),thereexistsξ i∈iwith ξi−ξ i≤ε.Nowforeveryplayeri, let the set of actions be Ai  =i∪Ti.Writeχi(ai)for the belief over T−i×naturally associated with ai,soχi:Ai→(T−i×) is defined by χi(ai) =λi(·|ti)if ai=ti∈Ti ξiif ai=ξi∈i Now let the payoff function of each player ibe ui((aia−i)θ)  =2χi(t−iθ|ai)− t−i∈T−i θ∈(χi( t−i θ|ai))2if a−i=t−i∈T−i 0if a−i/∈T−i Thus if others’ actions are within T−i, utility function uigives player ian incentive to choose an action associated with his true beliefs via a quadratic scoring rule. More precisely, suppose player iassigns probability 1to his opponents choosing a−i∈T−iand, in particular, for some ξi∈(T−i×), assigns probability ξi(t−iθ) to his opponents choosing a−i=t−i∈T−iand the state being θ. The expected payoff to player iwith this belief over A−i×of choosing an action aiwith χi(ai)=ξ iis  t−i∈T−iθ∈ 2ξ i(t−iθ)ξ i(t−iθ)−  t−i∈T−i θ∈ (ξ i( t−i θ))2 =2 t−i∈T−iθ∈ ξi(t−iθ)ξ i(t−iθ)−  t−i∈T−i θ∈ (ξ i( t−i θ))2 =2 t−i∈T−iθ∈ ξi(t−iθ)ξ i(t−iθ)− t−i∈T−iθ∈ (ξ i(t−iθ))2 =(ξi2−ξ i−ξi2) 506 Bergemann and Morris Theoretical Economics 11 (2016) Now the game (GS) has, by construction, a “truth-telling” BCE where each type tialways chooses action ti. This gives rise to an outcome ν∗where ν∗(a|θ) =π(a|θ) if a=t∈T 0otherwise. So ν∗is a BCE outcome of (GS).Forν∗to be a BCE outcome of (GS), there must exist aBCEof(GS),σ:×T→(T), inducing ν∗. Note that formally, the BCE of (GS) is specified by σ:×T→(A),butifa∈Ais such that a/∈T, then the fact that σ induces ν∗implies that  t∈T π(t|θ)σ(a|tθ)=0 Therefore, if π(t|θ) > 0, then σ(a|tθ)=0. Now setting π∗(tt|θ) =π(t|θ)σ(t|tθ) information structure S∗=(T ×Tπ∗)is a combined information structure for Sand S. Obedience constraints imply that λπ∗ i(·|tit i)−λi(·|ti)≤ε Thus Sis ε2-individually sufficient for S. But now Sbeing more incentive constrained than Srequires that BCE(GεS) ⊆ BCE(GεS)for all such games Gε, and thus that Sbe ε2-individually sufficient for S for all ε>0. But because the set of mappings of combined information structures, π∗:→(T ×T),isacompactset,ifSis ε2-individually sufficient for Sfor each ε>0,Sis individually sufficient for S. 5. Discussion 5.1 Obedience and incomplete information correlated equilibrium Aumann (1974, 1987) introduced the notion of correlated equilibrium for complete information games. A correlated equilibrium is a joint distribution over actions such that each player’s action is optimal for that player if all the player knew is the action he is playing and the joint distribution over actions. Bayes correlated equilibrium is the natural incomplete information generalization where we (i) add incomplete information and (ii) require that players’ actions be optimal when they condition on their type as well as their equilibrium action. This is the obedience condition. Thus Bayes correlated equilibrium captures only the role of information in tightening obedience constraints. Theorem 1 formalizes this motivation for studying Bayes correlated equilibrium: the solution concept captures rational behavior given that players have access to the signals in the information structure, but may have additional information. The existing literature on incomplete information correlated equilibrium has focussed on additional restrictions on behavior that capture the idea that players are constrained by what information is available to them. To put our solution concept in context, we report some key feasibility restrictions imposed in the literature. A decision Theoretical Economics 11 (2016) Bayes correlated equilibrium 507 rule σis belief invariant if, for each player i, the probability distribution over player i’s actions that it induces depends only on player i’s type, and is independent of other players’ types and the state. Writing σi:T×→(Ai)for the probability distribution over player i’s actions induced by σ, σi(ai|(tit−i)θ)  = a−i σ((aia−i)|(tit−i)θ) decision rule σis belief invariant for (GS) if, for each player i,σi(ai|(tit−i) θ) is independent of t−iand θ. An equivalent statement is that player i’s beliefs about (t−iθ) conditional on tido not depend on ai. In the language of mediation it says that the mediator’s recommendation does not give a player any additional information about the state and other players’ types. The condition of belief invariance was introduced in this form and so named by Forges (2006). If a decision rule σis belief invariant for (GS), then players have no less but also no more information under σand Sthan under information structure S. If we impose belief invariance as well as obedience on a decision rule, we get a solution concept that was introduced in Liu (2015). Definition 8 (Belief invariant BCE). Decision rule σis a belief invariant Bayes correlated equilibrium of (GS) if it is obedient and belief invariant for (GS). Belief invariant BCE captures the implications of common knowledge of rationality and that players know exactly the information contained in S(and no more) if the common prior assumption is maintained.10 The set of Bayes correlated equilibria of (GS) is the union of all belief invariant BCE of (GS)for all information structures Sthat are individually sufficient for S.Liu (2015) showed that if two information structures have the same canonical representation, then they have the same set of belief invariant Bayes correlated equilibria. This in turn implies that they have the same set of Bayes correlated equilibria. Much of the literature on incomplete information correlated equilibrium started from the premise that an incomplete information definition of correlated equilibrium should capture what could happen if players had access to a correlation device/mediator under the maintained assumption that the correlation device/mediator did not have access to information that was not available to the players. We can describe the assumption formally as follows. Definition 9 (Join feasible). Decision rule σis join feasible for (GS) if σ(a|tθ) is independent of θ. Thus the probability of a profile of action recommendations conditional on the players’ type profile is independent of the state. If join feasibility but not belief invariance is assumed, we get another solution concept as follows. 10Liu (2015) notes that this solution concept can be seen as the common prior analogue of the solution concept of interim correlated rationalizability discussed by Dekel et al. (2007);seeBergemann and Morris (2014b) for a discussion of the rationalizability notion that is the noncommon prior analogue of Bayes correlated equilibrium. 508 Bergemann and Morris Theoretical Economics 11 (2016) Definition 10 (Bayesian solution). Decision rule σis a Bayesian solution of (GS) if it is obedient and join feasible. Join feasibility was implicitly assumed in Forges (1993) and other works, because it was assumed that type profiles exhausted payoff relevant information;11 Lehrer et al. (2010, 2013) explicitly impose this assumption. The Bayesian solution was named by Forges (1993) and it is the weakest version of incomplete information correlated equilibrium she studies. Imposing both join feasibility and belief invariance, we get a solution concept that has been an important benchmark in the literature. Definition 11 (Belief invariant Bayesian solution). Decision rule σis a belief invariant Bayesian solution of (GS) if it is obedient, belief invariant, and join feasible. Forges (2006) introduced this name. The other incomplete information correlated equilibrium solution concepts for an incomplete information game in Forges (1993, 2006)—communication equilibrium, agent normal form correlated equilibrium, and strategic form correlated equilibrium—are all strictly stronger than the belief invariant Bayesian solution, by imposing additional truth-telling constraints (for communication equilibrium), feasible correlation structure constraints (for agent normal form correlated equilibrium), and a combination of the two (for strategic form correlated equilibrium). Forges (1993) also discusses a “universal Bayesian solution,” which corresponds to Bayes correlated equilibrium in the case where Sis degenerate, i.e., there is no prior information structure (beyond the common prior over payoff states). 5.2 Alternative orderings on many-player information structures and their uses If we fix a pair of information structures Sand S, a combined information structure for these two information structures, and a prior on states, we generate a probability distribution on the space T×T×. We can identify a variety of conditional independence properties that we might be interested in on that space: (i) The distribution of t iconditional on tiis independent of θfor each i. (ii) The distribution of t iconditional on tiis independent of (t−iθ)for each i. (iii) The distribution of t iconditional on tiis independent of (t−it −iθ)for each i. (iv) The distribution of tconditional on tis independent of θ. In the one-player case, these four conditions are all equivalent to each other (and to Blackwell’s order). In the many-player case, they are all different from each other. Intuitively, condition (i) requires only that information structure Sconveys no new information to any player about the state; condition (ii) requires that information structure 11The issue is discussed in Section 4.5 of Forges (1993), where she notes how analyzing a “reduced form” game is not innocuous in general. But in many natural economic settings, type profiles do exhaust payoff relevant information and in those cases, there is an equivalence between Bayes correlated equilibria and Bayesian solutions. See Bergemann and Morris (2014b) for a discussion of this issue and economic settings (such as private values environments) where join feasibility is a natural maintained assumption. Theoretical Economics 11 (2016) Bayes correlated equilibrium 509 Sconveys no new information to any player about the state and higher-order beliefs about the state; condition (iii) requires that information structure Sconveys no new information to any player about the state, higher-order beliefs about the state, and redundant signals that other players may be observing; condition (iv) requires that information structure Sconveys no new information about the state to the players collectively (combining their information) that they did not collectively possess before. The exact relation between them is subtle: one can verify that (iii) ⇒(ii) ⇒(i) and (iii) ⇒(iv) but there are no further implications relating these conditional independence properties. In particular, in Example 2 in Section 3, information structure Swas individually sufficient for S, and thus conditional independence (ii) was satisfied, but one can verify that (iv) fails not only in the particular combined information structure used to establish individual sufficiency, but also in any other combined information structure. We can understand the related literature by comparing which conditional independence properties are required and in which combined experiments. We showed that information structure Sgives rise to fewer Bayes correlated equilibrium outcomes than information structure Sif and only if there exists a combined information structure such that (ii) holds. Gossner (2000) asked when information structure Sgives rise to fewer Bayes Nash equilibrium outcomes than information structure S.12 He showed that this is true if and only if there exists a combined information structure such that two conditions hold, namely (ii) t iconditional on tiis independent of (t−iθ) for each player i, and the stronger condition (iii) in its reverse form, i.e., ticonditional on t iis independent of (t−it −iθ) for each i.13 This combination of conditions implies that S and Smust have the same canonical representation. Intuitively, this is because feasibility considerations (implicit in the definition of Bayes Nash equilibrium) require that information structure Scontain at least as much information about beliefs and higherorder beliefs as S, and incentive considerations require that Scontain at least as much information about beliefs and higher-order beliefs as S. However, Gossner’s characterization also requires that Shas more information about redundant information than S, i.e., more correlation possibilities. Thus Gossner does not order information structures with distinct canonical representations and shows how more redundant information, i.e., correlation possibilities, must lead to a larger set of Bayes Nash equilibrium outcomes. We show that redundant information does not affect the set of Bayes correlated equilibrium outcomes and that more payoff relevant information must lead to a smaller set of Bayes correlated equilibrium outcomes. It is an implication of Gossner (2000) that two information structures give rise to the same set of Bayes Nash equilibrium outcomes if and only if there is a single combined information structure where (iii) holds (t iconditional on tiis independent of (t−it −iθ) for each i) and its reverse holds, i.e., ticonditional on t iis independent of (t−it −iθ)for each i.Lehrer et al. (2013) show that this result remains true if the conditional independence properties hold in distinct combined information structures, i.e., there exists one combined information structure where t iconditional on tiis independent of (t−it −iθ) for each i,14 and another combined information structure where ticonditional on t iis 12Gossner and Mertens (2001) and P˛eski (2008) characterize the value of information in zero sum games. 13In this case, Gossner (2000) says that “there is a faithful and compatible interpretation from Sto S.” 14In this case, Lehrer et al. (2013) say that “there is an independent garbling from Sto S.” 510 Bergemann and Morris Theoretical Economics 11 (2016) independent of (t−it −iθ) for each i.ThusLehrer et al. (2013) show that it is without loss of generality to require that the conditional independence properties hold in the same combined information structure.15 Lehrer et al. (2013) also establish analogous results for solution concepts that are intermediate between Bayes Nash equilibrium and Bayes correlated equilibrium. Thus they show that two information structures give rise to the same set of belief invariant Bayesian solution outcomes if and only if there exists a combined information structure where (ii) and (iv) hold,16 and another combined information structure where the reverse properties hold. 5.3 The one-player special case and many-player Bayesian persuasion Our results apply to the case of one player. In the one-player case, a basic game reduces to a decision problem, mapping actions and states to a payoff of the decision maker. An information structure corresponds to an experiment in the sense of Blackwell (1951, 1953). A decision rule in now a mapping from state and signals to probability distributions over actions. A decision rule is a Bayes correlated equilibrium if it is obedient. To interpret obedience, consider a decision maker who observed a signal under the experiment and received an action recommendation chosen according to the decision rule. The decision rule is obedient if he would have an incentive to follow the recommendation. Theorem 1 states that the set of Bayes correlated equilibria for a fixed decision problem and experiment equals the set of decision rules from a decision maker choosing an optimal action with access to that experiment and possibly more information (an expanded experiment). Thus Bayes correlated equilibria capture all possible optimal behavior if the decision maker had access to the fixed experiment and perhaps some additional information. Now consider the case where the original information structure is degenerate (there is only one signal, which represents the prior over the states of the world). In this case, the set of Bayes correlated equilibria correspond to joint distributions of actions and states that could arise under rational choice by a decision maker with any information structure. Kamenica and Gentzkow (2011) consider a problem of Bayesian persuasion. Suppose a “sender” could pick the experiment that the decision maker, the “receiver,” could observe. Kamenica and Gentzkow (2011) characterize the set of joint distributions over states and actions that the sender could induce through picking an experiment and having the decision maker choose optimally. This set is exactly what we label Bayes correlated equilibria. They can then analyze which (in our language) Bayes correlated equilibrium the sender would prefer to induce in a variety of applications. Thus if we want extend Bayesian persuasion to the case of many receivers who have some prior information, the set of Bayes correlated equilibria is the set of outcomes that 15As we noted in footnote 7, this argument can be adapted to show that if Sis individually sufficient for S and Sis individually sufficient for S, we can without loss of generality establish both directions of individual sufficiency using the same combined information structure, and thus the two information structures have the same canonical representation. 16In this case, Lehrer et al. (2013) say that “there is a noncommunicating garbling from Sto S.” Theoretical Economics 11 (2016) Bayes correlated equilibrium 517 Finally, if qfalls below the lower threshold established in (19), then the second-best decision rule σprescribes investment only by one player, but never by both players simultaneously: αG=γG=1α B=ψ(1+yG) (1−ψ)(1−q)γ B=0(20) As expected, we find that both the probability of investment by a player, given by αB,as well as the probability of a joint investment, γB, are increasing in the accuracy q. We ask again which expanded information structures decentralize these secondbest decision rules. As γB<α B, the decision rule σrequires with positive probability investment by one player only. This can only be achieved by private signals that lead to distinct choices by the players with positive probability. The expansion can still be achieved with two additional signals, t bt g, and as before the additional signals refine or split the posterior that each player held at tgin the information structure S.Butimportantly, now the signal realizations cannot be perfectly correlated across the players anymore. Thus if qis not too low, i.e., condition (19) prevails, then the following information structure decentralizes the second-best BCE: π∗(·|θB)t bt btgt btgt g tbt bq00 tgt b00 r tgt g0r1−q−2r π∗(·|θG)t bt btgt g tbt b00 tgt g01  By contrast, if qis sufficiently low, i.e., below the lower bound of (19), then the expanded information structure below decentralizes the BCE: π∗(·|θB)t bt btgt btgt g tbt bq00 tgt b01−q−2rr tgt g0r0 π∗(·|θG)t bt btgt g tbt b00 tgt g01  In either case, the expansion requires private signals in the sense that conditional on receiving a given signal, either (tgt g)or (tgt b), respectively, each player remains uncertain as to the signal received by the other player, i.e., either (tgt b)or (tgt g).Asrequired, the expanded information structure S∗preserves the likelihood distribution ψof the initial information structure S.20 The set of all symmetric Bayes correlated equilibria The above analysis focussed on second-best Bayes correlated equilibria that maximize welfare. We now visualize all symmetric Bayes correlated equilibria in a special case. 20An interesting question that we do not explore in any systematic manner in this paper is what we can say about the relation between Bayes correlated equilibria and the expansions that are needed to support them as Bayes Nash equilibria. Milchtaich (2014) examines properties of devices needed to implement correlated equilibria, and tools developed in his paper might be useful for this task. 518 Bergemann and Morris Theoretical Economics 11 (2016) Figure 1. BNE and set of BCE with low accuracy: q=1 5. Figure 2. BNE and set of BCE with intermediate accuracy: q=11 20 . We stay with a game of strategic substitutes, yj≤0, and the illustrations below are computed for the prior probability of the good state, ψ=1 3and z=2,yG=0,yB=−1 6.Because there is never investment conditional on bad signals, it is enough the focus on the probabilities αGand αBthat any player invests, conditional on good and bad states, respectively, after observing the positive signal tg. Figures 1–3show the set of all values of αGand αBcorresponding to symmetric BCE for low, intermediate, and high levels of accuracy q,namelyq=1 5,11 20 ,and4 5, respectively. The set of Bayes correlated equilibria for the binary games is completely characterized by the obedience constraints (15)and(16), given the parameterized decision rule σ; see (13). The detailed computations for the present example are recorded in Appendix B of Bergemann and Morris (2014a). For all values of q∈[01], the action profile that maximizes the sum of the payoffs is αB=αG=1, the first-best action profile. Every Bayes Nash equilibrium under the given information structure Shas to be located on the 45◦line, as each player cannot Theoretical Economics 11 (2016) Bayes correlated equilibrium 519 Figure 3. BNE and set of BCE with high accuracy: q=4 5. distinguish between the states θBand θGconditional on tg. In fact, the symmetric Bayes Nash equilibrium in the game with strategic substitutes is unique for all levels of q,and depending on the accuracy q, it is either a pure strategy equilibrium with no investment as in Figure 1, a mixed strategy equilibrium with positive probability of investment as in Figure 2, or a pure strategy equilibrium with investment as in Figure 3. By contrast, the second-best BCE, as computed by (20), always yields a strictly positive level of investment in the bad state θB, and one that is strictly higher than in any BNE, unless the BNE itself is a pure strategy equilibrium with investment (following tg); see Figure 3. If we consider an intermediate level of accuracy q, rather than a low level of accuracy q,asinFigure 2, then we find that there is unique mixed BNE, which provides investment with positive probability following tg. The BNE is therefore in the interior of the unit square of conditional investment probabilities (αGαB). By contrast, the secondbest BCE remains at the boundary of the unit square, and yields a strictly higher probability of investment in the bad state than the corresponding Bayes Nash equilibrium. Interestingly, the BNE is in the interior of the set of BCE, when expressed in the space of investment probabilities rather than an extreme point of the set of BCE. If the accuracy of the information structure increases even further (see Figure 3), then conditional on receiving the positive signal tg, it is sufficiently likely that the state is θG, and investment occurs with probability 1even in the Bayes Nash equilibrium. Essentially, the high probability of θG(and resulting high payoffs from investment) more than offsets the low probability of θB(and resulting low payoffs from investment). This first set of illustrations depicts the probabilities of investment conditional on the realization of the positive signal tgand the state θj,j=BG.Butaswevarytheaccuracy q, we are changing the probability of the signal tg, and hence the above figures do not directly represent the probabilities of investment βjconditional on the state θj only, which are simply given by βB=(1−q)αBand βG=αG. The resulting sets of investment probabilities are depicted in Figure 4 for all three levels of q. The set of BCE is shrinking as the information structure S, as represented by q, becomes more accurate. This comparative static illustrates Theorem 2. Because the set of BCE is shrinking, the 520 Bergemann and Morris Theoretical Economics 11 (2016) Figure 4. 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