Information Design in Multi-stage Games
aa r X i v : . [ ec on . T H ] F e b INFORMATION DESIGN IN MULTI-STAGE GAMES
MILTIADIS MAKRIS AND LUDOVIC RENOUA
BSTRACT . This paper generalizes the concept of
Bayes correlated equilibrium (Berge-mann and Morris, 2016) to multi-stage games. We demonstrate the power of our charac-terization results by applying them to a number of illustrative examples and applications.K
EYWORDS : Multi-stage games, information design, communication equilibrium, sequen-tial communication equilibrium, information structures, Bayes correlated equilibrium,revelation principle.JEL C
LASSIFICATION : C73, D82.
Date : March 1, 2021.Ludovic Renou gratefully acknowledges the support of the Agence Nationale pour la Recherche undergrant ANR CIGNE (ANR-15-CE38-0007-01) and through the ORA Project “Ambiguity in Dynamic Envi-ronments” (ANR-18-ORAR-0005). We thank Laura Doval, Stephen Morris, Sujoy Mukerji, Peter Norman,Alessandro Pavan, and Alex Wolitzky for insightful comments and the audiences at the many seminarswe have given. We are particularly indebted to Tristan Tomala for his generosity with time, pointeddiscussions and perspective comments.
1. I
NTRODUCTION
This paper generalizes the concept of
Bayes correlated equilibrium (Bergemann andMorris, 2016) to multi-stage games. In a multi-stage game, a set of players interact overseveral stages and, at each stage, players receive private signals about past and current(payoff-relevant) states, past actions and past signals, and choose actions. Repeatedgames and, more generally, stochastic games are examples of multi-stage games.Consider an analyst, who postulates a multi-stage game, which we call the base game ,but also acknowledges that players may receive additional signals, which can dependon past and current states, past actions, current and past signals (including the pastadditional ones). Which predictions can the analyst make if he does not want to assumea fixed expansion of the base game, i.e., a multi-stage game that differs from the basegame only in that players have fixed additional signals?Bergemann and Morris (2016) address that question within the class of static games.These authors show that the Bayes correlated equilibria of the (static) base game char-acterize all the predictions the analyst can make. (See below for an informal definitionof a Bayes correlated equilibrium.) In many economic applications, however, the interac-tion between the economic agents is best modeled as a dynamic game, where the agentsreceive information over time and have the opportunity to make multiple decisions.As an example, consider the refinancing operations of central banks. Typically, cen-tral banks organize weekly tender auctions to provide short-term liquidities to financialinstitutions. While extensive regulations carefully specify the auction formats centralbanks use, the information the financial institutions and the central banks receive overtime as well as the communication between them are substantially harder to model.An analyst may thus want to postulate a base game, which captures all that is knownto him – auction format, public annoucements, public statistics – and to remain agnos-tic about the private information the financial institutions and central banks have. Inother words, the analyst considers all possible expansions of the base game. Recentcontributions in the econometrics literature on partial identification have adopted suchan approach. See Bergemann, Brooks and Morris (2019), Gualdani and Sinha (2021),Magnolfi and Roncoroni (2017), Syrgkanis, Tamer, and Ziani (2018).
NFORMATION DESIGN IN MULTI-STAGE GAMES 3
Our main contribution is methodological. We derive several generalizations of the con-cept of Bayes correlated equilibrium, where each generalization corresponds to a solu-tion concept for multi-stage games. We focus primarily on the concept of Bayes-Nashequilibrium. While refinements are frequently used in applications, we do so for a sim-ple reason: the logical arguments do not differ from one solution concept to another. Ourmain theorem (Theorem 1) states an equivalence between (i) the set of all distributionsover states and actions induced by all
Bayes-Nash equilibria of all expansions of thebase game, and (ii) the set of all distributions over states and actions induced by all
Bayes correlated equilibria of the base game.At a Bayes correlated equilibrium of the base game, at each stage, an “omniscient” me-diator makes private recommendations of actions to players, conditional on past andcurrent states and signals, past actions and past recommendations. In other words,the mediator makes recommendations at each history of the base game. Moreover, ateach stage, players have an incentive to be obedient, if they have never disobeyed in thepast, and expect others to have been obedient in the past and to continue to be in thefuture. We stress here that the “omniscient” mediator is a metaphor, an abstract entity,which only serves as a tool to characterize all the equilibrium outcomes we can obtainby varying the information structures.The logical arguments are simple. Fix an expansion of the base game and an equilib-rium. We show that we can emulate the equilibrium of the expansion as an equilibriumof an auxiliary mediated game, where a dummy (additional) player makes reports toa mediator and the mediator sends messages to the original players. In that auxil-iary game, the dummy player knows the actions, signals and states and the messagesthe mediator sends are the additional signals of the expansion. We can then apply theclassical revelation principle of Myerson (1986) and Forges (1986) to replicate the equi-librium of the mediated game as a canonical equilibrium of the mediated game, whereplayers are truthful and obedient, provided they have been in the past. At that canonicalequilibrium, the mediator is “omniscient” at truthful histories and players are obedientprovided they have been in the past: we have a Bayes correlated equilibrium. The verysame logic generalizes to a variety of other solution concepts. All we need is a revelationprinciple.
MILTIADIS MAKRIS AND LUDOVIC RENOU
Finally, we provide two illustrations of the broad applicability of our results. In partic-ular, we generalize the characterization of de Oliveira and Lamba (2019). We refer thereader to Section 5 for more details.The closest paper to ours is Bergemann and Morris (2016), henceforth BM. These au-thors characterize the set of distributions over actions and states induced by all Bayes-Nash equilibria of all expansions of static base games, and show the equivalence withthe distributions induced by the Bayes correlated equilibria of the static base games.The present paper generalizes their work to dynamic problems. Three insights emergefrom our generalization.The primitives in BM are a set of payoff-relevant states Ω , a set of base signals S and adistribution p over Ω × S . There are two equivalent definitions of an expansion in staticproblems. The first definition states that an expansion is a set of additional signals M and a joint distribution π over Ω × S × M such that the marginal over Ω × S is p . Thesecond definition states that an expansion is a set of additional signals M and a kernel ξ from Ω × S to probability distributions over M . Both definitions have natural analoguesin multi-stage games, but they stop being equivalent. A first insight of our analysis isthat the work of BM generalizes to dynamic problems with the latter definition, but notwith the former. Intuitively, the latter definition induces a well-defined strategy for themediator in our auxiliary mediated game, while the former might not. (See Section 3.)A second insight of our analysis is that we genuinely need the mediator to make recom-mendations at all histories. To understand the need for this, note that even in dynamicgames where all the states and signals about the states are drawn ex-ante, it wouldnot be enough to have the mediator recommend strategies as a function of the realizedstates and signals at the first stage only. The reason is that players’ signals at interimstages may also provide private information about the actions taken by players in ear-lier stages. For instance, if the base game is a repeated game with imperfect monitoring,a possible expansion is to perfectly inform players of past actions. As a result, if the me-diator could not react to deviations that are unobserved by some players, it might not beable to induce the appropriate beliefs, and thereby actions, on the part of players in thebase game. In fact, as the introductory example (Section 2) demonstrates, applying thedefinition of BM on the strategic form of even the simplest multi-stage games does not characterize what we can obtain by considering all equilibria of all expansions. NFORMATION DESIGN IN MULTI-STAGE GAMES 5
The third insight is that the analysis of BM generalizes to any solution concept forwhich a revelation principle holds. In particular, this is true for the two versions ofperfect Bayesian equilibrium we consider. This is particularly important for many eco-nomic applications. Bargaining problems (e.g., Bergemann, Brooks and Morris, 2015),allocation problems with aftermarkets (e.g., Calzolari and Pavan, 2006, Giovannoni andMakris, 2014, and Dworczak, 2017), dynamic persuasion problems (Ely, 2017 and Re-nault, Solan and Vieille, 2017) are all instances of dynamic problems, where sequentialrationality is a natural requirement.Doval and Ely (2020) is another generalization of the work of BM and nicely comple-ments our own generalization. These authors take as given states, consequences andstate-contingent payoffs over the consequences, and characterize all the distributionsover states and consequences consistent with the players playing according to some extensive-form game. Our work differs from theirs in two important dimensions. First,we take as given the base game (and, thus, the order of moves). In some economic appli-cations, it is a reasonable assumption. For instance, if we think about the refinancingoperations of central banks, the auction format and their frequencies define the basegame. If a first-price auction is used to allocate liquidities, it would not make sense toconsider games, where another auction format is used. In other applications, this is amore problematic assumption. For instance, if we think about Brexit and the negotia-tions between the European Union and the United Kingdom, it was difficult to have awell-defined base game in mind. Second, unlike Doval and Ely, we are able to accom-modate dynamic problems, where the evolution of states and signals is controlled by theplayers through their actions. This is a natural assumption in many economic problems,such as mergers with ex-ante match-specific investments or inventory problems.Finally, this paper contributes to the literature on correlated equilibrium and its gen-eralizations, e.g., communication equilibrium (Myerson, 1986, Forges, 1986), extensive-form correlated equilibrium (von Stengel and Forges, 2008), or Bayesian solution (Forges,1993, 2006). The concept of Bayes correlated equilibrium is a generalization of all thesenotions. Solan (2001) is a notable exception. In stochastic games with players perfectlyinformed of past actions and past and current states, Solan considers general commu-nication devices, where the mediator sends messages to the players as a function ofpast messages sent and received, and the history of the game, i.e., the past actions, and The concept of extensive-form correlated equilibrium was first introduced in Forges (1986). The conceptintroduced in von Stengel and Forges (2008) differs from the one in Forges (1986).
MILTIADIS MAKRIS AND LUDOVIC RENOU the past and current states. Solan’s mediator is omniscient. For that class of games,Solan shows that the set of Bayes correlated equilibrium payoffs is equal to the set ofextensive-form correlated equilibrium payoffs. As we show in Example 4, this equiv-alence does not hold if players are not perfectly informed of past actions. See Forges(1985) for a related result. 2. A N I NTRODUCTORY E XAMPLE
This section illustrates our main results with the help of a simple example. The exampleillustrates a novel and distinctive aspect of information design in dynamic games: Inaddition to providing information about payoff-relevant states, the designer can choosethe information players have about the past actions of others. E.g., in voting problems,the designer can choose how much information to reveal about past votes.
Example 1.
There are two players, labelled 1 and 2, and two stages. Player 1 chooseseither T or B at the first stage, while player chooses either L or R at the second stage.In the base game, player has no information about player ’s choice. Figure 1 depictsthe base game, with the payoff of player 1 as the first coordinate. (2 , (0,1) (3,0) (1,1) T BL R L R F IGURE
1. The base gameSuppose that the information designer wants to maximize player 1’s payoff. Which in-formation structure(s) should it design?To address that question, we generalize the work of Bergemann and Morris (2016) todynamic games. Recall that Bergemann and Morris (2016) prove that the
Bayes cor-related equilibria of a static game characterize all the distributions over outcomes wecan induce by varying the information structure. At a Bayes correlated equilibrium,an omniscient mediator recommends actions, and the players have an incentive to beobedient.A naive idea is to apply the concept of Bayes correlated equilibrium to the strategicform of the base game. In our example, the unique Bayes correlated equilibrium of the
NFORMATION DESIGN IN MULTI-STAGE GAMES 7 strategic form is ( B, R ) with a payoff profile of (1 , . Working with the strategic formis, however, too restrictive. E.g., if the designer perfectly informs player 2 of player 1’saction, the induced game has an equilibrium with outcome ( T, L ) and associated payoff (2 , .Our approach is to have the omniscient mediator recommending actions to the playersnot only at the initial history, but at each history of the dynamic game. In addition,the players must have an incentive to be obedient, provided they have been obedientin the past. This approach generalizes the definition of Bayes correlated equilibria ofBergemann and Morris (2016) to multi-stage games and clearly demonstrates the needto work on the extensive-form games. (Myerson, 1986, has already pointed out theinsufficiency of the strategic form; see Section 2 of his paper.)We now illustrate how our approach works in our example. Since the mediator is omni-scient and makes recommendations at all histories, we need to consider two recommen-dation kernels. The first kernel specifies the probability of recommending an action toplayer 1 at the first stage. The second kernel specifies the probability of recommendingan action to player 2 at the second stage as a function of the action recommended andchosen at the first stage. Players must have an incentive to be obedient. We claim thatthere exist such recommendation kernels with a payoff profile of (5 / , .To see this, assume that the mediator recommends with probability 1/2 player to play T and with the complementary probability to play B at the first stage, and recommendsplayer to play L at the second stage if and only if player was obedient at the firststage. We now prove that the players have an incentive to be obedient.If player ’s recommendation is L , he believes that player has played T with probabil-ity / and thus expects a payoff of if he plays L . He therefore has an incentive to beobedient. If player ’s recommendation is R , we are off the equilibrium path and anyconjecture that puts probability of at least / on player having played B makes R optimal. As for player , he clearly has an incentive to be obedient when his recommen-dation is B since he gets his highest payoff. When his recommendation is T , a deviationto B is unprofitable because this leads player to play R . Thus, we indeed have a Bayescorrelated equilibrium with a payoff profile of (5 / , .To answer our initial question, we now argue that no information structures can giveplayer 1 a payoff higher than / . Since player can always play R , player ’s payoff MILTIADIS MAKRIS AND LUDOVIC RENOU cannot be lower than . Therefore, within the set of feasible payoff profiles, conditionalon player ’s getting a payoff of at least , player ’s highest payoff is / . See Section4.1 for a complete characterization of the Bayes correlated equilibria of our example.Finally, we now explain how we can use the Bayes correlated equilibrium to design aninformation structure, whose associated expansion generates an equilibrium payoff of (5 / , . The idea is simple: Think of recommendations as signals. Accordingly, supposethat there are two equally likely signals, t and b , at the first stage, and two signals l and r at the second stage. Player 1 privately observes the first signal, while player 2observes the signal l if only if either ( t, T ) or ( b, B ) is the profile of signal and actionat the first stage. With such information structure, players have an incentive to playaccording to their signals and, thus, we obtain the payoff profile (5 / , .3. M ULTI - STAGE G AMES AND E XPANSIONS
The model follows closely Myerson (1986). There is a set I of n players, who interactover T < + ∞ stages, numbered to T . (With a slight abuse of notation, we denote T theset of stages.) At each stage, a payoff-relevant state is drawn, players receive privatesignals about past and current states, past private signals and actions, and choose anaction. We are interested in characterizing the joint distributions over profiles of states,actions and signals, which arise as equilibria of “expansions” of the game, i.e., gameswhere players receive additional signals.3.1. The base game.
We first define the base game Γ , which corresponds to the gamebeing played if no additional signals are given to the players. At each stage t , a state ω t ∈ Ω t is drawn, player i ∈ I receives the private signal s i,t ∈ S i,t , which may de-pend probabilistically on the current and past states, past signals and actions, and thenchooses an action a i,t ∈ A i,t . All sets are non-empty and finite.We now introduce some notations. We write A t := × i ∈ I A i,t for the set of actions at stage t and A := × t ∈ T × i ∈ I A i,t for the set of profiles of actions. We let H i,t = A i,t − × S i,t be theset of player i ’s new information at the beginning of stage t ∈ { , . . . , T } , H i, = S i, theset of initial information, and H i,T +1 = A i,T the set of terminal information.We denote p ( h , ω ) the joint probability of ( h , ω ) at the beginning of the first stage and p t +1 ( h t +1 , ω t +1 | a t , h t , ω t ) the joint probability of ( h t +1 , ω t +1 ) at stage t + 1 given that a t isthe profile of actions played at stage t and ( h t , ω t ) is the history of actions played, signals NFORMATION DESIGN IN MULTI-STAGE GAMES 9 received and states realized at the beginning of stage t . We assume perfect recall and,therefore, impose that p t +1 (( b t , s t +1 ) , ω t +1 | a t , h t , ω t ) = 0 if b t = a t .We denote H Ω the subset of × T +1 t =1 ( × i ∈ I H i,t × Ω t ) that consists of all terminal histories ofthe game, with generic element ( h, ω ) . The history ( h, ω ) is in H Ω if and only if thereexists a profile of actions a ∈ A such that p a ( h, ω ) := p ( h , ω ) · Y t ∈ T p t +1 ( h t +1 , ω t +1 | a t , h t , ω t ) > . For any vector ( h, ω ) , we can denote various sub-vectors: h i = ( h i, , . . . , h i,t , . . . , h i,T +1 ) theprivate (terminal) history of player i , h ti = ( h i, , . . . , h i,t ) the private history of player i atstage t , h t = ( h ,t , . . . , h n,t ) the profile of actions played at stage t − and signals receivedat stage t , h t = ( h , . . . , h t ) the history of signals and actions at stage t , ω = ( ω , . . . , ω T ) the profile of realized states, and ω t = ( ω , . . . , ω t ) the profile of states realized up tostage t , with corresponding sets H i = { h i : ( h, ω ) ∈ H Ω for some ω } , H ti = { h ti : ( h, ω ) ∈ H Ω for some ω } , H t = { h t : ( h, ω ) ∈ H Ω for some ω } , H t = { h t : ( h, ω ) ∈ H Ω for some ω } , Ω = { ω : ( h, ω ) ∈ H Ω for some h } , Ω t = { ω t : ( h, ω ) ∈ H Ω for some h } . We write H t Ω t forthe restriction of H Ω to the first t stages. We let b H := × i ∈ I H i and b H t := × i ∈ I H ti . Similarnotations will apply to other sets. If there is no risk of confusion, we will not formallydefine these additional notations.The payoff to player i is u i ( h, ω ) when the terminal history is ( h, ω ) ∈ H Ω . We assumethat payoffs do not depend on the signal realizations, i.e., for any two histories h = ( a, s ) and h ′ = ( a ′ , s ′ ) such that a = a ′ , u i ( h, ω ) = u i ( h ′ , ω ) for all ω , for all i . Throughout, werefer to the signals in S as the base signals.3.2. Expansions.
In an expansion of the base game, at each stage, players receive ad-ditional signals, which may depend probabilistically on past and current states, pastand current signals (including the past additional ones), and past actions. Thus, play-ers can receive additional information not only about the realization of current andpast (payoff-relevant) states (such as the valuations for objects in auction problems) butalso about the past realization of actions (as in repeated games with imperfect moni-toring). Throughout, we use the same notation as in the base game to denote relevantsub-vectors and their corresponding sets. The sets S T +1 and Ω T +1 are defined to be a singleton. This is without loss of generality as we can always redefine the states to include the signals.
Formally, an expansion is a collection of sets of additional private signals ( M i,t ) i,t andprobability kernels ( ξ t ) t such that all sets of additional signals are non-empty and finite, ξ : H × Ω → ∆( M ) , and ξ t : H t × M t − × Ω t → ∆( M t ) for all t ≥ . Intuitively, at eachstage t , player i receives the additional private signal m i,t ∈ M i,t , with ξ t ( m t | h t , m t − , ω t ) the probability of m t when ( h t , m t − , ω t ) is the history of actions, base signals, states, andpast additional signals at the beginning of stage t . We write M for the collection ( M i,t ) i,t and ξ for ( ξ t ) t .Together with the base game Γ , an expansion ( M, ξ ) induces a multi-stage, where ateach stage t , a payoff-relevant state ω t is realized, player i receives the private signal ( s i,t , m i,t ) and takes an action a i,t . To complete the description of the induced multi-stagegame, we let π ( h , m , ω ) := ξ ( m | h , ω ) p ( h , ω ) be the probability of ( h , m , ω ) at thefirst stage and π t +1 ( h t +1 , m t +1 , ω t +1 | a t , h t , m t , ω t ) := ξ t +1 ( m t +1 | h t +1 , m t , ω t +1 ) p t +1 ( h t +1 , ω t +1 | a t , h t , ω t ) the probability of ( h t +1 , m t +1 , ω t +1 ) , when a t is the profile of actions played at stage t and ( h t , m t , ω t ) is the history of actions, signals and states at the beginning of stage t . Witha slight abuse of language, we use the word “expansion” to refer to the collection ofadditional signals and kernels ( M, ξ ) as well as to the multi-stage game Γ π induced byit.We denote HM Ω the set of all terminal histories with ( h, m, ω ) ∈ HM Ω if and only ifthere exists a profile of actions a ∈ A such that π a ( h, m, ω ) := π ( h , m , ω ) · Y t ∈ T π t +1 ( h t +1 , m t +1 , ω t +1 | a t , h t , m t , ω t ) > . We stress that the set of terminal histories HM Ω depends on the expansion chosen.Thus, for a fixed base game and fixed sets of additional messages, different kernels ( ξ t ) t may induce different sets of terminal histories HM Ω . In particular, if ξ t +1 ( m t +1 | h t +1 , m t , ω t +1 ) =0 , then the history ( h t +1 , m t +1 , ω t +1 ) of signals, actions and states up to period t + 1 is notpart of any terminal history in HM Ω . The set M i,T +1 is a singleton. NFORMATION DESIGN IN MULTI-STAGE GAMES 11
In closing, it is worth noting that an expansion ξ induces a collection of kernels π , withthe property that marg H Ω π a = p a for all a ∈ A , that is, X ( m ,...,m T ) (cid:16) π ( h , m , ω ) · Y t ∈ T π t +1 ( h t +1 , m t +1 , ω t +1 | a t , h t , m t , ω t ) (cid:17) = p ( h , ω ) · Y t ∈ T p t +1 ( h t +1 , ω t +1 | a t , h t , ω t ) . ( † )We call this property consistency. In static problems, the converse is also true, i.e., anyconsistent kernel π induces an expansion ξ . However, this equivalence breaks down indynamic problems, as the following example illustrates. Example 2.
There are two stages, Ω = Ω = { , } , and no private signals and actions,i.e., A , S , A and S are singletons (for simplicity, we omit them). The states areuniformly and independently distributed, that is, p ( ω ) = p ( ω | ω ) = 1 / for all ( ω , ω ) .Consider now the following sets of additional signals and kernels: M = { , } , M isa singleton, π ( m , ω ) = 1 / for all ( m , ω ) and π ( ω | m , ω ) = 1 if and only if ω =( ω + m ) (mod 2) . We can think of the second-stage state as the first-stage state plus ashock.We now verify consistency. We have X m π ( m , ω ) π ( ω | m , ω ) = π (( ω − ω ) (mod 2) , ω ) π ( ω | ( ω − ω ) (mod 2) , ω ) = 1 / . The collection of kernels is therefore consistent. Yet, there are no kernels ( ξ , ξ ) suchthat π ( m , ω ) = ξ ( m | ω ) p ( ω ) and π ( ω | m , ω ) = p ( ω | ω ) for all ( m , m , ω , ω ) . Theissue here is that m is not just an additional piece of information about the first-stagestate ω ; it actually causes the second-stage state.It is worth noting that the precise definition of the base game is important in deter-mining whether a consistent kernel induces an expansion or not. To demonstrate this,suppose that both states are drawn at the first stage, so that the additional signal m can now be made contingent on the new first-stage state ω ∗ = ( ω , ω ) . With this reinter-pretation of the base game, we have a well-defined expansion, and our analysis applies.For completeness, we formally write down this alternative multi-stage representation.Let Ω ∗ := Ω × Ω ; all other sets except M are singletons. Let p ∗ ( ω , ω ) = 1 / forall ( ω , ω ) and π ∗ ( m , ( ω , ω )) = 1 / if and only if ω = ( ω + m ) (mod 2) . Finally,observe that if we let ξ ∗ ( m | ( ω , ω )) = 1 if and only if m = ( ω − ω ) (mod 2) , then π ∗ ( m , ( ω , ω )) = ξ ( m | ( ω , ω )) p ∗ ( ω , ω ) for all ( m , ω , ω ) . However, such a reinterpre-tation is not always possible (see, for instance, Example 3). (cid:4)
4. E
QUIVALENCE THEOREMS
This section contains our main results. It provides characterization theorems, whichdiffer by the solution concepts adopted. In section 4.1, we consider first the concept ofBayes-Nash equilibrium. This allows us to present our first characterization theorem inthe simplest possible terms, without cluttering the analysis with issues such as consis-tency of beliefs, sequential rationality, or truthfulness and obedience at off-equilibriumpath histories. As we will see, the main arguments extend almost verbatim to other so-lution concepts. In addition, if we are interested in proving an impossibility result, e.g.,whether efficiency obtains, the weaker the solution concept, the stronger the result. Insection 4.2, we then extend our analysis to two refinements of the concept of Bayes-Nashequilibrium, which all impose sequential rationality.4.1.
A first equivalence theorem.
We first define the concepts of Bayes-Nash equi-librium and Bayes correlated equilibrium. Throughout, we fix an expansion Γ π of Γ . Bayes-Nash equilibrium.
A behavioral strategy σ i is a collection of maps ( σ i,t ) t ∈ T , with σ i,t : H ti M ti → ∆( A i,t ) . A profile σ of behavioral strategies is a Bayes-Nash equilibrium of Γ π if X h , m , ω u i ( h , ω ) P σ,π ( h , m , ω ) ≥ X h , m , ω u i ( h , ω ) P ( σ ′ i ,σ − i ) ,π ( h , m , ω ) , for all σ ′ i , for all i , with P ˜ σ,π ∈ ∆( HM Ω) denoting the distribution over profiles of actions,signals and states induced by ˜ σ and π . We let BN E (Γ π ) be the set of distributions over H Ω induced by the Bayes-Nash equilibria of Γ π .We now state formally the main objective of our paper: we want to provide a characteri-zation of the set S Γ π an expansion of Γ BN E (Γ π ) , i.e., we want to characterize the distributionsover the outcomes H Ω of the base game Γ that we can induce by means of some expan-sion Γ π of the base game, without any reference to particular expansions. To do so, weneed to introduce the concept of Bayes correlated equilibrium of Γ . NFORMATION DESIGN IN MULTI-STAGE GAMES 13
Bayes correlated equilibrium.
Consider the following mediated extension of Γ , de-noted M (Γ) . At each period t , player i observes the private signal h i,t , receives a privaterecommendation ˆ a i,t from an “omniscient” mediator and chooses an action a i,t . We let τ i,t : H ti × A ti → ∆( A i,t ) be an action strategy at period t and write τ ∗ i,t for the obedientstrategy. It is best to view the “omniscient” mediator as an abstraction, which makesit possible to characterize all distributions over outcomes that an information designercan induce.A Bayes correlated equilibrium is a collection of recommendation kernels µ t : H t Ω t × A t − → ∆( A t ) such that τ ∗ is an equilibrium of the mediated game, that is, X h , ω, ˆ a u i ( h , ω ) P µ ◦ τ ∗ ,p ( h , ω , ˆ a ) ≥ X h , ω , ˆ a u i ( h , ω ) P µ ◦ ( τ i ,τ ∗− i ) ,p ( h , ω , ˆ a ) for all τ i , for all i , with P µ ◦ ˜ τ,p denoting the distribution over profiles of actions, basesignals, states and recommendations induced by µ ◦ ˜ τ and p . We let BCE (Γ) be the set ofdistributions over H Ω induced by the Bayes correlated equilibria of Γ . The set BCE (Γ) isconvex.It is instructive to compare the concept of Bayes correlated equilibrium and communi-cation equilibrium (Forges, 1986, Myerson, 1986). In a communication equilibrium, themediator relies on the information provided by the players to make recommendations,while in a Bayes correlated equilibrium it is as if the mediator knows the realized states,actions and base signals prior to making recommendations. Let CE (Γ) be the distribu-tions over H Ω induced by the communication equilibria of Γ . For all multi-stage games Γ , we have that CE (Γ) ⊆ BCE (Γ) since the omniscient mediator can always replicate theForges-Myerson mediator. Since we also have that BN E (Γ) ⊆ CE (Γ) , we have the inclu-sion
BN E (Γ) ⊆ BCE (Γ) . However, it is a priori not clear whether
BN E (Γ π ) ⊆ BCE (Γ) forall expansions Γ π of Γ since players have additional signals in Γ π , while the omniscientmediator of Γ has no additional signals. A consequence of our main result, Theorem 1,is that it is indeed the case. Theorem 1.
We have the following equivalence:
BCE (Γ) = [ Γ π an expansion of Γ BN E (Γ π ) . Theorem 1 states an equivalence between (i) the set of distributions over actions, basesignals and states induced by all Bayes correlated equilibria of Γ and (ii) the set of dis-tributions over actions, base signals and states we can obtain by considering all Bayes-Nash equilibria of all expansions of Γ . It is a revelation principle for information design.Indeed, Theorem 1 states that any distribution over actions, base signals and states adesigner can implement by committing to an information structure is a Bayes corre-lated equilibrium distribution of the base game. We can therefore focus on the Bayescorrelated equilibrium distributions and abstract from the particular information struc-tures implementing them. This mirrors the focus on incentive compatible social choicefunctions in mechanism design theory.Theorem 1 generalizes the work of BM to multi-stage games. It is also worth emphasiz-ing again that our definition of a Bayes correlated equilibrium is weaker than applyingthe definition of BM on the strategic form of the base game, which would amount tomaking recommendations of strategies at the first stage, as a function of the realizedstates and base signals. See the introductory example for an illustration.Unlike BM’s constructive proof, our proof is non-constructive. This approach has twomain advantages: (i) it reveals the main logical arguments, which are somewhat hid-den in constructive proofs, and (ii) its generalization to many other solution conceptsis straightforward. The central arguments are the following. Consider an expansion Γ π of Γ and and equilibrium distribution µ d ∈ BN E (Γ π ) . By definition, there exists aBayes-Nash equilibrium σ of Γ π , which induces µ d . The main idea is to replicate theexpansion Γ π and its equilibrium σ as a Bayes-Nash equilibrium of an auxiliary medi-ated game M ∗ (Γ) , which we now describe. The game M ∗ (Γ) has one additional player,called player 0, and a Forges-Myerson mediator. Player 0 is a dummy player. At the firststage, Nature draws ( h , ω ) with probability p ( h , ω ) , player i observes h i, and player observes ( h , ω ) . Player then reports (ˆ h , ˆ ω ) to the mediator; all other players do notmake reports. The mediator then draws the message m with probability ξ ( m | ˆ h , ˆ ω ) and sends m i, to player i . Player 0 does not receive a message. Finally, player i choosesan action a i, ; player 0 does not take an action. Consider now a history ( a t − , h t − , ω t − ) ofpast actions, signals and states and a history ((ˆ h t − , ˆ ω t − ) , m t − ) of reports and messages.Stage t unfolds as follows: In a supplementary document, we also prove the equivalence with the set of distributions over actions,base signals and states we can obtain by considering all communication equilibria of all expansions of Γ . We provide a fully constructive proof in the supplementary material.
NFORMATION DESIGN IN MULTI-STAGE GAMES 15 - Nature draws ( h t , ω t ) with probability p t ( h t , ω t | a t − , h t − , ω t − ) .- Player i ∈ I observes the signal h i,t and player observes ( h t , ω t ) .- Player reports (ˆ h t , ˆ ω t ) to the mediator. All other players do not make reports.- The mediator draws the message m t with probability ξ t ( m t | ˆ h t , m t − , ˆ ω t ) and sendsthe message m i,t to player i . Player 0 does not receive a message.- Player i takes an action a i,t . Player 0 does not take an action.If player 0 is truthful and each player i ∈ I follows σ i , we clearly have a Bayes-Nashequilibrium of the mediated game M ∗ (Γ) , with equilibrium distribution µ d . From therevelation principle of Forges (1986) and Myerson (1986), there exists a canonical equi-librium, where the mediator recommends actions and players are truthful and obedientprovided that they have been in the past, which implements µ d . At truthful histories,the mediator is omniscient and players have an incentive to be obedient provided theyhave been in the past: this is the Bayes correlated equilibrium.Before applying Theorem 1, two additional remarks are worth making. First, the abovearguments are not limited to the concept of Bayes-Nash equilibrium. The same ar-guments apply to all solution concepts, such as weak perfect Bayesian equilibrium orconditional probability perfect Bayesian equilibrium, which admit a revelation princi-ple. We formally state these equivalences below. Second, the above arguments clearlydemonstrate the role our definition of an expansion plays. It makes it possible for themediator to replicate any expansion as the kernels ξ t are assumed measurable withrespect to the mediator’s histories. With the alternative and weaker definition of an ex-pansion as a consistent information structure, i.e., marg π a = p a for all a , it is no longerguaranteed that the mediator, despite being omniscient, can simulate any expansion, asthe next example illustrates. If at all possible, we would need an even more powerfulmediator. Example 3.
This example is an elaboration on Example 2. The main difference is thatthe second-stage state ω is partially controlled by a single player through his first-stageaction a .We first define the base game. There are a single player, two stages, two actions A = { , } at the first stage, two states Ω = { , } at the second stage, and all other setsare singletons. The probabilities are: p ( ω = 1 | a = 1) = 5 / and p ( ω = 1 | a = 0) =1 / . The player’s payoff is one (resp., zero) if the second-stage state is zero (resp., one),regardless of his action. Consider now the following information structure: M = { , } , M is a singleton, π ( m =1) = 1 / , π ( ω = 1 | a = 1 , m = 1) = 2 / , π ( ω = 1 | a = 0 , m = 1) = 1 , π ( ω = 1 | a =1 , m = 0) = 1 , and π ( ω = 1 | a = 0 , m = 0) = 0 . This information structure is con-sistent, but as in Example 2, there are no kernels ( ξ , ξ ) that induce this informationstructure from the base game.Player’s optimal payoff is / in the game Γ π : the optimal strategy consists in playing a = 1 (resp., a = 0 ) when m = 1 (resp., m = 0 ). The player’s optimal strategyconsists in choosing the action that maximizes the likelihood of the second-stage statebeing . The induced distribution µ over actions and states is µ ( a = 0 , ω = 0) = 1 / , µ ( a = 0 , ω = 1) = 0 , µ ( a = 1 , ω = 0) = 1 / , µ ( a = 1 , ω = 1) = 1 / . This is not a Bayescorrelated distribution. In any Bayes correlated equilibrium, the probability of ( a , ω ) is µ ( a ) p ( ω | a ) and there is no µ that induces the distribution µ . (cid:4) We close this section with an example illustrating how we can apply Theorem 1.
Example 4.
There are two players and two stages. Player 1 is active in the first stageand chooses an action a ∈ A ; player is inactive. Player 2 is active in the secondstage and chooses an action a ∈ A ; player is inactive. There are no base signalsand states, i.e., Ω , S , Ω and S are singletons. We are interested in characterizing thedistributions µ ∈ ∆( A × A ) as we vary the information players have. In particular, thisimplies varying the information player has about the action chosen by player beforechoosing his own action. Formally, we consider expansions ( ξ , ξ ) , where ξ ∈ ∆( M ) and ξ : M × A → ∆( M ) . In words, player receives the additional signal m at the firstperiod and player 2 receives the additional signal m at the second stage, which maydepend on the first-period signal and action ( m , a ) .From Theorem 1, we can restrict attention to the Bayes correlated equilibria of thegame. By definition, ( µ , µ ) is a Bayes correlated equilibrium if: X a ,a , ˆ a , ˆ a u ( a , a ) µ (ˆ a ) τ ∗ ( a | ˆ a ) µ (ˆ a | a , ˆ a ) τ ∗ ( a | ˆ a ) ≥ X a ,a , ˆ a , ˆ a u ( a , a ) µ (ˆ a ) τ ( a | ˆ a ) µ (ˆ a | a , ˆ a ) τ ∗ ( a | ˆ a ) , NFORMATION DESIGN IN MULTI-STAGE GAMES 17 for all τ : A → ∆( A ) , and X a ,a , ˆ a , ˆ a u ( a , a ) µ (ˆ a ) τ ∗ ( a | ˆ a ) µ (ˆ a | a , ˆ a ) τ ∗ ( a | ˆ a ) ≥ X a ,a , ˆ a , ˆ a u ( a , a ) µ (ˆ a ) τ ∗ ( a | ˆ a ) µ (ˆ a | a , ˆ a ) τ ( a | ˆ a ) , for all τ : A → ∆( A ) , with τ ∗ and τ ∗ the obedient strategies. Any Bayes corre-lated equilibrium ( µ , µ ) induces a distribution µ ∈ ∆( A × A ) , given by µ ( a , a ) = µ ( a ) µ ( a | a , a ) for all ( a , a ) . Moreover, it is easy to verify that a distribution µ ∈ ∆( A × A ) is a Bayes correlated distribution if and only if the following two constraintsare satisfied: ( i ) For all a such that P a µ ( a , a ) > , we have X a u ( a , a ) µ ( a | a ) ≥ max a ∈ A min a ∈ A u ( a , a ) . ( ii ) For all a such that P a µ ( a , a ) > , we have X a u ( a , a ) µ ( a | a ) ≥ X a u ( a , a ′ ) µ ( a | a ) , for all a ′ .Condition ( i ) states that if player 1 is recommended to play a , but plays a ′ = a in-stead, the mediator may recommend player to punish player , i.e., to play a ∈ arg min a ′ ∈ A u ( a ′ , a ′ ) . Consequently, any recommendation made to player , which givesplayer a payoff higher than his (pure) maxmin payoff, can be sustained as a Bayes cor-related equilibrium. Condition ( ii ) states that all recommendations the mediator makesto player must be best responses to player 2’s belief about player ’s action.For a concrete example, let us revisit Example 1. The strategic-form game is (player 1is the row player): L RT , , B , , The set of Bayes correlated distributions is given by { µ : µ ( T, L ) ≥ µ ( B, L ) , µ ( B, R ) ≥ µ ( T, R ) , µ ( T, L ) ≥ µ ( T, R ) } . Indeed, if L (resp., R ) is recommended to player 2, player must conjecture that player1 played T (resp., B ) with probability at least / for L (resp., R ) to be a best response. We therefore need µ ( T, L ) ≥ µ ( B, L ) and µ ( B, R ) ≥ µ ( T, R ) . Moreover, the maxminpayoff to player is 1. Therefore, if action T is recommended to player , it must bethat µ ( T, L ) / ( µ ( T, L ) + µ ( T, R )) ≥ , i.e., µ ( T, L ) ≥ µ ( T, R ) . The associated payoffs aredepicted in the figure below (the dark gray triangle): u u F IGURE
2. Feasible payoffs (light grey) and Bayes correlated equilibriumpayoffs (dark grey)For instance, the payoff (5 / , corresponds to the following signaling structure andequilibrium strategies. There are two equally likely signals t and b at the first stage;player is privately told the first-stage signal. There are two signals at the second-stage l and r ; player is privately told the second-stage signal. Player receives l if andonly if ( T, t ) and ( B, b ) are the first-stage profiles of signal and action. An equilibrium ofthat extended game consists in players playing according to their signals. This gives usthe distribution µ ( T, L ) = µ ( B, L ) = 1 / and its associated payoff (5 / , , as required.Finally, note that if we apply the definition of BM to the strategic-form of the game, µ ( B, R ) = 1 is the unique outcome. Indeed, if the mediator recommends a strategy toboth players as a function of the realized signals and states, we simply obtain the corre-lated equilibria of the game, since there are no signals and states (and the strategies arethe actions). This is also the unique distribution induced by the communication equilib-ria of the game. To see this, note that it is never optimal for player 1 to play T whenrecommended to do so. Player 1 can disobey and play B , and report to have played T tothe mediator. (cid:4) Additional Equivalence Theorems.
The objective of this section is to enrich ouranalysis by requiring rational behavior both on and off the equilibrium path. The mainmessage is that Theorem 1 generalizes to stronger solution concepts. This is also the unique distribution induced by the extensive-form correlated equilibria of the game, asdefined by von Stengel and Forges (2008).
NFORMATION DESIGN IN MULTI-STAGE GAMES 19
Weak Perfect Bayesian Equilibrium.
Throughout, we fix an expansion Γ π of Γ .We denote P σ,π ( ·| h t , m t , ω t ) the distribution over HM Ω induced by the profile of behav-ioral strategies σ and the expansion π , given the history ( h t , m t , ω t ) . The distribution P σ,π ( ·| h t , m t , ω t ) is well-defined even if ( h t , m t , ω t ) has zero probability under P σ,π , and itis equal to P σ,π ( ·| h t , m t , ω t ) when P σ,π ( h t , m t , ω t ) > . Intuitively, this distribution repre-sents the beliefs an outside observer has at ( h t , m t , ω t ) if it is conjectured that playerscontinue to follow their equilibrium strategies even after deviations. We adopt the con-vention that P σ,π ( h , m , ω ) := P σ,π ( h , m , ω | h , m , ω ) . At any given history ( h t , m t , ω t ) ,player i ’s expected payoff is U i ( σ | h t , m t , ω t ) := X h , m , ω u i ( h , ω ) P σ,π ( h , m , ω | h t , m t , ω t ) . To complete the description, we need to specify the belief player i has at any privatehistory ( h ti , m ti ) . To do so, we specify a belief system β . Player i believes that the historyis ( h t , m t , ω t ) with probability β ( h t , m t , ω t | h ti , m ti ) at the private history ( h ti , m ti ) . At theprivate history ( h ti , m ti ) , player i ’s expected payoff is therefore: U i ( σ, β | h ti , m ti ) := X h t , m t , ω t U i ( σ | h t , m t , ω t ) β ( h t , m t , ω t | h ti , m ti ) . A profile σ of behavioral strategies is a weak perfect Bayesian equilibrium of Γ π if thereexists a belief system β on H Ω M such that:(i) Sequential rationality:
For all t , for all i , for all ( h ti , m ti ) , U i ( σ, β | h ti , m ti ) ≥ U i (( σ ′ i , σ − i ) , β | h ti , m ti ) , for all σ ′ i .(ii) Belief consistency:
The belief system β is consistent with σ , that is, for all ( h, m, ω ) ∈ HM Ω , for all ( i, t ) , β ( h t , m t , ω t | h ti , m ti ) = P σ,π ( h t , m t , ω t ) P σ,π ( h ti , m ti ) , whenever P σ,π ( h ti , m ti ) > .We let w PBE (Γ π ) be the set of distributions over H Ω induced by the weak perfect Bayesequilibria of Γ π . As before, the objective is to provide a characterization of the sets S Γ π an expansion of Γ w PBE (Γ π ) , i.e., we want to characterize the distributions over the out-comes H Ω of the base game Γ that we can induce by means of some expansion Γ π of the base game, without any reference to particular expansions. To do so, we need tointroduce the concept of weak perfect Bayes correlated equilibrium of Γ . Weal Perfect Bayes Correlated Equilibrium.
We consider mediated extensions M (Γ) of the game Γ , where at each stage the set of recommendations made to a player maybe a strict subset of the set of actions available to the player. Formally, for each privatehistory ( h ti , ˆ a t − i ) of past and current signals s ti , past actions a t − i and past recommenda-tions ˆ a t − i , R i,t ( h ti , ˆ a t − i ) ⊆ A i,t is the set of possible recommendations to player i . We referto the function R i,t as the mediation range of player i at stage t . We denote H ( R ) the setof all terminal histories consistent with the mediation ranges in the mediated extension M (Γ) , i.e., ( h, ω, ˆ a ) ∈ H ( R ) if and only if ( h, ω ) ∈ H Ω and ˆ a i,t ∈ R i,t ( h ti , ˆ a t − i ) for all i , forall t .We denote P µ ◦ τ,p ( ·| h t , ω t , ˆ a t ) the distribution over H ( R ) induced by the profile of strate-gies τ , the recommendation kernels µ and the kernels p , given the history ( h t , ω t , ˆ a t ) . Atany history ( h t , ω t , ˆ a t ) , player i ’s expected payoff is U i ( µ ◦ τ | h t , ω t , ˆ a t ) := X h , ω , ˆ a u i ( h , ω ) P µ ◦ τ,p ( h , ω , ˆ a | h t , ω t , ˆ a t ) . Finally, at any private history ( h ti , ˆ a ti ) , player i ’s expected payoff is: U i ( µ ◦ τ, β | h ti , ˆ a ti ) := X h t , ω t , ˆ a t U i ( µ ◦ τ | h t , ω t , ˆ a t ) β ( h t , ω t , ˆ a t | h ti , ˆ a ti ) , where β is a belief system. We write T ∗ ,ti for the subset of action strategies of player i , where player i is obedient up to (including) stage t . We are now ready to define theconcept of weak perfect Bayes correlated equilibrium.A weak perfect Bayes correlated equilibrium of Γ is a collection of mediation ranges R i,t : H ti × A t − i → A i,t \ {∅} for all ( i, t ) , a collection of recommendation kernels µ t ( h t , ω t , ˆ a t − ) : × i ∈ I R i,t ( h ti , ˆ a t − i ) → [0 , , where P ˆ a t ∈× i ∈ I R i,t ( h ti , ˆ a t − i ) µ t ( h t , ω t , ˆ a t − )[ˆ a t ] = 1 , for all ( h t , ω t , ˆ a t − ) in H ( R ) and a belief system β such that:(i) Obedience:
For all t , for all i , for all private histories ( h ti , ˆ a ti ) such that ˆ a i,t ′ ∈ R i,t ′ ( h t ′ i , ˆ a t ′ − i ) for all t ′ ≤ t , U i ( µ ◦ τ ∗ , β | h ti , ˆ a ti ) ≥ U i ( µ ◦ ( τ i , τ ∗− i ) , β | h ti , ˆ a ti ) , for all τ i ∈ T ∗ ,t − i . NFORMATION DESIGN IN MULTI-STAGE GAMES 21 (ii)
Belief consistency: β is consistent with ( τ ∗ , µ, p ) , that is, for all ( h, ω, ˆ a ) ∈ H ( R ) ,for all ( i, t ) , β ( h t , ω t , ˆ a t | h ti , ˆ a ti ) = P µ ◦ τ ∗ ,p ( h t , ω t , ˆ a t ) P µ ◦ τ ∗ ,p ( h ti , ˆ a ti ) , whenever P µ ◦ τ ∗ ,p ( h ti , ˆ a ti ) > .It is worth pausing over the role of the mediation ranges. A weak perfect Bayes corre-lated equilibrium constrains the mediator to only recommend actions consistent withthe mediation ranges, that is, the only recommendations the mediator can make toplayer i are in R i,t ( h ti , ˆ a t − i ) at history ( h t , ω t , ˆ a t − ) . In addition, players must have anincentive to be obedient at all histories consistent with the mediation ranges. The roleof mediation ranges is precisely to insure that players can be obedient at all historiesof the mediated game. Without constraining the recommendations the mediator canmake, it wouldn’t be possible to insure that players are obedient at all histories. E.g., noplayer would ever have an incentive to play a strictly dominated action. An equivalentformulation is to consider weak perfect Bayesian equilibria of the mediated game, wherethe mediator is omniscient and unconstrained in its recommendations, and players areobedient on path. The drawback of this alternative formulation is that players do nothave to be obedient off path and, therefore, requires to explore all possible behaviors offpath. The advantage is that no mediation ranges are required. We let w PBCE (Γ) be theset of distributions over H Ω induced by the weak perfect Bayes correlated equilibria of Γ .With all these preliminaries done, we can now state our second equivalence result. Theorem 2.
We have the following equivalence: w PBCE (Γ) = [ Γ π an expansion of Γ w PBE (Γ π ) . Theorem 2 states an equivalence between (i) the set of distributions over actions, basesignals and states induced by all weak perfect Bayes correlated equilibria of Γ , and (ii)the set of distributions over actions, base signals and states we can obtain by considering all weak perfect Bayesian equilibria of all expansions of Γ . This explains why the domain of µ t ( h t , ω t , ˆ a t − ) is × i ∈ I R i,t ( h ti , ˆ a t − i ) in our definition. The logic behind Theorem 2 is identical to the the one behind Theorem 1. We can repli-cate any weak perfect Bayesian equilibrium of Γ π as a weak perfect Bayesian equilib-rium of the auxiliary game M ∗ (Γ) and then invoke the revelation principle, which wasrecently proved by Sugaya and Wolitzky (2018, Proposition 2.)We conclude with few additional remarks. First, the set w PBCE (Γ) is convex. To seethis, take two distributions ν and ν ′ in w PBCE (Γ) . It follows from Theorem 2 that thereexist two expansions Γ π and Γ π ′ and two associated weak perfect Bayesian equilibria ( σ, β ) and ( σ ′ , β ′ ) , which induce ν and ν ′ , respectively. Take α ∈ [0 , and consider theexpansion Γ απ +(1 − α ) π ′ , where the information structure π (resp., π ′ ) is drawn with proba-bility α (resp., − α ) and the players are informed about the draw. If players coordinateon σ (resp., σ ′ ) when the drawn information structure is π (resp., π ′ ), we obtain thedistribution αν + (1 − α ) ν ′ . From Theorem 2, it is in w PBCE (Γ) .Second, despite its theoretical shortcomings, we have considered the concept of weakperfect Bayesian equilibrium as our solution concept. We did so for two two main rea-sons. First, it is simple and indeed widely used in applications. Second, it generalizes tocontinuous games, a common assumption in applications. In what follows, we presentanother solution concept, which alleviates some of the theoretical shortcomings of weakperfect Bayesian equilibrium. However, it comes at a cost: it is “harder” to use in appli-cations.4.2.2.
Perfect Bayesian Equilibrium.
An important tool in modeling off-equilibrium pathbeliefs is the concept of conditional probability systems (henceforth, CPS). Fix a finitenon-empty set X . A conditional probability system β on X is a function from X × X \ {∅} to [0 , , which satisfies three properties: for all X, Y, Z with X ⊆ X , Y ⊆ X and ∅ 6 = Z ⊆ X ,(i) β ( Z | Z ) = 1 and β ( X | Z ) = 1 ,(ii) if X ∩ Y = ∅ , then β ( X ∪ Y | Z ) = β ( X | Z ) + β ( Y | Z ) ,(iii) if X ⊆ Y ⊆ Z and Y = ∅ , then β ( X | Z ) = β ( X | Y ) β ( Y | Z ) .Conditional probability systems capture the idea of “conditional beliefs” even after zero-probability events. In particular, if X is the set of terminal histories of a game, a con-ditional probability system induces a belief system, i.e., a belief over histories at eachinformation set of a player. A conditional probability system also captures the beliefs It is well-known that weak perfect Bayesian equilibria may not be subgame perfect, may rely on “irra-tional” beliefs, and may not satisfy the one-shot deviation principle.
NFORMATION DESIGN IN MULTI-STAGE GAMES 23 players have about the strategies and beliefs of others. Finally, using a conditionalprobability system to represent the players’ beliefs imposes that all differences in be-liefs come from differences in information. We refer the reader to Myerson (1986) formore on conditional probability systems. Conditional probability perfect Bayesian equilibrium (Sugaya and Wolitzky(2020)).
We first give an informal definition. A conditional probability perfect Bayesianequilibrium is a profile of strategies and a conditional probability system such that (i) se-quential rationality holds given the belief system induced by the conditional probabilitysystem and (ii) the conditional probability system is consistent with the profile of strate-gies and the data of the game. It is a stronger concept than the concept of weak perfectBayesian equilibrium and a weaker concept than the concept of sequential equilibrium.We now turn to a formal definition.In what follows, we use notation, which parallel the one used in previous definitions,and thus do not rehash formal definitions. A conditional probability perfect Bayesianequilibrium of Γ π is a profile σ of behavioral strategies and a CPS β on HM Ω , whichsatisfy:(i) Sequential rationality:
For all t , for all i , for all ( h ti , m ti ) , U i ( σ, β | h ti , m ti ) ≥ U i (( σ ′ i , σ − i ) , β | h ti , m ti ) , for all σ ′ i .(ii) CPS consistency:
The CPS β is consistent with ( σ, p, ξ ) , that is, for all ( h, m, ω ) ∈ HM Ω , for all ( i, t ) , β ( a t | h t , m t , ω t ) = Y i ∈ I σ i,t ( a i,t | h ti , m ti ) ,β ( h t +1 , ω t +1 | a t , h t , m t , ω t ) = p t +1 ( h t +1 , ω t +1 | a t , h t , ω t ) ,β ( m t +1 | h t +1 , m t , ω t +1 ) = ξ t +1 ( m t +1 | h t +1 , m t , ω t +1 ) . Few comments are worth making. First, to ease notation, we have written β ( a t | h t , m t , ω t ) for β (cid:16)n ( h , m , ω ) ∈ HM Ω : ( a t , h t , m t , ω t ) =( a t , h t , m t , ω t ) o(cid:12)(cid:12)(cid:12)n ( h , m , ω ) ∈ HM Ω : ( h t , m t , ω t ) = ( h t , m t , ω t ) o(cid:17) . Myerson shows that for any conditional probability system β , there exists a sequence of probability mea-sures P n on X such that (i) P n ( { x } ) > for all x ∈ X and (ii) β = lim n P n , that is, β ( X | Y ) = lim n P n ( X ∩ Y ) P n ( Y ) for all X , for all Y = ∅ . We use similar abuse of notation throughout; this should not create any confusion. Sec-ond, the consistency of the CPS implies that β ( h t , m t , ω t | h ti , m ti ) = P σ,π ( h t , m t , ω t ) P σ,π ( h ti , m ti ) , whenever P σ,π ( h ti , m ti ) > . Third, a conditional probability perfect Bayesian equilibriumis subgame perfect. Fourth, since the belief a player has is induced by the CPS, twoplayers with the same information have the same belief. However, the CPS does notimpose a “don’t signal what you don’t know” condition. To do so, we would need torequire the CPS to maintain the relative likelihood of any two histories before and afterplayers taking actions.We let CPPBE (Γ π ) be the set of distributions over H Ω induced by the conditional proba-bility perfect Bayesian equilibria of Γ π .As before, the objective is to provide a characterization of the set S Γ π an expansion of Γ CPPBE (Γ π ) ,i.e., we want to characterize the distributions over the outcomes H Ω of the base game Γ that we can induce by means of some expansion Γ π of the base game, without any refer-ence to particular expansions. To do so, we need to introduce the concept of sequentialBayes correlated equilibrium of Γ . Sequential Bayes correlated equilibrium.
As in the previous section, we considermediated extensions M (Γ) of the game Γ , where at each stage the set of recommenda-tions made to a player may be a strict subset of the set of actions available to the player.We use the same notation and do not rehash them.A feedback rule f := ( f , . . . , f T ) is a deterministic recommendation kernel, which rec-ommends the action f t ( h t , ω t ) at history ( h t , ω t ) ∈ H t Ω t . Note that given f , the history ( h t , ω t ) encodes the profile of recommendations ˆ a t as ( f ( h , ω ) , f ( h , ω ) , . . . , f t ( h t , ω t )) .A feedback rule f is consistent with the mediation ranges R if f i,t ( h t , ω t ) ∈ R i,t ( h ti , ˆ a t − i ) for all i , for all ( h t , ω t ) , for all t , where ˆ a t − is the profile of recommendations encoded by f at ( h t − , ω t − ) . We let F be the set of feedback rules and F ( R ) the subset of feedbackrules consistent with the mediation ranges R .We denote P f ◦ τ,p ( ·| h t , ω t ) the distribution over H ( R ) induced by the profile of strategies τ , the feedback rule f and the kernels p , given the history ( h t , ω t ) . At any history ( h t , ω t ) , NFORMATION DESIGN IN MULTI-STAGE GAMES 25 player i ’s expected payoff is U i ( f ◦ τ | h t , ω t ) := X h , ω u i ( h , ω ) P f ◦ τ,p ( h , ω | h t , ω t ) , when the feedback rule is f . Finally, at any private history ( h ti , ˆ a ti ) , player i ’s expectedpayoff is: U i ( τ, β | h ti , ˆ a ti ) := X h t , ω t ,f U i ( f ◦ τ | h t , ω t ) β ( f, h t , ω t | h ti , ˆ a ti ) , where β is a CPS on F ( R ) × H Ω . Here, we write β ( f, h t , ω t | h ti , ˆ a ti ) for: β (cid:16)n ( f , h , ω ) : ( f , h t , ω t = f, h t , ω t ) o(cid:12)(cid:12)(cid:12)n ( f , h , ω ) : ( f i, ( h , ω ) , . . . , f i,t ( h t , ω t )) = ˆ a ti , h ti = h ti o(cid:17) A communication mechanism µ ∈ ∆( F ) is a sequential Bayes correlated equilibrium ifthere exist mediation ranges R and a conditional probability system β on F ( R ) × H Ω such that:(i) Obedience:
For all t , for all i , for all private histories ( h ti , ˆ a ti ) such that ˆ a i,t ′ ∈ R i,t ′ ( h t ′ i , ˆ a t ′ − i ) for all t ′ ≤ t , U i ( τ ∗ , β | h ti , a ti ) ≥ U i (( τ i , τ ∗− i ) , β | h ti , a ti ) for all τ i ∈ T ∗ ,t − i .(ii) CPS consistency:
For all f, h, ω, t , β ( f, h, ω ) = µ ( f ) P f ◦ τ ∗ ,p ( h, ω ) β ( f, h, ω | ( f , . . . , f t ) , ( h t , ω t )) = β ( f | ( f , . . . , f t ) , ( h t , ω t )) P f ◦ τ ∗ ,p ( h, ω | h t , ω t ) . Few remarks are worth making. First, in a sequential Bayes correlated equilibrium,players have an incentive to be obedient at all histories consistent with the mediationranges. Second, unlike previous definitions, the definition asserts that the omniscientmediator selects a feedback rule f with probability µ , i.e., as if the mediator chooses amixed strategy (and not a behavioral strategy). In addition, the conditional probabilitysystem is required to be consistent with µ . We may wonder whether an equivalent for-mulation exists where the mediator chooses recommendation kernels ( µ t ) t (behavioralstrategies) and consistency is imposed with respect to ( µ t ) t , as we did in the definition ofa weak perfect Bayes correlated equilibrium. As Sugaya and Wolitzky (2020) show, theanswer is unfortunately no. Intuitively, the current formulation allows more flexibilityin choosing beliefs, which is needed for a revelation principle to hold. Third, sequential Bayes correlated equilibria are sequential communication equilibria (Myerson, 1986)of mediated games, where the mediator is omniscient. We let
SBCE (Γ) be the set ofdistributions over H Ω induced by the sequential Bayes correlated equilibria of Γ . Theorem 3.
We have the following equivalence:
SBCE (Γ) = [ Γ π an expansion of Γ CPPBE (Γ π ) . Theorem 3 states an equivalence between (i) the set of distributions over actions, basesignals and states induced by all sequential Bayes correlated equilibria of Γ , and (ii) theset of distributions over actions, base signals and states we can obtain by considering all conditional probability perfect Bayesian equilibria of all expansions of Γ . The logicbehind Theorem 3 and proof are the same as in previous sections. The set SBCE (Γ) isconvex. 5. A
PPLICATIONS
This section presents two simple applications, which are suggestive of the usefulness ofour characterization results.5.1.
Rationalizing dynamic choices.
Suppose that an analyst observes the choices ofa decision-maker over a finite number of periods, but does not observes the informationthe decision-maker had. Suppose, furthermore, that the analyst assumes that the statedoes not change over time. Which profiles of choices can be rationalized?de Oliveira and Lamba (2019) have recently addressed that question. These authorsassume, however, that the information the decision-maker receives over time is inde-pendent of his past actions. Thanks to Theorem 1, we are able to generalize theirresult with little difficulty. Throughout, we follow the terminology of de Oliveira andLamba.We say that the profile of choices ( a ∗ , . . . , a ∗ T ) is rationalizable if there exist a probability p ∈ ∆(Ω) , sets of signals M t and kernels ξ t : A t − × M t − × Ω → ∆( M t ) such that thedecision-maker chooses optimally and ( a ∗ , . . . , a ∗ T ) is optimal for some realizations ( ω, m ) Sequential Bayes correlated equilibria are the subsets of Bayes correlated equilibria, where the media-tor never recommends co-dominated actions, a generalization of the concept of dominance. We refer thereader to Myerson (1986) for more details. With our notation, this is equivalen to requiring that ξ t ( ·| ( a t − , s t ) , m t − , ω t ) = ξ t ( ·| (˜ a t − , s t ) , m t − , ω t ) for all ( a t − , ˜ a t − ) , for all ( s t , m t − , ω t ) . NFORMATION DESIGN IN MULTI-STAGE GAMES 27 of states and signals. We assume the decision-maker payoff function u is known to theanalyst.From Theorem 1, the profile of choices ( a ∗ , . . . , a ∗ T ) is rationalizable if there exists a prob-ability p ∈ ∆(Ω) and a Bayes correlated equilibrium µ such that P µ ◦ τ ∗ ,p ( a ∗ ) > . Recallthat µ is a Bayes correlation equilibrium if: X a, ˆ a,ω u ( a, ω ) P µ ◦ τ ∗ ,p ( a, ˆ a, ω ) ≥ X a, ˆ a,ω u ( a, ω ) P µ ◦ τ,p ( a, ˆ a, ω ) , for all τ . The objective is to derive conditions on the primitives, which guarantee theexistence of such a Bayes correlated equilibrium.We say that D : A → ∆( A ) is a deviation plan if there exists τ such that D ( a , . . . , a T | ˆ a , . . . , ˆ a T ) := T Y t =1 τ t ( a t | (ˆ a , . . . , ˆ a t ) | {z } recommendations , ( a , . . . , a t − ) | {z } choices ) for all (ˆ a, a ) . A deviation plan specifies what the decision-maker would do if he were toface a fixed sequence of recommendations. Definition 1.
The profile a ∗ is surely dominated if there exists a deviation plan D suchthat for all ω , for all a , for all a ′ : u ( a ∗ , ω ) < T X t =1 X b ∈ B ta ∗ u ( b, ω ) D ( b | a ∗ , . . . , a ∗ t , a ′ t +1 , . . . , a ′ T ) ,u ( a, ω ) ≤ T X t =1 X b ∈ B ta u ( b, ω ) D ( b | a , . . . , a t , a ′ t +1 , . . . , a ′ T ) , where B a := ( A \ { a } ) × A × . . . A T , B ta := { ( a , . . . , a t − ) } × ( A t \ { a t } ) × A t +1 × . . . A T forall t ∈ { , . . . , T − } and B Ta = { a } . The set B ta is the set of all profiles of choices, which coincide with a up to period t anddiffer from a at period t . Note that S Tt =1 B ta = A for all a . Intuitively, a profile of choicesis surely dominated if the decision-maker has a deviation plan which guarantees animprovement regardless of the state ω , the period t at which the decision-maker is firstdisobedient, and the subsequent recommendations ( a ′ t +1 , . . . , a ′ T ) . We have the followingcharacterization. Theorem 4.
The profile of choices a ∗ is rationalizable if and only it is not surely domi-nated. To understand Theorem 4, we first rewrite the obedience constraint. Let f = ( f , . . . , f T ) be a feedback rule, i.e., f t : A t − × A t − × Ω → A . A feedback rule specifies a deterministicrecommendation at each history of past actions, recommendations and states. Notethat we voluntarily include past recommendations in the definition of a feedback rule tostress that it is a pure strategy. Let F be the finite set of all feedback rules and F ∗ benon-empty subset of feedback rules, which recommend a ∗ on path.Similarly, we associate a pure strategy τ with an action rule g = ( g , . . . , g T ) , with g t : A t − × A t − × A → A . The action rule g specifies a deterministic pure action at eachhistory of past actions and past and current recommendations. We associate τ ∗ with therule g ∗ , where g ∗ t ( a t − , ˆ a t − , ˆ a ) = ˆ a . Let G be the set of action rules. Thanks to Kuhn’stheorem, we can rewrite the condition for rationalization as: there exists µ ∈ ∆( F × Ω) such that µ ( F ∗ ) > and X f,ω,g X a, ˆ a u ( a, ω ) (cid:16) P f,ω,g ∗ ( a, ˆ a ) − P f,ω,g ( a, ˆ a ) (cid:17) µ ( f, ω ) ν ( g ) ≥ , for all ν ∈ ∆( G ) , where P f,ω,g is the degenerate distribution over actions and recommen-dations induced by the feedback rule f and the action rule g when the state is ω . Withthis rewriting, it is clear that if a profile of choices is surely dominated, then it is notrationalizable. Indeed, sure dominance implies that regardless of f and ω , the decision-maker is better off following the deviation plan D than being obedient. More precisely,since the deviation plan D is induced by a behavioral strategy τ , Kuhn’s theorem statesthat there exists an outcome-equivalent mixed strategy ν , which is a profitable deviationfrom obedience.As for necessity, suppose that a ∗ is not rationalizable. For all µ such that µ ( F ∗ ) > ,there exists ν such the obedience constraint is violated, i.e., sup µ : µ ( F ∗ ) > min ν X f,ω,g X a, ˆ a u ( a, ω ) (cid:16) P f,ω,g ∗ ( a, ˆ a ) − P f,ω,g ( a, ˆ a ) (cid:17) µ ( f, ω ) ν ( g ) < . Since the set of µ such that µ ( F ∗ ) > is non-empty and convex (but not compact) andthe objective is bi-linear in ( µ, ν ) , we can apply Proposition I.1.3 from Mertens, Sorinand Zamir, (2015, p. 6) to obtain: min ν sup µ : µ ( F ∗ ) > X f,ω,g X a, ˆ a u ( b, ω ) (cid:16) P f,ω,g ∗ ( a, ˆ a ) − P f,ω,g ( a, ˆ a ) (cid:17) µ ( f, ω ) ν ( g ) < . Naturally, as we did earlier, we could restrict attention to the histories, which are not excluded by thefeedback rule. E.g., ˆ a = ( f ( ω ) , f ( f ( ω ) , g ( f ( ω )) , ω ) , ... ) and a = ( g ( f ( ω )) , g ( f ( f ( ω ) , g ( f ( ω ))) , ω ) , ... ) NFORMATION DESIGN IN MULTI-STAGE GAMES 29
Hence, there exists ν such that for all µ with µ ( F ∗ ) > , X f,ω,g X a, ˆ a u ( b, ω ) (cid:16) P f,ω,g ∗ ( a, ˆ a ) − P f,ω,g ( a, ˆ a ) (cid:17) µ ( f, ω ) ν ( g ) < . The result follows then immediately by constructing the behavioral strategy τ inducedby ν and its associated deviation plan D and considering all ( f, ω ) .As already alluded to, Theorem 4 generalizes a recent result by de Oliveira and Lamba(2019). Their main result states that the profile a ∗ is rationalizable if and only if it is not truly dominated, with a ∗ being truly dominated if there exists a deviation rule D suchthat: u ( a ∗ , ω ) < P b u ( b, ω ) D ( b | a ∗ ) ,u ( a, ω ) ≤ P b u ( b, ω ) D ( b | a ) , for all ω , for all a .Clearly, if a profile a ∗ is surely dominated, then it is truly dominated. Indeed, if wechoose ( a ′ t +1 , . . . , a ′ T ) to be equal to ( a t +1 , . . . , a T ) for all ( a, t ) , then we recover the condi-tion for true dominance. However, the converse is not true as the example in Table 1demonstrates. There are two states, ω and ω ′ , three actions, ℓ ( eft ) , c ( enter ) , r ( ight ) , andtwo periods. The inter-temporal payoff is the sum of the per-period payoff in Table 1.T ABLE ( ℓ, c ) is rationalizable and truly dominated ℓ c rω ω ′ We now argue that ( ℓ, c ) is truly dominated. Intuitively, since ℓ is strictly dominated,the decision-maker benefits from playing a mixture of c and r instead of ℓ in the firstperiod. More formally, consider the behavioral strategy τ given by τ ( c | ℓ ) = τ ( r | ℓ ) = 1 / , τ ( r | r ) = τ ( c | c ) = 1 and τ = τ ∗ . The induced deviation rule is D ( cℓ | ℓℓ ) = D ( rℓ | ℓℓ ) = D ( cc | ℓc ) = D ( rc | ℓc ) = D ( cr | ℓr ) = D ( rr | ℓr ) = 1 / and D ( a a | ˆ a ˆ a ) = 1 for all other profiles ( a , a ) and (ˆ a , ˆ a ) such that ( a , a ) = (ˆ a , ˆ a ) . It is then easy to verify that ( ℓ, c ) is indeedtruly dominated.Yet, it is not surely dominated and, therefore, is rationalizable. Intuitively, if the decision-maker learns the state after playing ℓ in the first period but does not get any additionalinformation otherwise, he has an incentive to play ℓ . A Bayes correlated equilibrium is as follows: the mediator recommends ℓ at the first period, regardless of the state, andrecommends c (resp., r ) at the second period if and only if the decision-maker has beenobedient and the state is ω (resp., ω ′ ). If the decision-maker disobeys the recommenda-tion, the mediator recommends then either c or r , independently of the state.To conclude, this application illustrates how we can apply our results to derive testableimplications in dynamic decision problems. We stress that our results apply equally todynamic games, including games with evolving states, and thus offer a wider scope forapplications.5.2. Bilateral Bargaining.
We consider a variation on the work of Bergemann, Brooksand Morris (2013). There are one buyer and one seller. The seller makes an offer a ∈ A ⊂ R + to the buyer, who observes the offer and either accepts ( a = 1 ) or rejects ( a = 0 )it. If the buyer accepts the offer a , the payoff to the buyer is ω − a , while the payoffto the seller is a , with ω being the buyer’s valuation (the payoff-relevant state). Weassume that ω ∈ Ω ⊂ R ++ . If the buyer rejects the offer, the payoff to both the seller andthe buyer is normalized to zero. The buyer and the seller are symmetrically informedand believe that the state is ω with probability p ( ω ) > . We assume that the set of offersthe seller can make is finite, but as fine as needed. For future reference, we write ω L for the lowest state, ω − L for the largest offer a strictly smaller than ω L , and ω H for thehighest state.This model differs from Bergemann, Brooks and Morris (2013) in one important aspect.In our model, both the seller and the buyer have no initial private information about thestate, while Bergemann, Brooks and Morris assume that the buyer is privately informedof the state ω . The base game of Bergemann, Brooks and Morris thus corresponds to aparticular expansion of our base game. Similarly, Roesler and Szentes (2017) considerall information structures, where the buyer has some signals about his own valuation(and the seller is uninformed). Unlike these papers, we consider all information struc-tures. In particular, the information the buyer receives may depend on the informationthe seller has received as well as the offer made. In addition, the seller can be betterinformed than the buyer in our model.We characterize the set of sequential Bayes correlated equilibria . A communicationsystem µ is a sequential Bayes correlated equilibrium if there exist mediation ranges ( R , R ) and a conditional probability system β , which jointly satisfy the following con-straints. First, if the omniscient mediator recommends f ( ω ) ∈ R to the seller, the seller NFORMATION DESIGN IN MULTI-STAGE GAMES 31 must have an incentive to be obedient, i.e., X f,ω f ( ω ) f ( f ( ω ) , ω ) β ( f, ω | f ( ω )) ≥ X f,ω a f ( a , ω ) β ( f, ω | f ( ω )) for all a . Second, if the offer made to the buyer is a and the mediator recommends f ( a , ω ) ∈ R ( a ) to the buyer, the buyer must have an incentive to be obedient, i.e., X f,ω ( ω − a ) f ( a , ω ) β ( f, ω | a , f ( a , ω )) ≥ X f,ω ( ω − a )(1 − f ( a , ω )) β ( f, ω | a , f ( a , ω )) . Third, the conditional probability system must be consistent, that is, for all f ∈ F ( R ) ,for all a , a , ω , β ( f, a , a , ω ) = µ ( f ) p ( ω ) { f ( ω ) = a , f ( f ( ω ) , ω ) = a } ,β ( f, a , a , ω | f , a , ω ) = β ( f | f , a , ω ) { ( f ( a , ω ) , ω ) = a } . There are immediate bounds on the equilibrium payoffs: the sum of the buyer andseller’s payoffs is bounded from above by E ( ω ) = P ω p ( ω ) ω , the buyer’s payoff is boundedfrom below by , and the seller’s payoff is bounded from below by ω − L . The followingproposition states that there are, in fact, no other restrictions on equilibrium payoffs. Proposition 1.
The set of sequential Bayes correlated equilibrium payoffs is co { (0 , ω − L ) , (0 , E ( ω )) , ( E ( ω ) − ω − L , ω − L ) } . The set of equilibrium payoffs is depicted in the figure below.seller’s payoff buyer’s payoff ( E ( ω ) − ω − L , ω − L )(0 , E ( ω ))(0 , ω − L ) F IGURE
3. Payoffs at all sequential Bayes correlated equilibriaWe prove this proposition in what follows. As a preliminary observation, note that theconditional probability system puts no restriction on the buyer’s beliefs after observing an off-path offer a , i.e., an offer such that P f,ω µ ( f ) p ( ω ) { f ( ω ) = a } = 0 . To see this,for any conditional probability system, β ( a , ω ) = β ( ω, a | a ) β ( a ) . Moreover, from theconsistency of β , we have that β ( a , ω ) = P f β ( f, a , ω ) = P f µ ( f ) p ( ω ) { f ( ω ) = a } = 0 .Since β ( a ) = 0 , β ( ω, a | a ) is arbitrary and, thus, we can assume that the buyer believesthat the state is ω L with probability one. We refer to those beliefs as the most pessimisticbeliefs. Similarly, there are no restrictions on the buyer’s beliefs after observing an off-path offer a and a recommendation f ( a , ω ) .We are now ready to state how to obtain the payoff profile ( E ( ω ) − ω − L , ω − L ) . We first startwith an informal description. The mediator recommends the seller to offer ω − L , regard-less of the state. If the offer ω − L is made, the mediator recommends the buyer to accept,regardless of the state. If any offer a > ω − L is made, the mediator recommends the buyerto reject the offer, regardless of the state. Since any such offer is off-path, the buyer hasan incentive to be obedient when he believes that the state is ω L with probability one. Aswe have just argued, we can choose a well-defined conditional probability system cap-turing such beliefs. Finally, if any offer a < ω − L is made, the mediator recommends thebuyer to accept, regardless of the state. Formally, the communication system puts prob-ability one to f , given by f ( ω ) = ω − L , f ( a , ω ) = 0 if a > ω − L and f ( a , ω ) = 1 if a ≤ ω − L for all ω . The mediation ranges are R = { ω − L } , R ( a ) = { } if a < ω L , R ( ω L ) ⊆ { , } ,and R ( a ) = { } if a > ω L .We now turn our attention to the two other payoff profiles (0 , E ( ω )) and (0 , ω − L ) . The pro-file (0 , E ( ω )) corresponds to full surplus extraction, which can be obtained with f ( ω ) = ω for all ω and f ( a , ω ) = 1 whenever a ≤ ω (and zero, otherwise). The mediation rangesare R = Ω , R ( a ) = { } if a > ω H , R ( a ) = { } if a < ω L , and R ( a ) = { , } if a ∈ Ω .Lastly, when E ( ω ) ∈ A (which we assume), the profile (0 , ω − L ) is implementable as fol-lows. Consider two feedback rules f and f ′ such that for all ω , f ( ω ) = f ′ ( ω ) = E ( ω ) , f ( a , ω ) = f ′ ( a , ω ) = 0 if a > E ( ω ) , f ( a , ω ) = f ′ ( a , ω ) = 1 if a < E ( ω ) , f ( E ( ω ) , ω ) = 1 while f ′ ( E ( ω ) , ω ) = 0 . Assume that µ ( f ) = ω − L / E ( ω ) , µ ( f ′ ) = 1 − µ ( f ) , and that R = { E ( ω ) } , R ( a ) = { } if a < ω L , R ( a ) = { , } if a = E ( ω ) , and R ( a ) = { } , otherwise.In effect, the mediator recommends the seller to offer E ( ω ) , regardless of the state, andthe buyer to accept that offer with probability ω − L / E ( ω ) , on path. Off-path, we again usethe most pessimistic beliefs to give the seller a payoff of zero, if he deviates. To completethe proof of Proposition 1, it is enough to invoke the bounds on the payoff profiles andthe convexity of the set of sequential Bayes correlated equilibrium payoffs. NFORMATION DESIGN IN MULTI-STAGE GAMES 33
6. C
ONCLUSION
This paper generalizes the concept of Bayes correlated equilibrium to multi-stage gamesand offers two applications, which are suggestive of the usefulness of our characteriza-tion results. The main contribution is methodological.The reader may wonder why we have not considered the concept of sequential equilib-rium. The main reason is that the revelation principle does not hold for this concept. Tobe more precise, Sugaya and Wolitzky (2020) show that the set of sequential communica-tion equilibria of a multi-stage game characterizes the set of equilibrium distributionswe can obtain by considering all mediated extensions of the multi-stage game, wherethe solution concept is sequential equilibrium. However, their definition of a sequen-tial equilibrium treats the mediator as a player and, thus, allows for the mediator totremble. When we consider an expansion and its emulation by a mediator with the me-diated game M ∗ (Γ) , players do not expect the mediator to tremble. If a player observesan unexpected additional signal, that player must believe with probability one that oneof his opponents has deviated. He cannot believe that none of his opponents deviated,but the mediator did. This would be inconsistent with the expansion being the gameactually played. Extending the analysis to other solution concepts such as sequentialequilibrium or rationalizability or to general extensive-form games is challenging andleft for future research. A PPENDICES A. Proof of Theorem 1. ( ⇐ . ) We first prove that S Γ π an expansion of Γ BN E (Γ π ) ⊆ BCE (Γ) .Throughout, we fix an expansion Γ π of Γ . Recall that there exist kernels ( ξ t ) t such that: π t +1 ( h t +1 , m t +1 , ω t +1 | a t , h t , m t , ω t ) = ξ t +1 ( m t +1 | h t +1 , m t , ω t +1 ) p t +1 ( h t +1 , ω t +1 | a t , h t , ω t ) , for all ( h t +1 , m t +1 , ω t +1 ) , for all t .Let σ ∗ be a Bayes-Nash equilibrium of Γ π . We now construct an auxiliary mediated game M ∗ (Γ) , which emulates the distribution P σ ∗ ,π as an equilibrium distribution.The game M ∗ (Γ) has one additional player, labelled player 0, and a (Forges-Myerson)mediator. Player 0 is a dummy player: his payoff is identically zero.The game unfolds as follows: At stage t = 1 , - Nature draws ( h , ω ) with probability p ( h , ω ) .- Player i ∈ I observes the signal h i, and player observes ( h , ω ) .- Player reports (ˆ h , ˆ ω ) to the mediator. All other players do not make reports.- The mediator draws the message m with probability ξ ( m | ˆ h , ˆ ω ) and sends themessage m i, to player i . Player 0 does not receive a message.- Player i takes an action a i, . Player 0 does not take an action.Consider now a history ( a t − , h t − , ω t − ) of past actions, signals and states and a history ((ˆ h t − , ˆ ω t − ) , m t − ) of reports and messages. At stage t :- Nature draws ( h t , ω t ) with probability p t ( h t , ω t | a t − , h t − , ω t − ) .- Player i ∈ I observes the signal h i,t and player observes ( h t , ω t ) .- Player reports (ˆ h t , ˆ ω t ) to the mediator. All other players do not make reports.- The mediator draws the message m t with probability ξ t ( m t | ˆ h t , m t − , ˆ ω t ) and sendsthe message m i,t to player i . Player 0 does not receive a message.- Player i takes an action a i,t . Player 0 does not take an action.In the above description, when we say that player i does not make a report, we implicitlyassume that the set of reports player i can make to the mediator is a singleton. Similarly,when we sat that player 0 does not take an action. In the rest of the proof, we omit thesetrivial reports and actions.At stage t , player i ’s private history is therefore ( h ti , m ti ) , which is also player i ’s privatehistory in Γ π . Thus, σ ∗ i is a valid strategy for player i in M ∗ (Γ) . Moreover, if player truthfully reports his private information ( h t , ω t ) at all histories (( h t , ω t ) , ( h t − , ω t − ) , (ˆ h t − , ˆ ω t − )) ,the conditional probability of the message m t is the same as in Γ π . It follows immediatelythat σ ∗ together with the truthful strategy for player is a Bayes-Nash equilibrium ofthe auxiliary mediated game M ∗ (Γ) .From the revelation principle of Forges (1986) and Myerson (1986), there exists a canon-ical equilibrium µ , where the mediator recommends actions and players are truthful andobedient, provided they have been in the past. At truthful histories, the mediator rec-ommends ˆ a t with probability µ t (ˆ a t | ( h t , ω t ) | {z } player 0 , ( h t , . . . , h tn ) | {z } players in I , ˆ a t − |{z} past recommendations ) . NFORMATION DESIGN IN MULTI-STAGE GAMES 35
It is then routine to verify that we have a Bayes-correlated equilibrium with the recom-mendation kernel µ t given by µ t (ˆ a t | h t , ω t , ˆ a t − ) := µ t (ˆ a t | ( h t , ω t ) , ( h t , . . . , h tn ) , ˆ a t − ) , for all ( h t , ω t , ˆ a t − ) for all t . ( ⇒ ) . We now prove that BCE (Γ) ⊆ S Γ π an expansion of Γ BN E (Γ π ) .Let µ be a Bayes correlated equilibrium with distribution P µ ◦ τ ∗ ,p . We now construct anexpansion Γ π and a Bayes-Nash equilibrium σ ∗ of Γ π , with the property that marg H Ω P σ ∗ ,π = marg H Ω P µ ◦ τ ∗ ,p .The expansion is as follows. Let M i,t = A i,t for all ( i, t ) , π ( h , m , ω ) = p ( h , ω ) µ (ˆ a | h , ω ) , with m = ˆ a , for all ( h , m , ω ) , and π t +1 ( h t +1 , m t +1 , ω t +1 | a t , h t , m t , ω t ) = p t +1 ( h t +1 , ω t +1 | a t , h t , ω t ) µ t +1 (ˆ a t +1 | h t +1 , ω t +1 , ˆ a t ) , with ( m t , m t +1 ) = (ˆ a t , ˆ a t +1 ) , for all ( a t , h t , m t , ω t , h t +1 , m t +1 , ω t +1 ) . Clearly, the expansion iswell-defined: ξ ( m | h , ω ) = µ (ˆ a | h , ω ) with m = ˆ a , and, for t > , ξ t +1 ( m t +1 | h t +1 , m t , ω t +1 ) = µ t +1 (ˆ a t +1 | h t +1 , ω t +1 , ˆ a t ) with ( m t , m t +1 ) = (ˆ a t , ˆ a t +1 ) .By construction, any strategy τ t : H t × A t → ∆( A t ) of M (Γ) is equivalent to a strategy σ t : H t × M t → ∆( A t ) of Γ π , i.e., σ t ( a t | h t , m t ) := × i σ i,t ( a i,t | h ti , m ti ) = × i τ i,t ( a i,t | h ti , ˆ a ti ) with m t = ˆ a t , with the property that P σ,π ( h t , m t , ω t ) = P µ ◦ τ,p ( h t , ˆ a t , ω t ) when m t = ˆ a t , for all ( h t , m t , ω t ) , for all t .To see this last point, note that the definition of π is clearly equivalent to P σ,π ( h , m , ω ) = P µ ◦ τ,p ( h , ω , ˆ a ) with m = ˆ a , for all ( h , m , ω ) . By induction, assume that P σ,π ( h t , m t , ω t ) = P µ ◦ τ,p ( h t , ω t , ˆ a t ) with m t = ˆ a t , for all ( h t , m t , ω t ) . We now compute the probability of ( h t +1 , m t +1 , ω t +1 ) . We have that P σ,π ( h t +1 , m t +1 , ω t +1 ) = P σ,π ( h t +1 , m t +1 , ω t +1 | h t , m t , ω t ) P σ,π ( h t , m t , ω t )= π t +1 ( h t +1 , m t +1 , ω t +1 | a t , h t , m t , ω t ) σ t ( a t | h t , m t ) P σ,π ( h t , m t , ω t )= p t +1 ( h t +1 , ω t +1 | a t , h t , ω t ) µ t +1 (ˆ a t +1 | h t +1 , ω t +1 , ˆ a t ) τ t ( a t | h t , ˆ a t ) P µ ◦ τ,p ( h t , ˆ a t , ω t )= P µ ◦ τ,p ( h t +1 , ω t +1 , ˆ a t +1 ) , with ˆ a t +1 = m t +1 . Finally, since µ is a Bayes correlated equilibrium of M (Γ) , the strategy σ ∗ ≡ τ ∗ is a Bayes-Nash equilibrium of Γ π and, thus, BCE (Γ) ⊆ [ Γ π an expansion of Γ BN E (Γ π ) . This completes the proof.B.
Proof of Theorem 2.
The proof is nearly identical to the proof of Theorem 1 and is,therefore, omitted. We only sketch the minor differences.( ⇐ .) Fix an expansion Γ π and a weak perfect Bayesian equilibrium ( σ ∗ , β ) of Γ π . We needto construct a weak perfect Bayesian equilibrium of the auxiliary game M ∗ (Γ) , whichreplicates the distribution P σ,π . To do so, we define a belief system β ∗ of the auxiliarygame M ∗ (Γ) as follows: β ∗ ( h t , m t , ω t , ( h t , ω t ) , ( h t , ω t ) | h ti , m ti ) := β ( h t , m t , ω t | h ti , m ti ) for all ( h t , m t , ω t ) , for all ( i, t ) . (In M ∗ (Γ) , player i also has beliefs about the signals ( h t , ω t ) player 0 receives and the reports (ˆ h t , ˆ ω t ) by player to the mediator.) It is immediateto verify that ( σ ∗ , σ ∗ , β ∗ ) is a weak perfect Bayesian equilibrium of M ∗ (Γ) , where σ ∗ isthe truthful reporting strategy of player 0. The proof then follows from the revelationprinciple for weak perfect Bayesian equilibrium. The mediation ranges and the beliefsystem come from the revelation principle.( ⇒ .) We construct the expansions as in the the proof of Theorem 1, i.e., defining theadditional signals as the recommendations. Since the additional signals player i canreceive are the recommendations, player i can only receive additional signals consistentwith the mediation ranges. Thus, we can use the belief system of the weak perfect Bayescorrelated equilibrium to construct the weak perfect Bayesian equilibrium of Γ π .C. Proof of Theorem 3.
The proof is yet again nearly identical to the proof of Theorem1. We only sketch the main differences.( ⇐ ). Fix an expansion Γ π and a conditional probability perfect Bayesian equilibrium ( σ ∗ , β ) of Γ π . As in the previous proofs, we construct a conditional probability perfectBayesian equilibrium of the mediated game M ∗ (Γ) , which replicates the distribution P σ ∗ ,π . As in the proof of Theorem 2, we construct a conditional probability system β ∗ ofthe mediated game M ∗ (Γ) from the conditional probability system β of the game Γ π suchthat (( σ ∗ , σ ∗ ) , β ∗ ) is a conditional probability perfect Bayesian equilibrium of M ∗ (Γ) , withplayer 0, the dummy player, truthfully reporting his private information ( h t , ω t ) at eachstage t . Since β is a conditional probability system, there exists a sequence β n of fully NFORMATION DESIGN IN MULTI-STAGE GAMES 37 supported probabilities such that β ( X | Y ) = lim n β n ( X ∩ Yβ n ( Y ) for all X and all non-empty Y .Consider now the sequence of fully supported kernels γ n : HM Ω → ∆( H Ω × H Ω) , where γ n converges to γ (( h, ω ) , ( h, ω ) | ( h, m, ω ) = 1 for all ( h, m, ω ) . The interpretation is thatplayer 0 learns and truthfully report ( h, ω ) , when the state the profile of actions, signals,and states is ( h, m, ω ) . Let β ∗ be the CPS resulting from taking the limit of β n × γ n .By construction, β ∗ ( h t , m t , ω t | h ti , m ti ) = β ( h t , m t , ω t | h ti , m ti ) , so that σ ∗ i remains sequentiallyrational for player i . Moreover, the newly constructed conditional probability system isconsistent with the kernels p and ξ . The rest of the proof follows from the revelationprinciple.( ⇒ ). Let ( µ, R, β ) be a sequential Bayes correlated equilibrium. The difference with theprevious proofs is that the definition of a sequential Bayes correlated equilibrium doesnot specify recommendation kernels ( µ , . . . , µ T ) , which can then be used as expansions.However, as in the proof of Kuhn’s theorem, we can construct such recommendationkernels from µ .The construction is iterative. In the sequel, we slightly abuse notation and write ( µ , . . . , µ T ) for the kernels. For all ( h , ω , ˆ a ) such that p ( h , ω ) > , µ (ˆ a | ˆ a , h , ω ) := X f µ ( f ) { f ( h , ω ) = ˆ a } . Note that µ (ˆ a | h , ω , ˆ a ) > and P ˆ a µ (ˆ a | h , ω , ˆ a ) = 1 . The kernel µ is thus well-defined. (Recall that ( h , ω ) = ( h , ω ) and that ˆ a is a singleton.)We proceed iteratively. For all ( h t , ω t , ˆ a t − ) such that p ( h , ω ) µ (ˆ a | ˆ a , h , ω ) × · · · × p t − ( h t − , ω t − | a t − , h t − , ω t − ) µ t − (ˆ a t − | ˆ a t − , h t − , ω t − t ) p t ( h t , ω t | a t , h t − , ω t − ) > for some ( a , . . . , a t ) , µ t (ˆ a t | ˆ a t − , h t , ω t ) := P f µ ( f ) { f ( h , ω ) = ˆ a , . . . , f t ( h t , ω t ) = ˆ a t } P f µ ( f ) { f ( h , ω ) = ˆ a , . . . , f t − ( h t − , ω t − ) = ˆ a t − } . It is immediate to verify that the kernel is well-defined.Two remarks are in order. First, since we consider histories ( h, ω ) ∈ H Ω , we alreadyhave that p ( h , ω ) µ (ˆ a | ˆ a , h , ω ) × · · · × p t − ( h t − , ω t − | a t − , h t − , ω t − ) p t ( h t , ω t | a t , h t − , ω t − ) > for some ( a , . . . , a t ) . Second, since we only consider feedback rules in the support of µ ,all the recommendations with positive probabilities are consistent with the mediationranges. Hence, players are obedient at these recommendations.We define the conditional probability system on H Ω A as β ( h, ω, ˆ a ) = X f β ( f, h, ω ) { f ( h, ω ) = ˆ a } . To complete the proof, we repeat the same steps as in the proof of Theorem 1, that is,the additional messages are the recommendations, the kernels ( ξ , . . . , ξ T ) are the rec-ommendation kernels ( µ , . . . , µ T ) , and the conditional probability system is the one on H Ω A defined above. 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ILTIADIS M AKRIS , D
EPARTMENT OF E CONOMICS , U
NIVERSITY OF K ENT , UK
Email address : mmakris.econ(at)gmail.com L UDOVIC R ENOU , Q
UEEN M ARY U NIVERSITY OF L ONDON , CEPR
AND U NIVERSITY OF A DELAIDE ,M ILES E ND , E1 4NS, L ONDON , UK
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