Information Spillover in Multiple Zero-sum Games

This paper considers an infinitely repeated three-player Bayesian game with lack of information on two sides, in which an informed player plays two zero-sum games simultaneously at each stage against two uninformed players. This is a generalization of the Aumann et al. [1] two-player zero-sum one-sided incomplete information model. Under a correlated prior, the informed player faces the problem of how to optimally disclose information among two uninformed players in order to maximize his long-term average payoffs. Our objective is to understand the adverse effects of \information spillover"from one game to the other in the equilibrium payoff set of the informed player. We provide conditions under which the informed player can fully overcome such adverse effects and characterize equilibrium payoffs. In a second result, we show how the effects of information spillover on the equilibrium payoff set of the informed player might be severe.


Introduction
In their seminal work, Aumann et al. [1] analyzed an undiscounted infinitely repeated game with one-sided incomplete information: one player (the informed) knows the stage game being played whereas the other (the uninformed) does not know and cannot observe payoffs, only actions.They showed that this game has a value and constructed optimal strategies for the players.Matters are more complicated if the informed player were to play against more than one uninformed player, as it would be the case of a military power (e.g., USA) negotiating with two different countries (e.g., Russia and Iran). 1 By observing what the informed player plays against some other uninformed player, an uninformed player can make inferences about the game he plays against the informed player.As a consequence, it may not be optimal for the informed player to play his unilaterally optimal strategy against some of the uninformed players.Put differently, the information spillover among the games played between the informed player and the uninformed players adds layers of complexity to the analysis.
Date: March 10, 2023.This paper subsumes a previous paper titled "Information Spillover in Bayesian Repeated Games".
I am grateful to Paulo Barelli and Hari Govindan for their guidance and encouragement.I would like to thank Sven Rady, Rida Laraki, Tristan Tomala, Heng Liu, Mathijs Janssen and two anonymous referees for comments and suggestions.I acknowledge financial support from the Hausdorff Center for Mathematics (DFG project no.390685813). 1 The USA may want to conceal from Russia the exact size of its arsenal, and at the same time may want to reveal it to Iran to leverage its bargaining position.More examples in this line can be found in Aumann et al. [1].
We consider a three-player undiscounted infinitely repeated game in which one of the players is informed of the two zero-sum stage games that he plays against each of the other two (uninformed) players.Each uninformed player only knows the prior probability distribution over the finite set of pairs of zero-sum finite-action stage games, and during the play of the game observes the profiles of actions (but not the payoffs).The informed player collects the sum of payoffs from the two component games.
In the absence of information spillover, for instance when each uninformed player cannnot observe the actions played in the other zero-sum game, our three-player game has a single expected payoff, namely, the sum of values of each of the two-player component games.However, when all players are able to observe the actions played across each zero-sum game, the information spillover kicks in and it is in principle unclear whether the informed player can attain the sum of values in equilibrium.This sum of values can actually be shown to be an upper bound on the equilibrium payoffs of the informed player in our three-player game.
Our first main result has two parts which together provide a condition under which the informed player can attain this upper bound in equilibrium, even in the presence of information spillover.More precisely, under such condition, we show the informed player can attain anything as an equilibrium payoff from his individually rational payoff to the above mentioned upper bound, thus characterizing the set of equilibrium payoffs in the three-player game.In particular, this result implies that the three-player model we analyse might have a continuum of equilibrium payoffs, even though it is a zero-sum model.
The method used to obtain this first result is also of interest to the model of two-player games studied by Aumann et al. [1].Under a sufficient condition on the stage payoffs, we show that different optimal strategies from those constructed by Aumann et al. exist. 2 The strategy of the informed player, in particular, does not involve any signalling on path of play, even when the standard optimal strategy constructed by Aumann et al. necessarily does.
In a second result we provide a necessary condition for the existence of equilibria yielding the upper bound to the informed player.We explore two consequences of this result.First, we show that a natural class of equilibria which involve signalling on equilibrium path never pays the upper bound to the informed player.Second, we present an example showing that the effects of information spillover might be very severe, in the sense that the informed player is not able to attain the upper bound in equilibrium.
Given the recent heightened interest in Bayesian Persuasion (BP) (since [9]'s seminal contribution) and its overtones with Aumann et al., it is interesting to compare the effects of information spillover in a multiple-receiver setting in BP with the results of this paper.We consider a model with one sender and two receivers under two alternative specifications: (i) the sender can send private messages to each receiver and (ii) the sender can only send public messages.We show that the difference in the sender's payoff from (i) to (ii) can be interpreted as a loss due to information spillover, but a similar intuition cannot be extended to our repeated-game model.
1.1.Related Literature.To the best of our knowledge, the model analyzed here is new.Although the model we analyse is zero-sum, the results and the techniques presented remain closer to the non-zero-sum literature, especially to Hart [10] and Sorin [18].We highlight here a few additional papers on discounted and undiscounted repeated games with incomplete information that have technical and thematic similarities to this project.A significant part of the literature on undiscounted repeated games with incomplete information analyses models under the assumption of "known own payoffs" (see Forges [6]).This is a reasonable assumption in several applications and allows for equilibrium-payoff characterizations which are especially tractable (see Shalev [16]).Under this assumption, Forges and Solomon [7] provide a simple characterization of Nash equilibrium payoffs in undiscounted repeated games of two players and incomplete information. 3This characterization is used to show that in a class of public good games, uniform equilibria might not exist.More closely related to our paper in terms of the information environment is Forges et al. [8].In this paper, among other results, cooperative solutions of one-shot games with two players and exactly one informed player are related to noncooperative solutions of two-player repeated games with exactly one informed player.More specifically, under the assumption of existence of uniform punishment strategies for the uninformed player, the joint plan equilibrium payoffs of the repeated game equal the set of cooperative solutions of the one-shot game.This folk theorem is not however an equilibrium payoff characterization, since it is known from Hart [10] that joint plans cannot account for the whole of equilibrium payoffs in general.
Our work was inspired by Huangfu and Liu [11] who considered the issue of information spillovers between markets.In their model, a seller holds private information about the quality of goods he sells in two different markets and buyers learn about the seller's private information from observing past trading outcomes not only in the market in which they directly participate, but also from observing the outcomes of the other market.The authors show that, under certain assumptions on the correlation of qualities between goods in different markets, information spillover mitigates the negative effects of adverse selection.
The literature on Bayesian Persuasion with multiple receivers is extensive.We highlight here a few papers that connect to our discussion in the last section.Wang [21] presents a Bayesian Persuasion model with multiple Receivers whose payoffs depend not only on the unobserved state drawn by Nature but also on the action choices of the other receivers.Concretely, the Receivers vote on the outcome after receiving a signal about the underlying state.The paper compares two environments: one in which all Receivers observe a public message drawn from the experiment chosen by the Sender, and another in which each Receiver observes a private message independently drawn from the same experiment set out by the Sender.The main difference with the models we analyse in our last section is the payoff-interdependencies of the receivers: in our models Receivers do not care about each other's action choices, but only about their own.This implies that there is in effect no game among Receivers in our model, only a decision problem.The payoff interdependencies among Receivers coupled with the assumption of the independence of the messages drawn from the Sender's experiment drive the result of that paper in which the best equilibrium payoffs for the Sender are higher in the public compared to the private message environment, a result that is the opposite of what we find.
Arieli and Babichenko [2] are interested in what happens with optimal experiments under different assumptions for the utility of the Sender as well as payoff interdependencies of the Receivers.One result is particularly reminiscent of the one we obtain in our analysis: a specification of the public Bayesian Persuasion model we define in our paper (say, with two perfectly correlated states for each Receiver and a well-chosen utility function for the Sender) yields [2]'s model, where the utility of the Sender is additive over the Receivers, and therefore supermodular in particular.Theorem 3 in that paper then applies and a public signaling experiment is optimal if and only if all Receivers have the same "persuasion level", i.e., are essentially identical from the point of view of the optimal persuasion policy.Therefore, if Receivers meaningfully differ, the optimal experiment must send private messages.In our words, this result could be read as saying that unless information spillover is not meaningful (i.e., Receivers have identical persuasion levels), public Bayesian Persuasion yields lower a equilibrium payoff for the Sender compared to private Bayesian Persuasion.Finally, Koessler, Laclau, and Tomala [12] generalize the standard Bayesian Persuasion model in a number of directions, most importantly by considering multiple Senders and multiple Receivers.
1.2.Organization.The remainder of the paper is organized as follows.Section 2 presents the model.Our first main result is divided in 2 parts: main result 1 and main result 2. Section 3 presents our main result 1. Section 4 presents our main result 2. Section 5 compares the equilibrium payoff set in our model with that of a Bayesian Persuasion model with multiple receivers and Section 6 concludes.The proofs of technical results are left to the Appendix.Additional results can be found in the Supplemental Appendix.

Model and Equilibrium Concept
2.1.Notation.Given a finite set K, ∆(K) is the set of probability distributions over K; given a topological space X the interior of X will be denoted by int(X).If X is a subspace, its boundary will be denoted by ∂(X).For a set Y ⊆ R m , its convex closure is denoted by co(Y ).For p ∈ ∆(K A × K B ), p A (resp.p B ) denotes its marginal on K A (resp.K B ), and supp(p) its support.We denote a product distribution on K A × K B by p A p B , and use ∆(K A ) ∆(K B ) to denote the set of all such distributions.

Model.
A three-player infinitely repeated zero-sum game with two-sided incomplete information, denoted G(p 0 ), is given by the following data: • Three players, namely player 1 (the informed player), player 2 and player 3 (the uninformed players).• Finite sets: I i , J i , K i , i = A, B with I A × I B (resp.J A and J B ) being the set of actions of player 1 (resp.players 2 and 3), and K A × K B being the set of states.
The play of the infinitely repeated game is as follows: drawn according to distribution p 0 and only player 1 knows the draw.• At each stage t = 1, 2, ..., the players independently choose an action in their own set of actions: player 1 chooses (i t A , i t B ) ∈ I A ×I B and players 2 and 3 choose j t A ∈ J A and j t B ∈ J B , respectively.The stage payoff to player 1 is then Monitoring is perfect, i.e., the chosen actions are observed by all players before starting stage t + 1. Realized payoffs are not observed by the players (though player 1 knows them, since he is fully informed).
Players are assumed to have perfect recall and the whole description of the game is common knowledge.
A behavior strategy for player 1 is a tuple σ = (σ ) t≥1 and σ is a mapping from the Cartesian product H t := (I A × J A × I B × J B ) t−1 (with H 0 := {∅}) to ∆(I A × I B ), giving the lottery on actions played by player 1 at a stage t, when the state is (k A , k B ).Because players 2 and 3 do not know the state, a behavior strategy for player 2 (resp.player 3) is an element τ A = (τ A,t ) t≥1 (resp.τ B = (τ B,t ) t≥1 ), where τ A,t (resp.τ B,t ) is a mapping from H t to ∆(J A ) (resp.∆(J B )), giving the lottery on actions to be played by player 2 (resp.player 3) on stage t.The set of behavior strategies of player 1 is denoted by Σ; for player 2, it is denoted by T A and for player 3, it is denoted by T B .
A behavior strategy profile (σ, τ A , τ B ) induces, for every state (k A , k B ) and stage T > 0, a probability distribution on H T +1 .Also, (σ, τ A , τ B ) and p 0 induce a probability distribution over K A × K B × H T +1 .We can thus define the expected average payoffs (with κ being a random variable taking values on K A × K B distributed according to p 0 , κ A the random variable obtained from projecting κ on K A and κ B the random variable obtained from projecting κ on K B ): ) .
The number ) is the expected average payoff (up to time T ) of player 2 and β B T (σ, τ A , τ B ) is the expected average payoff (up to time T ) of player 3.
The model defined is therefore a "combination" of two zero-sum games: at each stage, player 1 plays simultaneously a zero-sum game against player 2 and another zero-sum game against player 3, collecting the sum of the payoffs of each of these games; players 2 and 3 are the minimizers in each of the zero-sum game they play against player 1.One distinctive and important aspect of this model is the fact that each uninformed player can observe not only the actions played in his own zero-sum game, but also the actions played in the other zero-sum game.
(2) For each > 0, there exists a positive integer T 0 such that for all T ≥ T 0 , (σ, τ A , τ B ) is an -Nash equilibrium in the finitely repeated game with T stages, i.e., (a) For each (k Uniform equilibrium is a standard equilibrium concept for the analysis of undiscounted repeated games.It contains a strong requirement, namely (2), which posits that the profile (σ, τ A , τ B ) must generate an -Nash equilibrium on the finite but sufficiently long-horizon (T ≥ T 0 , where T < ∞) version of our model. 5 Unless explicitly stated otherwise, from now on whenever we refer to uniform equilibrium or equilibria we will use simply equilibrium or equilibria. 4Remark 3.3 in Section 3 shows that assuming the prior p 0 ∈ int(∆(KA × KB)) (as customary in the literature) is not without loss of generality for the results in this paper.This is why we present the definition requiring convergence of (α k A ,k B T ) T ≥1 , (kA, kB) ∈ supp(p 0 ).The same reasoning applies to condition (2). 5 One notable aspect of this equilibrium notion is that uniform equilibria in our model are approximate Nash equilibria in the discounted version of our model: if (σ, τA, τB) is a uniform equilibrium, then (σ, τA, τB) is a ε-Nash equilibrium of the discounted versions of our model for a sufficiently high discount factor.See Theorem 13.32 in Maschler et al. [15].
, is the vector of payoffs of (σ, τ A , τ B ). Also α(σ, τ A , τ B )•p 0 (where • is the standard scalar product in Euclidean space) is the ex-ante equilibrium payoff of the informed player.

2.3.
Preliminaries on the Aumann et al. model.Our analysis of the equilibrium payoff set of the game G(p 0 ) in the next section will rely on certain properties of each of the two-player, infinitely repeated zero-sum games that the informed player plays against each uninformed player.For this reason we now recall some of the main results in Aumann et al. [1], which is the original reference for this two-player model.Let K be the finite set of states and M = (M k ) k∈K a collection of zero-sum payoff matrices where M k ∈ R I×J for each k ∈ K. Denote by G M (p) the infinitely repeated, two-player, zero-sum game with one-sided incomplete information with prior p ∈ ∆(K) and undiscounted payoffs (see Sorin [19], Chapter 3, for a detailed description of this model or Aumann et al. [1]).Let M (p) = k∈K p k M k and define v M (p) = min t∈∆(J) max s∈∆(I) sM (p)t = max s∈∆(I) min t∈∆(J) sM (p)t, where s is a row vector and t a column vector.The function q ∈ ∆(K) → v M (q) ∈ R is called the non-revealing value function.Let Cav(v M ) be the (pointwise) smallest concave function g from ∆(K) to R such that g(q) ≥ v M (q) for all q ∈ ∆(K).Alternatively, one can define Cav(v M )(q) := sup{ k i=1 α i v M (q i )|∃k ∈ N, ∀i ∈ {1, ..., k}, α i ≥ 0, k i=1 α i q i = q, k i=1 α i = 1}.Aumann et al. [1] proved that a (uniform) value of G M (p) exists and equals Cav(v M )(p).They also showed how to construct (uniformily) optimal strategies for both players.
Given the model G(p 0 ), the infinitely repeated, two-player, zero-sum game with one-sided incomplete information with prior p 0 A defined by states K A and payoff matrices (A k A ) k A ∈K A with undiscounted payoffs will be denoted G A (p 0 A ) -this game is played by players 1 (informed) and 2 (uninformed).Analogously, we define G B (p 0 B ) as the two-player, infinitely repeated, zero-sum game with one-sided incomplete information played between players 1 and 3.The two-player, infinitely repeated zero-sum game with one-sided incomplete information and prior p 0 ∈ ∆(K A × K B ) with undiscounted payoffs, where stage payoff matrices are ( ).This two-player game will be used as an auxiliary game to construct strategies in the three-player game G(p 0 ).

2.4.
Example.We would like to illustrate the new strategic difficulties that arise in the model G(p 0 ) in comparison to the two-player zero-sum model of Aumann et al. [1].Specifically, we would like to show that in this example that if the informed player plays the optimal strategies constructed by Aumann et al. in each game G A (p 0 A ) and G B (p 0 B ), he cannot guarantee the ex-ante expected payoff equal to the sum of values Cav(v A )(p 0 A ) + Cav(v B )(p 0 B ).
In the matrix p 0 , an entry p 0 ij corresponds to the probability with which Nature chooses A i and B j .So, the stage-payoffs in the two zero-sum games are given by A 1 and B 1 with probability 1/2, and they are given by A 2 and B 2 with probability 1/2.Since the prior assigns perfect correlation between states, there are only two states to consider effectively: states (1, 1) and (2,2).
In figures 1 and 2, q denotes the probability of state (1, 1) and 1− q the probability of state (2, 2).Each row of A i and B j , i, j ∈ {1, 2} corresponds to a stage-game action of the informed player: call the first row "U " and the second row "D".By computation, we get that the non-revealing values are: These imply that the concavification of these values are: We present the optimal strategy of the informed player in the game G B (1/2).The optimal strategy of the informed player in the two-player repeated zero-sum game G B (1/2) is defined as follows: in case the state drawn by Nature is 1, the informed player plays "U " with probability 1/4 and after that plays at each stage, independently, the optimal action of the one-shot zero-sum 1 0 game whose matrix is B(1/4); in case the state drawn by Nature is 2, the informed player plays "U " with probability 3/4 and, after that, plays the optimal action of the one-shot zero-sum game whose matrix is B(3/4).
After observing the realized action of the informed player in the first stage, the uninformed player updates his beliefs about the states to posteriors about states 1 and 2: in our example, the uninformed player, after observing "U ", assigns probability 1/4 to the state being 1.After observing "D", the uninformed player assigns probability 3/4 to the state being 1.The strategy just described for the informed player is an example of a signalling strategy: the informed player uses his actions to signal information about the underlying state.
After the first stage, according to the construction described, no more information is signaled and the uninformed player plays the optimal action of the one-shot zero-sum game with matrix B(1/4) or B(3/4) forever, depending on whether U or D was realized, respectively.Playing the signalling strategy guarantees to the informed player an ex-ante payoff of ( Now, in game G A (1/2) the non-revealing value function of the informed player is strictly concave, which implies that his optimal strategy in this game is non-revealing (at any prior): one optimal strategy for the informed player is to play at each stage the optimal action of the one-shot zero-sum game with matrix A(1/2) independently forever, which generates no uptading of the beliefs on the part of the uninformed players.
If the informed player uses the signalling strategy described in G B (1/2), because of perfect correlation between κ A and κ B , this strategy also induces the same updating on the part of the (uninformed) player 2, inducing, similarly, posteriors 1/4 and 3/4 in the two-player zero-sum repeated game G A (1/2).By using that strategy in game G B (1/2), the informed player in game G A (1/2) can now only guarantee Therefore, if the informed player plays the strategies described above in games G A (1/2) and G B (1/2), he cannnot guarantee in G(p 0 ) the sum of the uniform values of each zero-sum game i.e., Cav(v A )(p 0 A ) + Cav(v B )(p 0 B ).

Main Result 1: Equilibrium Payoff-Set Characterization
Our first main result (main result 1) has two parts: The first part of main result 1 is Theorem 3.2 which provides a sufficient condition under which the ex-ante equilibrium payoffs of the informed player in G(p 0 ) permits a simple characterization.The second part of our main result 1 is Theorem 3.6, which provides a general class of games under which that sufficient condition holds.The two results provide a condition under which G(p 0 ) has a continuum of equilibrium payoffs.
This section is subdivided in three subsections.The first of these, subsection 3.1, is dedicated to Theorem 3.2 and the main ideas of its proof.Subsection 3.2 is dedicated to Theorem 3.6.Finally, subsection 3.3 highlights a by-product for the theory of two-player zero-sum games of the equilibrium constructions used in Theorems 3.2 and 3.6.
3.1.First Part of Main Result 1.We now introduce the necessary concepts and state our main result 1 in full generality.Paralleling the definitions of the previous section, we denote the set of histories at stage t ≥ 1 for a two-player, zero-sum, infinitely repeated, undiscounted game with onesided incomplete information G A (p 0 A ) by H A t with generic element h A t .The notation for a behavior strategy of the informed player in G A (p 0 A ) is exactly analogous to the one defined for player 1 in the three-player game G(p 0 ): In non-revealing equilibria the informed player makes no use of his private information on-path of the equilibrium play.Therefore, the uninformed player can infer nothing from the actions played at each stage, which leaves the "posterior" unchanged and equal to the prior. In ) is an upper bound on the ex-ante equilibrium payoffs of player 1, because each uninformed player can always play the optimal strategy of his repeated zero-sum game, holding the payoffs of the informed player at most at Cav(v A )(p A ) + Cav(v B )(p B ).On the other hand, letting h(p) := v A (p A ) + v B (p B ), a lower bound on the ex-ante equilibrium payoffs of the informed player is given by the concavification of h evaluated at p 0 . 6 For p 0 ∈ ∆(K A × K B ), let We call Cav(h)(p 0 ) the lower end of I(p 0 ) and Cav(v A )(p 0 A ) + Cav(v B )(p 0 B ) the upper end of I(p 0 ).The interval I(p 0 ) might be degenerate as well as non-degenerate.Whenever I(p 0 ) is nondegenerate, for any sufficiently small perturbation of the stage-game payoff matrices, the resulting model G(p 0 ) also has an associated interval I(p 0 ) which is non-degenerate.A proof of this robustness property can be found in the Supplemental Appendix (Proposition 9.4).As an example, the interval ).There are cases where I(p 0 ) is degenerate (for example, when p 0 ∈ ∆(K A ) ∆(K B )).In these cases only one ex-ante equilibrium payoff exists (without any assumptions on the games G A (p 0 A ) and G B (p 0 B )).The first part of our main result can now be stated: Then the set of ex-ante equilibrium payoffs of the informed player is I(p 0 ).
On the proof of Theorem 3.2.First, the lower end of I(p 0 ) is always an ex-ante equilibrium payoff for the informed player in G(p 0 ) -i.e., no additional assumption is required.A proof of this result can be found in the Supplemental Appendix (Proposition 9.11). 7When the upper end of I(p 0 ) is also an ex-ante equilibrium payoff for the informed player, then we can use a straightforward application of jointly controlled lotteries developed in Aumann et al. [1] in order to obtain that the whole interval I(p 0 ) can be attained as an ex-ante equilibrium payoff to the informed player. 8 Therefore, the only remaining task to obtain a proof of Theorem 3.2 is to show that the upper end of I(p 0 ) is an ex-ante equilibrium payoff for the informed player.
We define the set of non-revealing equilibrium payoffs (of the informed player), denoted N R(p 0 ), 6 Consider the two-player infinitely repeated zero-sum game GA+B(p 0 ).It is straightforward to check that the non-revealing value of GA+B(p 0 ) is h(p 0 ).From Aumann et al. [1], the value of GA+B(p 0 ) is Cav(h)(p 0 ), so the informed player can guarantee Cav(h)(p 0 ). 7We do not include this proof here because it derives essentially from a long application of the work of Simon et al. [17] to our three-player environment. 8A jointly controlled lottery is a public randomization device that is endogenously generated by the players so that the players can coordinate.We can define a jointly controlled lottery that randomizes between equilibria paying the upper and lower ends of I(p 0 ) with any given probability, in order to obtain any given number in I(p 0 ) as an ex-ante equilibrium payoff of the informed player.See Aumann et al. [1] p. 272 for details.For completeness, the explicit construction of the equilibrium involving jointly controlled lotteries is included in the Supplemental Appendix (Proposition 9.19).
(1) (Feasibility) where (2) (Individual rationality for player 1) ) (Individual rationality for players 2 and 3) The three conditions above defining the set of non-revealing equilibrium payoffs parallel the conditions defined for two-player repeated games with a single informed player (see Hart [10]).We briefly recall the reason why (φ A , φ B ) ∈ N R(p 0 ) implies we can construct equilibria in G(p 0 ) that reveal no information on path of play and have φ A (resp.φ B ) as the vectors of payoffs of the informed player in where Consider then a sequence of (pure) actions ((i t A , j t A )) t≥1 and define a function When players 1 and 2 play the sequence of actions Obviously, a similar reasoning applies to φ B ∈ F B , and φ k B B is the payoff achieved by a deterministic sequence of actions played at each stage.Conditions ( 2) and ( 3) now imply that this deterministic path of play can be supported as an equilibrium path of play: condition (2) guarantees that, in case player 1 deviates from the deterministic sequence, players 2 and 3 can punish him.So it guarantees that under no possible state (k A , k B ) player 1 could obtain more that φ (k A k B )10 ; condition (3) guarantees, in turn, that 9 Lemma 2 in Sorin [18] shows such a sequence exists for any such λi,j. 10Condition (2) implies that there is an optimal strategy of the uninformed player in the game GA+B(p 0 ) which guarantees the informed player will not obtain more than φ (k A ,k B ) for each state (kA, kB).In the Supplemental Appendix, Proposition 9.18, we show this strategy can indeed be played by players 2 and 3 in G(p 0 ).any deviation of player 2 (resp.player 3) can be punished by player 1 with an optimal strategy of game G A (p 0 A )(resp.G B (p 0 B )).We now define the set N R A (p 0 A ) of non-revealing equilibrium payoffs of the two-player repeated game This is the set of equilibrium payoffs for which no signalling occurs on path.This set is essentially the specification for a two-player, zero-sum infinitely repeated game with one-sided incomplete information of the set of non-revealing equilibrium payoffs (called "G") defined for nonzero-sum two-player repeated games with lack on information on one side in Hart [10] A ) generate the same ex-ante payoff, i.e., the (uniform A , but there might be several vectors generating this payoff.It is now easy Remark 3.3.We remark that property (2) of N R(p 0 ) differs from the usual assumption present in the literature, namely, that supp(p 0 ) = K A ×K B .The reason why we do not adopt this assumption is as follows: as shown above, ), we just have to check that property (i) defining the sets N R A (p A ) and N R B (p B ) is satisfied; the other conditions are immediate.Suppose by contradiction there exists qA One can see immediately that the argument above relies on the product structure of the set of states supp(p 0 ) = K A × K B ; if one assumes a prior p 0 for which supp(p 0 ) is not a cartesian product, then the argument above cannot be repeated and, in fact, the claim is not true. 11The example below shows that for a certain Example 3.4.We follow the notation of Example 2.4.For ε > 0 and q 0 ∈ (0, 1), consider the game G(p 0 ) given by the following data: 11 This reasoning also justifies why the assumption that is adopted throghout is p 0 A ∈ int(∆(KA)) and p 0 B ∈ int(∆(KB)) and not p 0 ∈ int(∆(KA × KB)): if, for instance, p 0 A does not have full support, then one can eliminate from the set of states in KA the ones that have 0 probability under p 0 A and the resulting space of states is also a product subset of KA × KB.
One can now show that N R(p 0 ) = ∅.We sketch the proof for completeness: fix q 0 = 1 5 .The game G A (p 0 A ) trivially satisfies the condition N R at p 0 A (see Definition 3.5), because v A is strictly concave and smooth in the interval (0, 1).Therefore, by Theorem 3.6, there exists Taking ε > 0 sufficiently small, the vector φ ∈ R K A ×K B whose entries are given by

Second Part of Main Result 1.
The second part of our main result provides a general sufficient condition for the non-emptyness of N R A (p 0 A ) and N R B (p 0 B ).We introduce a few preliminary definitions in order to state the condition.
Recall that the non-revealing value function v A (q) is defined by min τ max σ σA(q)τ , where σ is a row vector and τ a column vector of the one-shot, two-player, zero-sum game with payoff matrix A(q).For q in the affine hull H A of the simplex ∆(K A ), one can consider the immediate extension of v A (q) to H A given by the same min-max formula.Denote this extension by v e A .As we will need to make considerations about the derivative of v e A at points in ∆(K A ), we will define once and for all a parametrization for the affine space H A .Let T : R |K A |−1 → H A be defined as follows: Let e i+1 and 0 → e , where S is an injective linear transformation; we will also denote by S the matrix representation of S according to the canonical basis.The function (v e A • T ) : ), for all k ∈ N. 13 For notational convenience, we write ∂v for all h with x + h ∈ P and T (x) = p.Below, φ A S denotes the pre-multiplication of the row vector φ A by the matrix S.
Definition 3.5.The two-player infinitely repeated zero-sum game with one-sided incomplete information The properties (1)-( 3) in the Definition of N R at p 0 A (Definition 3.5) are properties of v A : (1) states that p A is a point of identity between v A , Cav(v A ) and the affine function q ∈ ∆(K A ) → φ A •q; (1) also states that p 0 A is a point of identity between the same affine function and Cav(v We are now ready to state the sufficient condition. Then the set of non-revealing equilibrium payoffs N R A (p 0 A ) is nonempty.Evidently, the analogous statement holds for G B (p 0 B ).
A proof of this Theorem can be found in Appendix A. Evidently, for satisfies N R at p 0 B , then it follows from Theorem 3.2 that I(p 0 ) is the ex-ante equilibrium payoff set of the informed player.
Remark 3.7.As both definitions of the property N R at p 0 A and the set N R A (p 0 A ) concern the game G A (p 0 A ), we compare their content in detail.If a vector φ A satisfies (3) and the first two equalities of (1) of Definition 3.5, then this implies that φ A satisfies (i) in the definition of N R A (p 0 A ).If φ A satisfies the last equality of (1) of Definition 3.5, then it is immediate that it satisfies (ii) of N R A (p 0 A ). Property (iii) in the definition of N R A (p 0 A ) bears no immediate relation with the nonrevealing value function v A .But, as the proof of Theorem 3.6 shows, it is implied by a geometric property of v A , namely, property (2).Therefore, information about N R A (p 0 A ) can be infered from properties of v A only.The precise way to do this inference is presented in the proof of Theorem 3.6. 13Notice that to define the generalized gradient, one needs to take limits from all possible directions in That is why one needs to extend the non-revealing value function "outside" of the simplex.

Our aim at introducing the condition N R at p 0
A is to highlight that the attainability of the upper end of I(p 0 ) as an ex-ante equilibrium payoff of the informed player is not purely an "information problem", i.e., it does not depend exclusively on the existence of correlation between states in K A and K B .The property shows how the attainability of the upper end of I(p 0 ) also fundamentally relies on the payoff structure of the component games, which determines the geometry of the nonrevealing value function.Though the sufficient condition presented might not be easier to check computationally than the direct non-emptyness of N R A (p 0 A ), it isolates the aspects of this geometry which determine the existence of equilibria with such payoffs.
An Interpretation for N R.Even though the property N R at p 0 A is not straightforward to interpret, as it mainly describes certain geometric properties of the non-revealing value function, there is a class of infinitely repeated two-player zero-sum games with one-sided incomplete information in which this property can be interpreted straightforwardly and in which this property is always satisfied.
In this section, whenever a two-player, zero-sum infinitely repeated game Definition 3.8.A two-player, infinitely repeated zero-sum game with lack of information on one-side and undiscounted payoffs G A (p) is locally non-revealing at p14 whenever there exist k ∈ The definition above implicitly describes an optimal strategy for the informed player in the game G A (p) for which there is signalling on path of play: the informed player "splits" the prior p into finitely many posteriors . This is the typical optimal strategy constructed by Aumann et al..The local non-revelation property tells us that whenever the informed player in G A (p 0 A ) has an Aumman et al.'s strategy under which he does not exclude some state (i.e., when there is at least one induced posterior in the interior of the simplex of states), then there exists an equilibrium of the game G A (p 0 A ) where the informed player does not signal on path of play.equal weights of 1/4 and 3/4 (the "optimal splitting" which determines the Cav(v B )).Note that at the interior posterior q = 1/4 in the figure, the vectors N 2 and N 1 generate the normal cone to the graph of v B at q = 1/4.Each vector of this normal cone is uniquely associated with a supergradient of v B at q = 1/4, i.e., N m • q = (q, n m • q), m ∈ {1, 2}, so and N belongs to that normal cone at q = 1/4, and therefore n can be given by a convex combination of n 1 and n 2 .
Proposition 3.9.Suppose G A (p 0 A ) is locally non-revealing at p 0 A .Then G A (p 0 A ) satisfies N R at p 0 A .Evidently, the same statement holds for G B (p 0 B ).
Proposition 3.9 shows that the local non-revelation property implies the N R property.For a proof of this proposition, see Appendix A. Example 3.10 below shows that the N R property is strictly more general than the local non-revelation property.As an example, the games G A (p 0 A ) and G B (p 0 B ) defined in Example 2.4 are both locally non-revealing at their respective priors.Note that in that example, the game G B (p 0 B ) is such that the optimal strategy of the informed player constructed by Aumann et al. involves signalling on path (this is the signalling strategy we briefly described in the example).
Example 3.10.Let K A = {1, 2} be the set of states.Let q denote the probability of state 1 and p 0 A = 1/2 (prior of state 1).Consider the following game G A (p 0 A ): For a row vector v ∈ R m , denote by v the transposed column vector.The figure depicts the graphs of v A (q) = −q(1 − q) and Cav(v A )(q) = 0, ∀q ∈ [0, 1].Consider the actions σ A = (1, 0) and τ A = (0, 1) .Then First note that the local non-revelation property at p 0 A is not satisfied in G A (p 0 A ).We show that the property N R at p 0 A is satisfied in the example.The linear transformation S in this example is defined by Sx = (x, −x).Notice that the only candidates for p and φ A satisfying the conditions of property N R at p 0 A are p = 0 or p = 1 and α A .Notice that at p = 0, just by looking at the graph depicted above, one can see that conditions (1) and (3) of property N R are satisfied for the vector α A .Now, lim q→0 − ∇(v e A • T )(q) = 0, since v e A • T is constant and equal to 0 in (−∞, 0).Using the notation defined above, we have that α A S = 0, so that α A S ∈ ∂v A (p). Observe that because of the strict convexity of the non-revealing value function v A , the optimal strategy of the informed player as constructed in Aumann et al. necessarily involves signalling on path, namely, inducing posteriors at the boundary of the 1-simplex of states.However, property N R at p 0 A guarantees that an equilibrium exists for which no signalling occurs on path.
The next example shows a game G A (p 0 A ) on which N R at p 0 A is not satisfied.
Example 3.11.Let G A (p 0 A ) be defined from the payoff matrices below and q denote the probability of state 1 with prior p 0 A = 1/2.
Figure 5 depicts the non-revealing value function and its concavification.The only two candidates for p A and φ A are p A = 0 and p A = 1 and (1, 1).Notice that v e A • T has a derivative at p A = 0, which is equal to −2, so ∂v A (p A ) = {−2}, whereas φ A S = (1, 1)S = 0. Therefore, φ A S / ∈ ∂v A (p A ).The same reasoning applies to show that φ A S / ∈ ∂v A (p A ), when p A = 1.Therefore, property (2) of Definition 3.5 is not satisfied.Remark 3.12.It is possible to prove directly that if The proof is illustrative of how the geometry of v A determines the existence of equilibria in G A (p 0 A ) that involve no signalling on path of play and therefore we include it here.The proof is divided in two cases: ).We first construct the candidate vector of payoffs φ A ∈ R |K A | for the informed player.Then we show that φ A ∈ N R A (p 0 A ).
, λ s ≥ 0, s∈S λ s = 1 and let p s 0 be the posterior which is interior to the simplex of states.Let is not an extremum point of H (cf. [20], Section 18).Therefore, there exists a face F of H, with dimension d ≥ 1, such that F contains (p 0 A , Cav(v A )(p 0 A )) and each point (p s , v A (p s )).Take now a supporting hyperplane H to H which contains F .The hyperplane H intersected with ∆(K A ) × R is the graph of an affine function , where φ A is a vector in R |K A | .We now claim the following: Claim 3.13.The vector φ A belongs to N R A (p 0 A ).
Proof.As the graph of the affine function , where [p s 0 , p 0 A ] denotes the segment between p s 0 and p 0 A .In particular, Cav(v A )(p 0 A ) = φ A • p 0 A is satisfied, and so φ A satisfies (ii).From the fact that H supports H at F , it follows that φ A • q ≥ v A (q), ∀q ∈ ∆(K A ).This proves φ A satisfies (i).Thus it only remains to show φ A ∈ F A .Let σ A be the Aumann et al. optimal strategy of the informed player in G A (p s 0 ), which is state-independent since Cav(v A )(p s 0 ) = v A (p s 0 ).For the uninformed player, there exists a Blackwell strategy 15 τ A and a constant L > 0 such that .34 in Sorin [19]), where 15 Cf. the next subsection for a discussion and definition of Blackwell strategies.
For each k A ∈ K A , let v k A be this limit.It follows from the inequality of the Blackwell strategy given above that ∀k A , for all k A ∈ K A .Since F A is compact and convex and neither σ A nor τ A condition on realized states, it follows that ( This shows φ A ∈ F A and concludes that φ A ∈ N R A (p 0 A ), as claimed.The proof of case (b) is immediate, since the Aumann et al. optimal strategy of the informed player in G A (p 0 A ) is already state-independent.This concludes the proof of the claim.
We would like to highlight two things about the above remark.Notice that the first part of the proof does not rely on any assumption on G A (p 0 A ) being locally non-revealing at p 0 A , i.e., it does not rely on the assumption that an optimal strategy of the informed player exists inducing an interior posterior.The first paragraph just serves the purpose of pinning down the correct vector φ A .Notice that the assumption of p s 0 ∈ int(∆(K A )) is used in the proof above only when we show that φ A = v.Second, we did not show in the proof that the profile (σ A , τ A ) is an equilibrium of G A (p 0 A ); we used the fact that (σ A , τ A ) is a uniform equilibrium of G A (p s 0 ), inducing the vector of payoffs for the informed player equal to φ A , and obtained that φ A ∈ N R A (p 0 A ).

3.3.
Consequences for Two-player Zero-sum Repeated Games with Incomplete Information.Theorem 3.6 tells us that given a two-player zero-sum game with one-sided incomplete information that satisfies the property N R at the prior, there exists an equilibrium of the 2-player game for which no signalling occurs on path of play (i.e., the set of non-revealing equilibrium payoffs of this game is non-empty).Even for games for which Aumann et al. constructed strategies that necessarily involved signalling on path (see Example 3.10 or G B (p 0 B ) in Example 2.4), the N R property implies the existence of an equilibrium for which no signalling occurs on path.We would like to illustrate this message with an example.
Consider the two-player, zero-sum infinitely repeated game with one-sided incomplete information G B (1/2) between players 1 and 3, defined by the data in Example 2.4.Following Aumann et al.'s technique for constructing optimal strategies, the strategy of the informed player would be the strategy highlighted in Example 2.4, that is, a signalling strategy: the informed player uses his actions to signal information about the underlying state for finitely many stages and after that plays the (mixed) optimal action of the one-shot, zero-sum game given by the posterior at each state independently.Playing this strategy in G B (p 0 B ) guarantees to the informed player an ex-ante payoff of ( 1 Now, the optimal strategy for the uninformed player in G B (1/2) is a so-called approachability strategy or Blackwell strategy.For a general game G B (p 0 B ), a Blackwell strategy can be defined as follows: given φ is a Blackwell strategy (for φ B and p 0 B ), if for each ε > 0, there exists T 0 ∈ N such that ∀T ≥ T 0 , σ B a strategy of the informed player and k B ∈ K B , where E k B σ B ,τ B ,p 0 is the conditional expectation given κ B = k B .So τ B precludes the informed player from achieving more than φ k B B + ε in a sufficiently long (but finitely repeated) game, for any state k B ∈ K B .This implies that the ex-ante expected payoff to the informed player in a sufficiently long game is not larger than Cav(v B )(p 0 B ) + ε.For our example, take φ B = (1, 1) and consider τ B the Blackwell strategy for φ B and p 0 B as defined in the example.It follows the pair (σ B , τ B ) is a (uniform) equilibrium with associated payoff Cav(v B )(p 0 B ) = 1.We call the strategies just defined standard optimal strategies.In contrast to these standard optimal strategies, for which there is revelation of information on path of play, we now construct equilibrium strategies for both players for which no information is revealed on path.The idea for the construction of these strategies is simple.Both players play a deterministic sequence of actions as long as the other plays it.This deterministic sequence is supported by punishment strategies, in the sense that if any player deviates from his sequence of actions, the other player starts to play the punishment strategy forever.Let us first define the deterministic path of play for this example: the deterministic sequence of actions is defined by ((i t B , j t B )) ∞ t=1 = ((U, R), (U, M ), (U, R), (U, M ), ...); so the uninformed player alternates between R (right column) and M (middle column), whereas the informed player plays U at every stage.For each state k B ∈ {1, 2}, the payoff (to the informed player) obtained from this path of play is: lim This implies the ex-ante payoff to the informed player is also 1 = Cav(v B )(1/2).In case any player deviates from his prescribed sequence of actions, the other player can observe the deviation and play from the next stage onwards his standard optimal strategy in G B (1/2), which guarantees that a deviation is not profitable.As mentioned, the strategies just defined are also a (uniform) equilibrium in G B (1/2), but reveal no information on path of play.4. Main Result 2: Necessary Condition for the Upper End of I(p 0 ) to be Attained in Equilibrium Theorem 4.1 is the main result of this section.Intuitively, it shows that equilibria which pay the upper end of I(p 0 ) to the informed player imply a particular type of signalling procedure.Theorem 4.1 can therefore be viewed as a constraint to the signalling strategies an informed player might play in an equilibrium paying the upper end of I(p 0 ).Presenting the formal statement of the Theorem requires some preliminary definitions.
Endow I A , I B , J A , J B , K A and K B with the discrete topology.For each t ∈ N, let H t be the discrete field over the induced product topology and let H ∞ and F(Ω) be the Borel sigma-fields over H ∞ and Ω, respectively.For notational convenience, we will also denote by H t the field generated by H t on Ω.
Let (σ, τ A , τ B ) be a profile of strategies in G(p 0 ).Let P σ,τ A ,τ B ,p 0 be the probability induced by (σ, τ A , τ B , p 0 ) on (Ω, F(Ω)).We define the martingale of posteriors obtained through Bayesian The sequence (p t ) t∈N is a ∆(K A × K B )-valued martingale with respect to (H t ) t∈N , satisfying: (i) p 1 = p 0 ; (ii) there exists p ∞ such that p t → p ∞ a.s. as t → +∞.The a.s.limit p ∞ of the process (p t ) t∈N is called the asymptotic posterior.For an intuition on Theorem 4.1, note that from Jensen's inequality, it is immediate to see that Cav(v A )(p t ) is a supermartingale w.r.t.(H t ) t∈N .If Cav(v A )(p t 0 ) is not a martingale for some 0 ∈ {A, B}, then this would imply that the expected payoffs induced by the equilibrium in game G 0 (p 0 0 ) are less than Cav(v A )(p 0 A ), contradicting the assumption of the Theorem.The next corollary of Theorem 4.1 is motivated by the following intuition.Take a game G(p 0 ) for which p 0 ∈ int(∆(K A × K B )) and assume that N R(p 0 ) = ∅.This last assumption implies that if the upper end of I(p 0 ) can be attained in equilibrium in G(p 0 ), then the informed player must use a state-dependent strategy on path, i.e., he must use signalling.An idea of how such signalling procedure could occur is as follows: the informed player could signal so as to induce posteriors that are product distributions; this would imply that, once the posterior realizes, from then onwards no correlation exists between the zero-sum games at the posteriors and therefore the informed player could play each of the zero-sum games without concern for information spillover; is it possible that there are equilibria of G(p 0 ) for which the described signalling procedure exists and the upper end of I(p 0 ) can be attained in equilibrium?Corollary 4.2 provides an answer to this question.Now the process of posteriors is a martingale, which implies that the expectation of p ∞ is p 0 .This implies that the following equation has a solution in λ ∈ [0, 1]: But this equation has no solution for λ ∈ [0, 1], which finally implies a contradiction.Hence, there is no equilibrium paying ex-ante to the informed player Cav(v A )(p 0 A ) + Cav(v B )(p 0 B ).The arguments above give us more: recall that we had , where the second inequality follows by definition of Cav(h) and the last inequality is given by Jensen's inequality.The number Cav(h)(p 0 ) is the lowest possible ex-ante equilibrium payoff to the informed player.This implies that every uniform equilibrium of the example pays Cav(h)(p 0 ) to the informed player.
Remark 4.7.We would like to provide an alternative proof of the claim that in the game of Example 4.4 only the lower end of I(p 0 ) is an equilibrium payoff. 18It is obvious that the matrix A 1 (respec.A 2 ) can be substituted by the following equivalent matrix A 1 r (respec.A 2 r ), by simply eliminating the redundant column action of player 2. So, The model G(p 0 ) defined by the data above is evidently equivalent to the model G(p 0 ) defined by the data of the previous example, as only redundant actions have been eliminated, which leaves the best-reply correspondences of all players unaltered.Player 2 now, evidently, is a dummy player.We can define therefore a two-player non-zero-sum game between players 1 and 3, whose equilibria immediately induce the equilibria of the three-player game G(p 0 ).To be precise, we define a two-player, non-zero-sum infinitely repeated game with one-sided incomplete information and undiscounted payoffs G(q 0 ), where the set of states will be K = {1, 2} with prior q 0 = 1/2 for state 1: the payoffs are given by the following bimatrix C k (k ∈ K), where the informed player plays row and the uninformed player 3 plays column: The rows and column labels in matrix C k should be read as follows: L and R stand for the stage-game actions of the uninformed player (i.e., player 3).For the row player (i.e., player 1), (U, D) corresponds to choosing the top row in game in A k and the bottom row in game The other entries are analogously constructed.
We can now modify the stage-game payoffs C k , k = 1, 2, so that the best-reply correspondence of both players remains unaltered and, after the modification, we obtain a zero-sum game between players 1 and 3. Define new stage-game payoff matrices D k , k = 1, 2 by: D k . Consider now the two-player, zero-sum infinitely repeated game with one-sided incomplete information and undiscounted payoffs where the stagegame payoff matrices are given by (D 1 , D 2 ) and the prior of state 1 is q 0 = 1/2.This modification leaves the payoffs of player 1 unaltered, and therefore does not change his best-reply correspondence when compared to G(q 0 ).Though the payoffs of player 3 are modified, his best-reply correspondence is not, which finally implies that the equilibria under this modification are the same as in G(q 0 ).Since the modified game is now zero-sum, it follows from Aumann et al. that it has a (uniform) value, which is the unique ex-ante uniform equilibrium payoff to player 1, and is then evidently the lower end of I(p 0 ) (cf. footnote 7).

Information Spillover in Bayesian Persuasion: A comparison with our results
The problem of information spillover can also be studied in the Bayesian Persuasion (BP ) setting.This possibility is briefly discussed in the paper by Kamenica and Gentzkow [9] in the section "Multiple Receivers".We would like to draw a comparison between the effects of information spillover over equilibrium payoffs in our model and over equilibrium payoffs in BP .We first describe the game form of the BP model we have in mind in detail.We refer to this model as public BP .
Let K A × K B be the set of states, with K A and K B finite sets.The set M A × M B is the set of messages, with M A and M B being finite sets and |M i | ≥ |K i |, i ∈ {A, B}.The actions of player 2 (respec.player 3) are denoted j A ∈ J A , (j B ∈ J B ), with both J A and J B being finite sets.At an ex-ante stage, player 1 chooses a state-dependent lottery or experiment x ∈ ∆(M A × M B ) K A ×K B (the set of pure actions of Player 1).Then Nature draws a state (k A , k B ) according to some prior If we specify payoffs to be zero sum, i.e., ν A = −u A and ν B = −u B , the public BP model yields that U A and U B are concave functions, therefore implying that Cav(U The next claims settles this result.
Claim 5.1.In the public BP zero-sum model, U A as well as U B are concave.Therefore, for Proof.We prove that U A is concave.The proof of concavity of U B is similar.For each p A ∈ ∆(K A ) and The map U A is therefore piecewise affine and concave in p A .The remainder of the claim is immediate from the definition of Cav.
Hence, the effect of information spillover in the zero-sum public BP model is inexistent from the perspective of equilibrium payoffs, but is relevant in the non-zero sum public BP model, since there might be loss to player 1 generated by information spillover.As we showed with our main result 1 in this paper, for the model ) cannot be interpreted similarly as the loss generated by information spillover, because Cav(v A )(p 0 A ) + Cav(v B )(p 0 B ) might be attained in equilibrium.

Conclusion
We studied a three-player generalization of the Aumann et al. model and analysed the effects of information spillover on the equilibrium payoff set of the informed player.Our first two results provided a sufficient condition under which a continuum of equilibrium payoffs exist in the model and which implies, in particular, the existence of equilibria where the informed player circumvents the adverse effects of information spillover.These equilibria involve no signalling on equilibrium path.This sufficient condition is implied by the more interpretable local non-revelation condition.Our second main result presented a necessary condition for equilibria to attain the upper end of I(p 0 ), which provides a restriction on the signalling processes that can be generated by such an equilibrium.A corollary of this result is that equilibria which "uncorrelate" the two two-player zero-sum games G A (p 0 A ) and G B (p 0 B )(whenever I(p 0 ) is, of course, non-degenerate) do not achieve the upper end of I(p 0 ).
Several questions remain unanswered with regards to the model G(p 0 ).What are the ex-ante equilibrium payoffs of the informed player that can be achieved through signalling on path of play?Is it possible, when N R(p 0 ) = ∅ and I(p 0 ) is non-degenerate, that the upper end of I(p 0 ) is achieved as an equilibrium payoff of the informed player?As our last example (Example 4.4) showed, it might be the case that only the lower end of I(p 0 ) is achievable as an equilibrium payoff.Is it possible that an example of G(p 0 ) exists for which the upper end of I(p 0 ) is not achievable, but something in the interior of I(p 0 ) is an equilibrium payoff?These questions remain to be answered in future work.(1), the left hand side of the last inequality can be written as

Therefore, we have that φ
It now remains to prove φ A ∈ F A .We will show that ∂v Let Σ(x) denote the set of optimal strategies of the maximizer in the one-shot zero-sum game with matrix A(T (x)) and analogously denote T (T (x)) for the optimal strategies of the minimizer in the same game.Applying Proposition 3.4.2 in [13]: Let B be the closed unit ball in Applying the minmax theorem gives now that ∇f (x) ∈ F A .Let T (x) = p A .By definition of the generalized gradient of f at x, we have that ∂v A (p A ) ⊆ F A , as we wanted to show.
Proof of Proposition 3.9.
Proof.The proof is divided in two-cases: ).We start with (a).We first construct the candidate vector of payoffs φ A ∈ R |K A | and a probability distribution p A and show the pair (φ A , p A ) satisfies the definition of N R at p 0 A .Let Cav(v A )(p 0 A ) = s∈S λ s v A (p s ), λ s ≥ 0, s∈S λ s = 1 and let p s 0 be the posterior which is interior to the simplex of states.Let . Since, by (a), ∀s ∈ S, p s = p 0 , it follows that (p 0 A , Cav(v A )(p 0 A )) is not an extremum point of H (cf. [20], Section 18).Therefore, there exists a face F of H, with dimension d ≥ 1, such that F contains (p 0 A , Cav(v A )(p 0 A )) and each point (p s , v A (p s )).Take now a supporting hyperplane H to H which contains F .The hyperplane H intersected with ∆(K A ) × R is the graph of an affine function . We now check that it satisfies the conditions of Definition 3.5.It is clear from the construction that φ A • q ≥ Cav(v A )(q), ∀q ∈ ∆(K A ). Using the definition of T , we can rewrite this in Definition 3.5 is satisfied.It remains to prove that φ A S ∈ ∂v A (p s 0 ).In order to show this, we are going to apply a result in Clark [4]. 21Let v A • T = f .Note that f has directional derivatives in all directions (see Proposition 3.4.2 in [13]).We denote the directional derivative of f at point x in the direction h by f (x; h).
Notice now that from (*) and (**) we have that for each , it follows immediately that φ A S ∈ ∂v A (p s 0 ).This concludes the proof of case (a).Now we prove case (b).Assume Cav(v A )(p 0 A ) = v A (p 0 A ). From Aumann et al. [1], an optimal strategy σ A of the informed player in G A (p 0 A ) is to play the mixed action which is optimal in the (one-shot) zero-sum game with matrix A(p 0 A ), independently at each stage, whereas the strategy of the uninformed player 2 is an approachability strategy τ A at p 0 A .By definition of the approachability strategy, there exists a vector and the profile (σ A , τ A ) is a uniform equilibrium, with associated vector of payoffs equal to φ A .We now observe that the pair (φ ) is clear.By the same reasoning as in the first paragraph of this proof A ), which is condition (3).The same reasoning as in the paragraph above (changing now x = T (p s 0 ) to x 0 = T (p 0 A )) gives that φ A ∈ ∂v A (p 0 A ), which is condition (2).This concludes the proof.

Appendix B
Proof of Lemma 4.3.Let (σ, τ A , τ B ) be an equilibrium profile of G(p 0 ).We start with the construction of the sequence of random variables (p t , β A,t , β B,t ) t∈N .The sequence (p t ) t∈N is the martingale of posteriors obtained from (σ, τ A , τ B ).It is therefore, immediately adapted to the sequence of increasing fields (H t ) t∈N .Fix now a Banach limit L : For notational convenience, we shall denote L((u t ) t∈N ) by L(u t ).A uniform equilibrium of the game G(p 0 ) automatically satisfies (1), ( 2) and (3) above, so it is an L-equilibrium.For each t ∈ N, We now show (1) of Lemma 4.3: we have already argued that (p t ) t∈N is a martingale and it 21 We thank an anonymous referee for pointing out a simplification of the proof of this step. 22For the definition of Banach limit, see Section 4.2 in Hart [10].
is immediate it is bounded.Fix now s < t and h s ∈ H s with P(h s ) > 0: 1 hs , we have that X := (X T ) T ∈N : Ω → ∞ is a random variable that takes finitely many values in ∞ .Therefore, by Lemma 4.6 in Hart [10], L commutes with E and we obtain: , where the last equality follows from L being linear.Now, L( 1 , from the definition of the conditional expectation on a finite field.As h s was arbitrarily chosen, we have ) t∈N is a martingale adapted to (H t ) t∈N (boundedness of this martingale is immediate from the fact that payoffs are bounded).The proof that (β B,t ) t∈N is a bounded martingale adapted to (H t ) t∈N is the same.We now show condition (2) of Lemma 4.3.Notice that where the last equality follows from the fact that payoffs of an equilibrium profile converge, by definition; the limit lim T →+∞ E[α T ] is precisely the expected payoff to player 1 in the game G A (p 0 A ) from the equilibrium.By the same reasoning, β B,1 is precisely the expected payoff of player 1 in game G B (p 0 B ).We now prove (3) from Lemma 4.3.Aiming for a contradiction, suppose there exists We will construct a profitable deviation for player 2. Let τ A be the following strategy.After h 0 t has occurred, τ A will be equal to a Blackwell strategy τA (p t ) for the uninformed player in G A (p tA ), where p t := p t (h 0 t ) is the posterior given h 0 t ; otherwise, τ A is equal to τ A .We have, where E is the expectation taken with respect to the p 0 , σ, τ A , τ B .We claim that , ∀t ∈ N. We will assume the claim for now and conclude the proof of (3).We then prove the claim.Taking L on the right-hand-side of the highlighted equation and using linearity of L, one gets: This implies a contradiction: since taking L on the left hand side we obtain the difference between the expected payoffs from (σ, τ A , τ B ) to player 1 and the expected payoffs from (σ, τ A , τ B ) to player 1.As the expected payoffs to player 1 decreased strictly, player 2 is strictly better, which is a contradiction with the fact that (σ, τ A , τ B ) is an equilibrium.
We now prove the claim: note that, 1. Nondegeneracy of I(p 0 ) is Robust to payoff perturbations.The next result shows that the property of the interval I(p) being nondegenerate is robust to stage-payoffs perturbations.Proposition 9.4.Let I A , J A , I B , J B , K A and K B be finite sets.Let G denote the set of vectors that define a G(p) for which Notice that (f + g) is continuous, by Lemma 9.2.
Let now (A 1 n , ..., A as the set of matrices (A 1 , ..., A |K| ) such that (A 1 , ..., A |K| , p) is locally nonrevealing at p. We will show that G p has non-empty interior in k∈K R I×J .We will consider wihtout loss of generality p ∈ int(∆(K)).If the prior p is not in the interior of ∆(K), then we drop the type that has probability zero from set K.
Let {f i } n i=1 be a finite collection of real affine functions defined on ∆(K) such that f i (p) := a i •p+ b i .This collection defines a concave piecewise linear function by letting H(p) := min{f 1 (p), ..., f n (p)}.This concave piecewise linear function H induces a polyhedral subdivision 25 on ∆(K) by projecting the faces of the graph of H over ∆(K).Consider a point p 0 in the interior of ∆(K) and consider the following polytope of R K × R: let y 0 = (p 0 , x) ∈ ∆(K) × R where x > 0 and let v k = (v k , 0) where v k is a vertex of ∆(K).Consider P = co{y 0 , v 1 , ..., v k }.The boundary of P minus int(∆(K)) × {0} is the graph of a concave piecewise linear function H y 0 , with each maximal proper face of P corresponding to the graph of an affine function.The polyhedral subdivision on ∆(K) induced by this concave piecewise linear function will be denoted P y 0 .Proposition 9.5.Assume |I|, |J| ≥ |K| and p ∈ int(∆(K)).Then there exists a game (A 1 , ..., A |K| ) ∈ G p .
Proof.We prove a slightly stronger result: we will show that there exists (A 1 , ..., A K , p) such that for any (p i ) n i=1 ⊂ ∆(K) with n i α i p i = p and Cav(v A )(p) = n i=1 α i v A (p i ), there exists i 0 such that p i 0 ∈ int(∆(K)).
Choose p 0 = p in the interior of ∆(K) and consider the polytope P constructed just like in the previous paragraph.As discussed in the previous paragraph, consider the finite collection of affine functions {f i } J i=1 defining the concave piecewise linear function H y 0 whose graph is the boundary of polytope P .We show that the function H y 0 is the nonrevealing value of a game in G p .Let a t j be t-th entry of vector a j such that f j (p) = a j • p + b j .Define the matrix A t = [v t 1 , ..., v t J ] and η = min j,t {a t j }, where v t j is a column vector of length I with entries from 1 to |K| all equal to a t j +b j and η < η everywhere else.Consider the vector (A 1 , ..., A |K| , p).Notice that for each t, the rows of the submatrix of A t composed by the first |K| rows and |J| columns are all equal.So in the one shot zero-sum game given by matrix A(p), the row player is indifferent between these rows, which are in turn strictly better than the rows from |K| + 1 to I, by construction.The column player, who is a minimizer, chooses therefore the columns that minimizes the row players' payoffs: each column j of A(p) is a vector such that the first |K| entries are equal to f j (p) by construction.So the column player chooses j 0 such that f j 0 (p) = min j∈J {f 1 (p), ..., f J (p)} = H y 0 (p) = v A (p). Therefore the nonrevealing value function v A equals H y 0 .Now, let P y 0 be the polyhedral subdivision induced by H y 0 .By definition p belongs to a maximal cell C of P y 0 .If p belongs to the interior of C, then any optimal strategy of the informed player at p induces a posterior at the relative interior of ∆(K), because there is one vertex of C in the interior of ∆(K), namely p 0 .If p is in the relative interior of the intersection of two or more cells, then the intersection also has a vertex at p 0 , which implies that any optimal strategy of the informed player induces at least one posterior in the interior of the simplex.Proposition 9.5 above provides conditions under which G p is nonempty.We use it to prove the following robustness result: Proposition 9.6.Let p ∈ int(∆(K)) and |I|, |J| ≥ K. Then G p has nonempty interior.
Proof.Consider (A 1 , ..., A |K| , p) such that for any (p i ) n i=1 ⊂ ∆(K) with n i α i p i = p and Cav(v A )(p) = n i=1 α i v A (p i ), there exists i 0 such that p i 0 ∈ int(∆(K)).The existence of such a game is guaranteed by the proof of Proposition 9.5.Assume by contradiction there exists a sequence (A . This is a contradiction, since p i ∈ ∂∆(K), ∀i = 1, .., |K| + 1. 9.3.Existence of Equilibria in G(p 0 ).In this section we construct a particular kind of uniform equilibrium of G(p 0 ) called a joint-plan (see Aumann et al. [1]) which pays the lower end of I(p 0 ).x k A ,k B is a probability distribution on S.
• (Contracts) γ = (γ A , γ B ), with γ i = (γ s i ) s∈S , and γ s i := σ s i τ s i , σ s i ∈ (I i ) and τ s i ∈ (J i ), for i ∈ {A, B}.We denote by γ s A (i A , j A ) the probability of moves (i A , j A ) and γ s B (i B , j B ) the probability of moves (i B , j B ).
Following Lemma 2 in [18], the independent distribution γ s A = σ s A τ s A over I A × J A (resp.γ s B ) can be induced through the play of a deterministic sequence of moves at each stage by players 1 and 2 (resp.players 1 and 3) with the appropriate frequency.This deterministic path of play is what the contract γ s A (resp.γ s B ) represents.
Notation.Let (S, x, γ) be an independent joint-plan in G(p).We define some notation necessary for the statement of Lemma 9.10.The prior p and signaling strategy x define a probability distribution P on We define the posterior probability of (k A , k B ) ∈ supp(p) given the realization of s ∈ S by p k A ,k B (s) := P (k A ,k B ,s) , where . This is the probability over states, obtained by Bayes rule, that players 2 and 3 may compute after observing signals.For s ∈ S, ∈ {A, B}, given a posterior probability p(s) ∈ ∆(K A × K B ), the marginal posterior over K is p(s) ∈ ∆(K ).Also, for each s ∈ S, (k A , k B ) ∈ supp(p), and , j ∈ {A, B}, we define: the expected payoff of player 1 after ; the highest payoff player 1 can obtain after any signal when states are (k ∈supp(p) is their vector; the marginal expected payoff after s of player 1 in game ∈ {A, B} is α k (s) = i ,j k i ,j γ s (i , j ); we define by β (s) := k p k (s) α k (s) the negative of the expected payoff of the uninformed player playing game after signal s, and by β := s∈S P (s)β (s).Definition 9.9.An independent joint-plan (S, x, γ) in G(p) is called safe if for each s ∈ S, τ s A is optimal in the one-shot game with matrix A(p(s) A ) := k A ∈K A p k A (s)A k A and τ s B is optimal in the one-shot zero-sum game B(p(s) B ). Lemma 9.10.Let (S, x, γ) be an independent joint-plan in G(p) satisfying: (3) α • q ≥ h(q), for all q ∈ ∆(supp(p)).

Then there exists an equilibrium (σ, τ
Proof.Let (S, x, γ) be a safe independent joint-plan.We construct an equilibrium (σ, τ A , τ B ) in G(p) inducing the required joint-plan vector of payoffs.The proof is exactly analogous to Proposition 1 in [18].We indicate the steps of the construction.If (k A , k B ) ∈ supp(p) realizes, player 1 uses a state-dependent lottery (see Lemma 1 in [18]), using finitely many stages of his play to induce one of the posteriors (p(s)) s∈S .For each s ∈ S, Lemma 2 of [18] implies there exists h s A,∞ = ((i n A , j n A )) n≥1 such that for each (i A , j A ) ∈ I as n → +∞.Assume the state-dependent lottery of player 1 draws signal s ∈ S. Then player 1 will play according to h s A,∞ , as long as player 2 plays according to h s A,∞ .The asymptotic frequency induced by this deterministic play is γ s A .The expected payoff to player 2 given that s realizes is then −β A (s).Similarly, players 1 and 3 will play a deterministic sequence of moves according to some h s B,∞ such that the induced asymptotic frequency is γ s B .The expected payoff to player 3 given that s realizes is then −β B (s).The payoff to player 1 from following the deterministic sequence of moves in both games is α k A ,k B (s) = α k A ,k B , ∀s ∈ S that realize with positive probability, by condition (2).If any player deviates from the deterministic sequence of moves they are supposed to play after s realizes, the deviation is detectable.If player 2 deviates, player 1 will punish him by playing the optimal strategy of the zero-sum game G A (p A (s)).If player 3 deviates, player 1 will punish him by playing the optimal strategy of the zero-sum game G B (p B (s)).If player 1 deviates from his deterministic sequence in either game he is playing, the deviation is detectable by players 2 and 3 and players 2 and 3 will play the approachability strategy given by Theorem 9.18.During signaling stages, players 2 and 3 can play anything, but if player 1 makes a detectable deviation during signaling stages -by not using a signal s -then players 2 and 3 will also play the approachability strategy given by Theorem 9.18.We show that the strategies defined form a uniform equilibrium: first, condition (2) of Theorem 9.10 prevents any undetectable deviation of player 1 at signaling stages from being profitable.After signal s realizes, if player 2 deviates, then player 1 plays the optimal strategy of G A (p A (s)).Since β A (s) ≤ Cav(v A )(p(s) A ), it implies that the deviation is not profitable for player 2. The same reasoning applies for a deviation of player 3. The inequality α • q ≥ h(q), ∀q ∈ ∆(supp(p)) of condition ( 3), shows that a deviation is not profitable for player 1, because M (α) is approachable by players 2 and 3, by Theorem 9.18.
Condition (1) of the uniform equilibrium definition is immediately satisfied, because the payoffs to each player given by the strategies defined above converge.Also, condition (2) of the definition follows immediately from the fact that deviations are punished with approachability strategies, in case player 1 deviates, and optimal strategies of the zero-sum games, in case players 2 or 3 deviate.
Given an independent joint-plan (S, x, γ), the vector (α, −β A , −β B ) ∈ R supp(p) × R × R will be called the vector of payoffs of the equilibrium joint-plan.
Condition (1) in Lemma 9.10 guarantees that players 2 and 3 do not deviate after signaling stages from the deterministic path of play induced by the joint-plan contracts: in case signal s realizes, the contract γ s A has expected payoff to player 2 given s of −β A (s) and player 1 can punish player 2 in case player 2 deviates from the deterministic path of play given by the contract γ s A by playing his optimal strategy at G A (p(s) A ), which guarantees the payoff of player 2 would not be larger than −Cav(v A )(p(s) A ) ≤ −β A (s).The analogous reasoning holds to prevent a deviation of player 3. Condition (2) prevents undetectable deviations from player 1 at signaling stages: it says that player 1 cannot profit from "lying" about a signal because he is indifferent to the payoffs under any contract that realizes with positive probability, for any pair of states chosen by Nature.Condition (3) implies the existence of strategies for the uninformed players to punish the informed player in case he makes an observable deviation.This is the approachability strategy of the uninformed player in the repeated game G A+B (p). 27In section 9.4, we show this strategy can indeed be played by players 2 and 3.This will be a simple consequence of the fact that the payoffs for the informed player have a "separable" structure -they are the addition of payoffs obtained in each zero-sum game separately.Proposition 9.11.There exists an independent and safe joint-plan in G(p 0 ) satisfying (1), ( 2) and (3) of Lemma 9.10.Also, if (σ, τ A , τ B ) is the equilibrium induced by this joint-plan, then (σ, τ A , τ B ) pays Cav(h)(p) as an ex-ante payoff to the informed player. 2827 Recall GA+B(p) is a two-player zero-sum infinitely repeated game with one-sided incomplete information and undiscounted payoffs. 28The generalization of Lemma 9.10 to a model of one informed player and n uninformed players -as can be readily checked in the proof in the Appendix -is straightforward.
Proof.We first state Theorem 9.12 and Lemma 9.13 below, which are the main tools for the proof of Proposition 9.11.
Theorem 9.12.[Simon et al. [17]] Let K be a finite set and p 0 ∈ int (K).Let a : (K) → R and h : (I) × (K) → R |K| be continuous functions such that: (1) The function h is affine with respect to the variable σ ∈ (I), for all p ∈ (K).
(5) For all p ∈ P 0 , k ∈ K we have φ k ≥ h k (σ p , p), with equality occuring in place of inequality whenever p k > 0.
We will also make use of the following simple version of a Lemma in [17]: Lemma 9.13.For every > 0 there exists a continuous map g Assume first p 0 ∈ int∆(K A ×K B ).Given > 0, applying Lemma 9.13 we have that σ where m A σ := marg I A σ and m B σ := marg I B σ. Since the marginal operator is affine, the function h is affine on σ.It is also continuous.Now, given p, q ∈ (K A × K B ), let σq A be the optimal strategy of the informed player in the one-shot zero-sum game with matrix A(q A ) and let σq B be the optimal strategy of the informed player in the one-shot zero-sum game with matrix B(q B ). Define σ := σq A σq B ∈ ∆(I A × I B ). Then we have that h(σ, p) • q ≥ a(q) := h(q) = v A (q A ) + v B (q B ). Applying Theorem 9.12, we have that there exists P 4) and (5).From ( 3) and (4) we have that there exists a nonnegative collection (λ p ) p∈P 0 and a vector φ such that p∈P 0 λ p p = p 0 and p∈P 0 λ p = 1, Notice that the solutions given by the application of Theorem 9.12 are all indexed by > 0. For each n ∈ N, we can therefore consider s λ p n s = 1; also, φ n such that φ n • q ≥ a(q), ∀q with (5) being satisfied.Passing to a subsequence if necessary, consider the (Hausdorff) limit P 0 of the sequence P n 0 .Notice that P 0 has finite cardinality (less than |K A × K B |).We can also consider limits of the associated solutions, since they all lie in compact sets. 29Therefore, consider S a finite set, P 0 = {p s } s∈S , (σ ps ) s∈S , (g A (p sA )) s∈S , (g B (p sB )) s∈S , (λ ps ) s∈S and φ to be the limits of those sequences.It is straightforward to check that the limit of the sequences of solutions satisfy (3), ( 4) and ( 5).The joint-plan is now defined as follows: consider as contracts γ .This signaling strategy satisfies condition (2) of Lemma 9.10 and induces the appropriate posteriors: in case s ∈ S is observed, the uninformed players will update their priors according to Bayes rule to posterios p s .
We now calculate the ex-ante payoffs of player 1 obtained from the joint-plan just defined.Using condition (5) of Theorem 9.12, we have that ps λ ps (h(σ ps , p s )•p s ) = φ•p 0 = Cav(h)(p 0 ).Rewriting the expression for h, ps λ ps (m A σ ps A(p s A )τ A ps + m B σ ps B(p s B )τ B ps ) = Cav(h)(p 0 ).This proves the result for p 0 ∈ int∆(K A × K B ).If now p 0 ∈ ∂∆(K A × K B ), then consider the model G(p) defined for p ∈ int∆(supp(p 0 )), where pk A ,k B = p 0,k A ,k B , ∀(k A , k B ) ∈ supp(p 0 ) and apply the result proved to this case.9.4.Approachability Strategies.In this section we construct an approachability strategy for the uninformed player in game G A+B (p 0 ) and show that this strategy can indeed be played by players 2 and 3 in game G(p 0 ).The proof is simple but requires several preliminary definitions.This section is taken from Sorin [19] and adapted to our setting.Fix p ∈ ∆(K A × K B ).The main result is Proposition 9.18.
Let C be a |I A × I B | × |J A × J B |-matrix with coefficient in R supp(p) , where where a k A i A ,j A ∈ R and b k B i B ,j B ∈ R. We define a vector payoff zero-sum game: at stage n, player 1 (resp.player 2) chooses a move (i n A , i n B )(resp.(j n A , j n B )).The corresponding vector payoff g n = C i n A ,j n A ,j n A ,j n B is announced.Denote by h n the sequence of vector payoffs at stage n.This is the information available to both players up to stage n.Let g n = 1 n n m=1 g m be the vector of average payoffs up to stage n.Let ||C|| = max 29 Property (5) of Theorem 9.12 guarantees then that the sequence of vectors (φn) n∈N is bounded, so it will also have an accumulation point.
Definition 9.14.A set P ⊂ R supp(p) is approachable by player 2 if for any > 0 there exists strategy τ of player 2 and N ∈ N such that for any strategy σ of player 1 and n ≥ N : where d n is the euclidean distance d(g n , P ). Let where C i A ,i B ,j A ,j B τ j A ,j B := (C k A ,k B i A ,i B ,j A ,j B τ j A ,j B ) (k A ,k B )∈supp(p) .
Definition 9.15.A closed set P ⊂ R supp(p) is a B-set for player 2 if for any z / ∈ P there exists a closest point y = y(z) in P to z and a mixed move τ = τ (z) ∈ ∆(J A ×J B ), such that the hyperplane through y orthogonal to the segment [y, z] separates z from Cτ .Theorem 9.16.Let P be a B-set for player 2. Then P is approachable by that player.More precisely with a strategy satisfying τ (h n+1 ) = τ (g n ), whenever g n / ∈ P , one has: and d n converges P σ,τ a.s. to 0.
Remark 9.17.The strategy τ obtained in the statement of Theorem 9.16 above will be called an approachability strategy.
For z ∈ R supp(p) , let M (z) := z − R supp(p) + .Proposition 9.18.Let z ∈ Z ∞ = {z ∈ R supp(p) |z • q ≥ h(q), ∀q ∈ ∆(supp(p))}.Then player 2 can approach M (z).Also, the approachability strategy τ for player 2 can be assumed to satisfy Proof.Approachability of M (z) follows from Theorem 3.33 in [19], where it is checked that M (z) is a B-set.Fix g n and let τ (g n ) be the mixed move associated with the B-set M (z) and g n in Theorem 9.16.Let τ A = marg J A τ and τ B = marg J B τ .We show Cτ = C(τ A τ B ): let x ∈ Cτ such that x := (i A ,i B ) λ i A ,i B j A ,j B C i A ,i B ,j A ,j B τ j A ,j B with i A ,i B λ i A ,i B = 1 and λ i A ,i B ≥ 0.Then, which implies that x ∈ C(τ A τ B ). Now, let y := (i A ,i B ) λ i A ,i B j A ,j B C i A ,i B ,j A ,j B τ A j A τ B j B ∈ C(τ A τ B ), with (i A ,i B ) λ i A ,i B = 1 and λ i A ,i B ≥ 0. Using the the same equalities above, it implies y ∈ Cτ .So, if player 2 uses τ t (h t ) := τ A (h t ) τ B (h t ) after each h t , τ is an approachability strategy.9.5.A Continuum of Equilibria in G(p 0 ).In subsection 3.1 of the main paper we showed how Cav(v A )(p 0 A ) + Cav(v B )(p 0 B ) is an equilibrium of G(p 0 ), provided some conditions are satisfied.Proposition 9.11 in this Appendix shows how Cav(h)(p 0 ) is always an equilibrium payoff of the informed player in G(p 0 ).The next proposition immediately implies that any element in I(p 0 ) is an ex-ante equilibrium payoff of the informed player, implying a continuum of equilibria exists in the model G(p 0 ).Proposition 9.19.Let (σ, τ A , τ B ) and (σ , τ A , τ B ) be two uniform equilibria of G(p 0 ) with associated ex-ante payoffs γ and γ for the informed player, respectively.For any α ∈ (0, 1), there exists a uniform equilibrium (σ, τA , τB ) whose ex-ante equilibrium payoff to the informed player is αγ + (1 − α)γ .
Proof.Fix α ∈ (0, 1).We show that αγ + (1 − α)γ is an ex-ante equilibrium payoff for the informed player in G(p 0 ).Since each of the uninformed players can play their optimal strategy in their repeated zero-sum games G A (p 0 A ) or G B (p 0 B ), it implies that Cav(v A )(p 0 A ) + Cav(v B )(p 0 B ) ≥ γ as well as Cav(v A )(p 0 A ) + Cav(v B )(p 0 B ) ≥ γ .Now, since the informed player has an optimal strategy in the zero-sum game G A+B (p 0 ) that guarantees him Cav(h)(p 0 ), it implies that γ ≥ Cav(h)(p 0 ) as well as γ ≥ Cav(h)(p 0 ).Consider a jointly controlled lottery that implements the equilibrium profile associated with γ with probability α and the equilibrium associated with γ with probability 1 − α.By the properties of the jointly controlled lottery, there cannot be profitable undetectable deviations at the stages where the jointly controlled lottery is played.For detectable deviations of the uninformed player 2 (respectively, player 3) at the lottery stages, the informed player plays the optimal strategy of the zero-sum game G A (p 0 A ) (respectively G B (p 0 B )) to punish.For detectable deviations of the informed player at the lottery stages, the uniformed players play the approachability strategy of Theorem 9.18 to punish.The strategy profile where a jointly controlled lottery is played at initial stages -with deviations punished as described -and, after that, the corresponding strategy profile paying γ or γ drawn from the lottery, is a uniform equilibrium of the game G(p 0 ).Indeed, we already showed that there are no profitable deviations during lottery stages.After lottery stages, players play a uniform equilibrium so there is no profitable deviation for any player, also.The ex-ante payoff of this equilibrium is αγ + (1 − α)γ .

Figure 1 . 1 0 Figure 2 .
Figure 1.Graphs of Cav(v A ) and v A

Figure 3
illustrates how the conditions of Definition 3.8 are related to the property of N R at p 0 B .Let G B (p 0 B ) be the game originating the non-revealing value function v B (whose graph is depicted in black).The data defining this game is the exact same as in Example 2.4.In this figure, let p 0 B = 1/2 denote the prior probability of state 1.One can obviously write p 0 B as a convex combination with

Figure 5 .
Figure 5. Graphs of Cav(v A )(dotted) and v A (continuous)

Theorem 4 . 1 .
Let (σ, τ A , τ B ) be an equilibrium of G(p 0 ) and let (p t ) t∈N be the martingale of posteriors induced by the equilibrium.Assume (σ, τ A , τ B ) pays ex-ante Cav(v A )(p 0 A )+Cav(v B )(p 0 B ) to the informed player.Then (Cav(v )(p t )) t∈N is a martingale, for each ∈ {A, B}.
) n∈N and (v Bn ) n∈N be the corresponding sequences of nonrevealing value functions.By Lemma 9.2, h n converges uniformly to h, v An converges uniformly to v A and v Bn converges uniformly to v B .It follows now by Theorem 9.3 that there exists n 0 ∈ N such that ∀n ≥ n 0 we have Cav(h n )(p) < Cav(v An )(p A ) + Cav(v Bn )(p B ). Since the sequence considered is arbitrary, it follows that there exists an open set around (A 1 , A 2 , ..., A |K A | , B 1 , B 2 , ..., B |K B | ) where I(p) is nondegenerate.9.2.The property of Local Nonrevelation at the Prior is not Non-Generic.Define G p

Definition 9 . 7 .
Let h 1 m := (i t A , i t B ) 1≤t≤m−1 .We define H 1 n := {h 1 n } and call it the set of individual histories of player 1. Definition 9.8.An independent joint-plan 26 in G(p) is a triple (S, x, γ) where: • (Signals) S is the set of signals, i.e., a subset of H 1 n , for some n ∈ N. • (Signaling Strategy) The vector x is a |supp(p)|-tuple where for each (k A , k B ) in supp(p), s A = m A σ ps g A (p sA ), γ s B = m B σ ps g B (p sB ).Let τ A ps = (g A (p sA )) T and τ B ps = (g B (p sB )) T .By construction, max σ σA(p sA )τ A ps = v A (p sA ) and max σ σB(p sB )τ B ps = v B (p sB ).Therefore it implies that (m A σ ps )A(p sA )τ A ps ≤ v A (p sA ) and (m B σ ps )B(p sB )τ B ps ≤ v B (p sB ), for each s ∈ S.This implies in particular (1) of Lemma 9.10 is satisfied.Condition (3) of Lemma 9.10 follows directly from the fact that condition (5) of Theorem 9.12 above is satisfied by φ.We can now define the signaling strategy x: let S ⊂ H 1 t 0 with|H 1 t 0 | ≥ |K A × K B |, for t 0 ∈ N. Define x k A ,k B (s) = λ ps p k A ,k B s p 0,k A ,k B be the vector of posteriors induced by an optimal strategy given by Lemma 9.1 for the zero-sum game with lack of information on one side with prior p defined by (A 1 s , ..., A |K| s ).By assumption, p s i ∈ ∂∆(K), for all i = 1, .., |K|+1.Passing to convergent subsequences if necessary, we can assume that p s i → p i ∈ ∂∆(K), for each i = 1, ..., |K| + 1.Similarly, assume α s i → α i , for each i = 1, ..., |K| + 1.