Long information design

Koessler, Frédéric; Laclau, Marie; Renault, Jérôme; Tomala, Tristan Article Long information design Theoretical Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Koessler, Frédéric; Laclau, Marie; Renault, Jérôme; Tomala, Tristan (2022) : Long information design, Theoretical Economics, ISSN 1555-7561, The Econometric Society, New Haven, CT, Vol. 17, Iss. 2, pp. 883-927, https://doi.org/10.3982/TE4557 This Version is available at: https://hdl.handle.net/10419/296373 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/4.0/ Theoretical Economics 17 (2022), 883–927 1555-7561/20220883 Long information design Frederic Koessler Paris School of Economics—CNRS Marie Laclau HEC Paris and GREGHEC—CNRS Jérôme Renault TSE (Université Toulouse 1 Capitole) Tristan Tomala HEC Paris and GREGHEC—CNRS We analyze information design games between two designers with opposite preferences and a single agent. Before the agent makes a decision, designers repeatedly disclose public information about persistent state parameters. Disclosure continues until no designer wishes to reveal further information. We consider environments with general constraints on feasible information disclosure policies. Our main results characterize equilibrium payoffs and strategies of this long information design game and compare them with the equilibrium outcomes of games where designers move only at a single predetermined period. When information disclosure policies are unconstrained, we show that at equilibrium in the long game, information is revealed right away in a single period; otherwise, the number of periods in which information is disclosed might be unbounded. As an application, we study a competition in product demonstration and show that more information is revealed if each designer could disclose information at a predetermined period. The format that provides the buyer with most information is the sequential game where the last mover is the ex ante favorite seller. Keywords. Bayesian persuasion, concavification, convexification, information design, Mertens–Zamir solution, product demonstration, splitting games, statistical experiments, stochastic games. JEL classification. C72, D82. Frederic Koessler: [email protected] Marie Laclau: [email protected] Jérôme Renault: [email protected] Tristan Tomala: [email protected] Frederic Koessler acknowledges the support of the ANR (Investissements d’Avenir program ANR-17-EURE001 and StratCom ANR-19-CE26-0010-01). Marie Laclau gratefully acknowledges the support of the ANR through the program Investissements d’Avenir (ANR-11-IDEX-0003/Labex Ecodec/ANR-11-LABX-0047) and under grant ANR CIGNE (ANR-15-CE38-0007-01). Jérôme Renault gratefully acknowledges funding from ANR-3IA Artificial and Natural Intelligence Toulouse Institute, grant ANR-17-EUR-0010 (Investissements d’Avenir program) and ANR MaSDOL. Tristan Tomala gratefully acknowledges the support of the HEC foundation and ANR/Investissements d’Avenir under grant ANR-11-IDEX-0003/Labex Ecodec/ANR11-LABX-0047. ©2022 The Authors. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at https://econtheory.org.https://doi.org/10.3982/TE4557 884 Koessler, Laclau, Renault, and Tomala Theoretical Economics 17 (2022) 1. Introduction In many environments, economic agents spend time acquiring information before making decisions, and information often comes from multiple and interested sources. This paper analyzes strategic interactions between two information providers (called the designers) who compete for influencing the action of a decision-maker (called the agent). The designers have opposite preferences and control the access to independent pieces of information. The first key feature of our model is that designers are able to disclose information in multiple stages, as long as they want. Specifically, at each stage, designers disclose information publicly and choose new disclosure policies at the next stage. Hence, in each stage, a designer can react to the information disclosed by himself and by the other designer. Once both designers have finished releasing information, the agent chooses an action. The payoffs of the designers and of the agent depend on the realized state parameters and on the action taken. The second feature of our model is that it introduces general technological restrictions on available information disclosure policies. The model encompasses the standard information design setting in which a designer can choose any public information policy, but it also covers more realistic environments in which a designer may be constrained to choosing information policies from an exogenous subset. For example, in most real world information design problems, a designer may be constrained to choose deterministic information structures, may run experimental tests with unavoidable false–positive or false–negative results, or may rely on imperfectly reliable experts and reviewers. As in the literature on Bayesian persuasion initiated by Kamenica and Gentzkow (2011), the term “information designer” refers to a player who is uninformed about a state parameter but is able to choose an information disclosure policy (a statistical experiment) about this parameter to modify agents’ information. Contrary to cheap-talk communication, there is no issue of credibility in information disclosure: the statistical experiment is publicly observable and verifiable. In addition, since an information designer is not privately informed about the state, the choice of the disclosure policy has no signaling effect. In practice, information designers can represent competing sellers who release information about a new product to influence buyers’ valuations. For example, sellers can offer free samples or trial periods to social media influencers, control the access to and content of online, press, or magazine reviews, organize product testing and trade fairs, or make announcements about future product development. Alternatively, information designers could be lobbyists who control the informativeness of some studies to influence a policymaker. In such contests, the value of releasing additional information depends on the information publicly revealed by the competing designer. The aim of the paper is to study designers’ equilibrium information disclosure policies and payoffs in multistage information design problems in which there is a priori no deadline, that is, no bound on the number of disclosure stages. We refer to the model with no deadline as “long” information design even though, under some conditions on the primitives of the game (see Proposition 2), information disclosure ends very quickly in equilibrium. We compare equilibrium disclosure strategies and players’ Theoretical Economics 17 (2022) Long information design 885 welfare of such long information design game, with those of information design games in which each designer can disclose information only at a predetermined stage, either simultaneously (the one-stage simultaneous-move game) or sequentially (the two-stage sequential-move game). Beyond the theoretical interest, our analysis is motivated by the question of how a decision maker should choose to acquire information from competing designers. For example, a committee could decide in which order (i) evidences should be presented, (ii) experts and reviewers should be consulted and whether to set a deadline to communication. Similarly, a buyer could decide how to acquire information from competing sellers by committing to a purchasing period or by choosing in which order to visit them. Contributions The main result of this paper is a characterization of equilibrium payoffs and strategies of long information design games in which designers have opposite preferences, noninformative policies are always feasible, and feasible policies are closed under iteration (i.e., a distribution of beliefs that can be obtained by a two-step combination of experiments can also be obtained by a single feasible experiment).1We show that the long information design game admits a stationary equilibrium, that is, such that designers’ strategies depend only on the current beliefs. In addition, if there is no constraint on the set of available information disclosure policies, there exists an equilibrium in which information is disclosed at the first stage only. That is, along the equilibrium path, at most one designer discloses some information at the first stage, information disclosure policies are uninformative thereafter, and the agent therefore takes his decision at the end of the second stage. If information disclosure policies are constrained, we provide an example in which, in every equilibrium, the number of disclosure stages is unbounded, even under the maintained assumption that feasible policies are closed under iteration (see Section 3.4.2). As in the usual case of one designer of Kamenica and Gentzkow (2011), the value of the game (the equilibrium payoff of Designer 1) is derived from the expected payoff of Designer 1, denoted by u(p1,p2), as a function of the beliefs (p1,p2),wherepiis the public belief about the information controlled by Designer i. Equilibrium strategies can be directly backed out from the characterization of the value of the game. First, assume that only Designer 1 is active (e.g., because Designer 2 is constrained to be silent); then, according to Kamenica and Gentzkow (2011), Designer 1 would “concavify” the payoff function with respect to p1, obtaining what we denote by cavp1u(p1,p2). Similarly, if only Designer 2 were active, we would obtain the “convexification” with respect to p2, which we denote by vexp2u(p1,q2). Consider now the two-stage sequential information design game in which Designer 1 can disclose information in the first stage only, and Designer 2 in the second 1Under this latter condition, which holds in standard Bayesian persuasion models, multiple stages of information disclosure is irrelevant if there is only one designer. However, with more than one designer, multiple stages of information disclosure become relevant because they allow each designer to react to the information released by the competitor. In a companion paper (Koessler, Laclau, Renault, and Tomala (2021)), we study particular cases where the assumption that feasible policies are closed under iteration can be relaxed. 886 Koessler, Laclau, Renault, and Tomala Theoretical Economics 17 (2022) stage. By backward induction, the value of the game is cavp1vexp2u(p1,p2).Similarly, if Designer 2 moves first and Designer 1 last, then the value of the game is vexp2cavp1u(p1,p2). In the one-stage simultaneous move game in which designers move simultaneously in a single stage, the value, which we call the splitting game value, is in between cavp1vexp2u(p1,p2)and vexp2cavp1u(p1,p2). In the long information design game, the values and equilibrium strategies are the same for games with simultaneous or alternating moves, and this value is the unique function v(p1,p2),whichwe call the Mertens–Zamir function, that satisfies the following system: v(p1,p2)=cavp1min(u(p1,p2),v(p1,p2)=vexp2max(u(p1,p2),v(p1,p2). This system is key to the study of discounted zero-sum repeated games with incomplete information on both sides (Mertens and Zamir (1971,1977)) and of zero-sum dynamic gambling games (Laraki and Renault (2020)). It allows simple optimal strategies to be derived directly: Designer 1 plays noninformatively if u(p1,p2)≥v(p1,p2)and Designer 1 concavifies min(u,v)if u(p1,p2)<v (p1,p2); Designer 2 plays noninformatively if u(p1,p2)≤v(p1,p2)and Designer 2 convexifies max(u,v)if u(p1,p2)>v (p1,p2). An intuition for this result is as follows. When a Mertens–Zamir function exists, for any profile of current posterior beliefs (p1,p2), there exists an information disclosure policy for Designer 1 such that, if the game would end after his move, then his payoff would be greater than or equal to the value of the Mertens–Zamir function at (p1,p2). Similarly, there exists an information disclosure policy for Designer 2 such that, if the game would end after his move, then the continuation payoff of Designer 1 would be less than or equal to the value of the Mertens–Zamir function at (p1,p2). Proceeding backward, this implies that at the prior beliefs, Designer 1 has an information disclosure strategy that gives him a continuation payoff greater than or equal to the value of the Mertens–Zamir function at the priors, and Designer 2 has an information disclosure strategy that gives Designer 1 a continuation payoff less than or equal to the value of the Mertens–Zamir function at the priors. Thus, the Mertens–Zamir value at the priors can be guaranteed by both designers; since the game is zero-sum between the two designers, it is the unique equilibrium payoff of the game. The value of the long information design game is between cavp1vexp2u(p1,p2)and vexp2cavp1u(p1,p2), so if these two quantities are equal, all variants of the multistage information design games have the same value. Indeed in such a situation, the Mertens– Zamir value at the priors can be guaranteed by Designer 1 even if he moves first and only once, and it can be defended by Designer 2 even if he moves first and only once, so the order of moves and the time horizon are irrelevant. Hence in this case, the information revealed to the agent in equilibrium is the same for any information disclosure protocol. When cavp1vexp2u(p1,p2)=vexp2cavp1u(p1,p2), it does not matter how the agent acquires information from competing designers.2 2The way the agent acquires information when the two designers have access to the same information (i.e., states are perfectly correlated) and all information disclosure policies are feasible is also irrelevant because full information disclosure is always an equilibrium in this case (see Section 5.3 where we discuss the general correlated case). Theoretical Economics 17 (2022) Long information design 887 We provide examples and economic applications in which cavp1vexp2u,vex p2cavp1u, the splitting game value and the Mertens–Zamir function are all different. In such situations, the protocol of information disclosure determines the information revealed to the agent in equilibrium. Hence, our equilibrium characterizations allow to derive directly the agent’s preference over disclosure protocols. This comparison is especially relevant if the agent can commit to an information acquisition strategy, or if the disclosure protocol can be regulated. Application to a product demonstration by two sellers We apply our results and methodology to a stylized model of competitive and public product demonstration in which two sellers disclose information about their respective products to a representative buyer. All players are initially uncertain about the buyer’s valuation (or “match”) for each product.3The buyer decides to buy from the seller for whom the expected match conditional on the public information is highest (see Boleslavsky and Cotton (2015)for the analysis of the one-stage simultaneous information design game). We show that regardless of the prior expected match values, sellers’ equilibrium strategies are always less informative in the long information design game than the the one-stage simultaneous and sequential information design games. In the long information design game, only one designer discloses information, and the expected payoff of the buyer is the same as without information disclosure. The best disclosure protocol for the buyer is the two-stage sequential game in which the seller with the highest ex ante value moves last. The intuition for this result is as follows. When seller iknows that he is the last mover, he has an incentive to choose an information policy that has the highest chance to strictly improve his expected match value compared to his competitor j=i.Inturn, this incentivizes seller jto reveal additional match information, especially if seller jhas a lower ex ante expected match. On the contrary, in the long information design game, where each seller has the opportunity to react to the information policy of his competitor by disclosing additional information later, no seller has an incentive to strictly improve his expected match compared to his competitor. Indeed, if seller ichooses an information policy that induces a strictly higher expected match than jwith positive probability, then seller jis able to react with an another information disclosure policy that improves his expected match value compared to seller i’s expected match value with positive probability. Such a deviation by seller iis actually strictly detrimental because it provides an informational advantage to seller j: if seller i’s information policy generates a bad signal, then seller jtakes over the market without revealing any information. Formally, in this application, the Mertens–Zamir function is strictly concave in the prior about seller 1 and strictly convex in the prior about seller 2. Related literature The methodology and results of Bayesian persuasion (Kamenica and Gentzkow (2011)) and information design (e.g., Bergemann and Morris (2016a,b), Mathevet, Perego, and Taneva (2020), and Taneva (2019)) are deeply related to repeated games with incomplete information on one side (Aumann and Maschler (1966,1967), Aumann, 3For a literature review of match advertising see, for example, Renault (2015). 888 Koessler, Laclau, Renault, and Tomala Theoretical Economics 17 (2022) Maschler, and Stearns (1995)) and to the literature on generalized principal-agent problems, correlated and communication equilibria (Aumann (1974), Myerson (1982,1986), Forges (1986,1993)). See, for example, the literature reviews in Kamenica (2019), Bergemann and Morris (2019)andForges (2020). Bayesian persuasion models with constraints on information disclosure policies appear in Perez-Richet (2014), in Boleslavsky and Kim (2018)andSalamanca (2021), where the designer chooses from Bayes-plausible distributions that satisfy some incentive constraints, in Le Treust and Tomala (2019), where the designer is constrained to sending noisy messages, in Wu (2020), where the accuracy of feasible test designs is bounded due to exogenous false–negative errors, and in Matyskova and Montes (2021), where the receiver is able to gather additional information. The strategic interaction between multiple information designers has been studied under the assumption of simultaneous and one-stage information disclosure by, among others, Gentzkow and Kamenica (2017), Albrecht (2017), Au and Kawai (2020,2021), Boleslavsky and Cotton (2015,2018), and Koessler, Laclau, and Tomala (2022). Gentzkow and Kamenica (2017) consider the case in which each designer is able to choose an information policy that is more informative than that of the other designer.4 Albrecht (2017), Au and Kawai (2020,2021), and Boleslavsky and Cotton (2015,2018)consider the case in which designers control independent pieces of information in applied examples. Koessler, Laclau, and Tomala (2022) provide existence results and properties of equilibria in games with multiple designers and multiple agents. This latter paper assumes that designers disclose information simultaneously, followed by agents making decisions simultaneously as well. Multistage information design with a single designer has been studied in dynamic decision problems by, among others, Doval and Ely (2020), Ely (2017), Renault, Solan, and Vieille (2017), and Makris and Renou (2021). Since we assume that the state of nature is persistent and the decision problem is static (the agent makes a decision only once), multistage information design would be irrelevant in our model if there were only one designer. The dynamics of information design is interesting in our setting precisely because there are multiple designers. Sequential information design with multiple information designers has been studied by Li and Norman (2021)(seealsoWu (2020)), but there are important differences with our work. First, in Li and Norman (2021), the time horizon is finite and commonly known. Second, as in Gentzkow and Kamenica (2017), Li and Norman (2021) assume that designers disclose information about a common state and therefore each designer is able to choose an information policy that is more informative than the one of the competitor. Under these assumptions, they show that the sequential game cannot generate a more informative equilibrium than the simultaneous game. This property is not true in our model.5 4In our model, if designers can both reveal all the information about the payoff-relevant state (i.e., if their private states are perfectly correlated and all information disclosure policies are available), then the value of every multistage information design game coincides with the expected payoff under full information for the agent. See Section 5.3. 5See, for example, Section 4. Theoretical Economics 17 (2022) Long information design 889 Our methodology and results are closely related to the contributions in the literature on repeated games with incomplete information on both sides, splitting games and acyclic gambling games. The Mertens–Zamir function has been introduced by Mertens and Zamir (1971)(seealsoMertens and Zamir (1977), Sorin (2002)andMertens, Sorin, and Zamir (2015)) for the value function u(p1,p2)of the one-stage incomplete information zero-sum game, which we replace by the indirect utility function of Designer 1. Mertens and Zamir (1971) have shown that the Mertens–Zamir function is the limit of the value of the infinitely repeated and discounted game as the discount factor tends to one.6It is also the value of zero-sum splitting games studied in Laraki (2001a,b)and Oliu-Barton (2018). Laraki and Renault (2020) have recently extended these results from splitting games to more general stochastic games, called acyclic gambling games. We consider the indirect utility function of designers, given the beliefs and sequentially rational actions of the agent, and obtain our results by adapting the methodology of Renault and Venel (2017)andLaraki and Renault (2020). One main difference with Laraki and Renault (2020) is that we consider terminal payoffs (when the agent makes the decision), while payoffs in splitting games and acyclic gambling games are cumulated and discounted. We also consider and compare various possible timings of the game. Finally, a technical contribution with respect to Laraki and Renault (2020) is that we introduce a continuity condition on the feasible disclosure policies correspondences, which is adapted to information design and is different from the nonexpansivity condition of Laraki and Renault (2020). The timing of our long information design game is inspired by that of long cheaptalk games (Forges (1990), Aumann and Hart (2003)). In one of our examples, the equilibrium martingale of posteriors does not reach its limit within a bounded number of disclosure stages, similar to the “four frogs” example in Aumann and Hart (1986)and Forges (1984,1990). The one-stage simultaneous game of our application to a competitive product demonstration by two sellers has been studied, among others, by Albrecht (2017), Au and Kawai (2020)andBoleslavsky and Cotton (2015,2018). Independently to our work, Whitmeyer (2020) considers a multistage extension of this example with a finite horizon Nand discounted payoffs. Differently from our setting, he assumes that there is a sequence of short lived buyers, each buyer takes a decision once at a predetermined period. He directly solves the game by backward induction and, similar to us, shows that less information is revealed by the designers in the long information design game than in the static game. Precisely, the solution of Whitmeyer (2020)ofthisexample,converges to the solution of our long information design game when the time horizon Ntends to infinity and the discount factor tends to one. Structure of the paper In Section 2, we present the model. The main results are in Section 3. The application to competition in product demonstration is developed in Section 4.Several generalizations and extensions are studied in Section 5:weprovide 6It is also the limit of the value of the undiscounted N-stage repeated game as N→∞.Ifcav p1vexp2 differs from vexp2cavp1, the undiscounted infinitely repeated game has no value (see Aumann, Maschler, and Stearns (1995)). 890 Koessler, Laclau, Renault, and Tomala Theoretical Economics 17 (2022) sufficient conditions under which our results apply with discontinuous indirect utility functions (Section 5.1); we consider alternative extensive form games with no or exogenous stopping rules (Section 5.2), allow for correlated private states (Section 5.3), and consider the case in which the messages generated by information disclosure policies are unobservable by the designers (Section 5.4). All proofs that are not in the text are in the Appendix. 2. Model 2.1 Environment There are two information designers and a single agent. There is a finite set of states 1×2that is endowed with a common prior probability distribution. At the start of the game, no player is informed about the state. For each i∈{1, 2}, Designer iis able to design public information about i. To simplify the exposition, we assume that the prior probability distribution is the product of its marginal distributions, p0 1⊗p0 2,where p0 i∈(i)for each i(in all the paper, (X)denotes the set of Borel probability measures over a compact set X). The extension to correlated priors, which includes the extreme case in which designers can choose information disclosure policies on a common state space, is studied in Section 5.3. Designers produce and disclose publicly independent pieces of information; thus, each Designer icontrols the public belief pi∈(i)in a Bayes-plausible way. 2.2 Information disclosure and admissible splittings An information disclosure policy for Designer iis a public information structure àla Blackwell for i. When the public belief about iis pi∈(i), the policy induces a probability distribution over posterior beliefs with expectation pi. Such a meanpreserving spread is called a splitting of pi. The set of splittings of pi∈(i)for Designer iis denoted: Si(pi)=s∈(i):˜ pi∈(i)˜ pids(˜ pi)=pi. Our analysis covers situations in which designers are able to induce any splitting, that is, any Bayes-plausible distribution of posteriors, as in the benchmark information design models. We also consider restricted sets of feasible information policies (for both or just one designer); this is important for the model to represent more realistic environments; restricting the set of feasible information policies also helps to illustrate our results with tractable examples. For instance, a simple restriction is to allow designers to choose between either fully revealing or nonrevealing policies, as in the simple illustrative example of Section 3.4.1.InSection2.6, we discuss richer cases of constrained information policies, including restrictions found in the literature. If designers face technological constraints when choosing information disclosure policies, then the sets of splittings of beliefs that designers are able to induce are constrained as well. We model those constraints as follows. For each i,letPi⊆(i)be Theoretical Economics 17 (2022) Long information design 897 Notice that cavp1vexp2u(p0 1,p0 2)is the value of the 2-stage sequential information design game in which Designer 1 plays first and Designer 2 second, and vexp2cavp1u(p0 1, p0 2)is the value of the sequential game in which Designer 2 plays first and Designer 1 second. Indeed consider the sequential game in which Designer 1 plays first and Designer 2 second. At the first stage, Designer 1 chooses a splitting s1∈S1(p0 1)that maximizes u(s1,p0 2). For any strategy of Designer 2, the expected payoff is Ecavp1u(p0 1,p20)≥ vexp2cavp1u(p0 1,p0 2). Hence, Designer 1 has a strategy that guarantees a payoff of at least vexp2cavp1u(p0 1,p0 2)irrespective of the strategy of Designer 2. Define a strategy σ2of Designer 2 as follows. At the second stage, Designer 2 chooses a splitting s2∈S2(p0 2)that minimizes Es2cavp1u(p1 1,p2). Regardless of the strategy of Designer 1, the expected payoff is at most Evexp2cavp1u(p1 1,p0 2)≤vexp2cavp1u(p0 1,p0 2) by S1-concavity of vexp2cavp1(p1,p2). Hence, Designer 2 has a strategy that guarantees that Designer 1’s payoff is at most vexp2cavp1u(p0 1,p0 2)for any strategy of Designer 1. Therefore, vexp2cavp1u(p0 1,p0 2)is the value of the game. The argument is analogous for the sequential game in which Designer 2 plays first and Designer 1 second. More generally, it can be shown that cavp1vexp2u(p0 1,p0 2)is the value of any multistage information design game in which Designer 2 moves last, and vexp2cavp1u(p0 1,p0 2)is the value of any multistage information design game in which Designer 1 moves last. In particular, the previous lemma implies that vexp2cavp1vexp2u(p0 1,p0 2)=cavp1vexp2u(p0 1,p0 2)and cavp1vexp2cavp1u(p0 1,p0 2)=vexp2cavp1u(p0 1,p0 2). 3.3 Values and equilibria of the long information design game The following definition is due to Mertens and Zamir (1971). Definition 3. Let u:P1×P2→Rbe a continuous function. A Mertens–Zamir (MZ) function for uis a function v:P1×P2→Rsuch that for every (p1,p2)∈P1×P2, v(p1,p2)=cavp1min(u(p1,p2),v(p1,p2)=vexp2max(u(p1,p2),v(p1,p2). By definition, if vis a MZ function for u, then vis S1-concave and S2-convex. Section 3.4 provides two simple illustrations of this definition. Below we show that when the Mertens–Zamir function exists, then its value at the prior beliefs is equal to the value of the long information design (Theorem 1) and lies in between the values of the 2-stage sequential games in which each designer plays only once (Lemma 3). We also provide an alternative formulation of the MZ function (Proposition 1) that allows simple optimal strategies to be derived directly. Then we provide general conditions on the splitting correspondances for the Mertens–Zamir function to exist. Lemma 3. If vis a MZ function for u, then for every (p1,p2)∈P1×P2, cavp1vexp2u(p1,p2)≤v(p1,p2)≤vexp2cavp1u(p1,p2). Proposition 1. A continuous function visaMZfunctionforuif and only if vis S1concave, S2-convex and satisfies the following properties for all (p1,p2)∈P1×P2: 898 Koessler, Laclau, Renault, and Tomala Theoretical Economics 17 (2022) (C1) There exists s1∈S1(p1)such that v(p1,p2)=v(s1,p2)and v(p 1,p2)≤u(p 1,p2), ∀p 1∈supp(s1). (C2) There exists s2∈S2(p2)such that v(p1,p2)=v(p1,s2)and v(p1,p 2)≥u(p1,p 2), ∀p 2∈supp(s2). This result is very useful since it allows deriving intuitive strategies from the Mertens–Zamir function (see the discussion after the next theorem). Theorem 1. If there exists a continuous MZ function of u, denoted by MZ(u)(p0 1,p0 2), then the long information design game G(p0 1,p0 2)has a stationary equilibrium and the value is V=MZ(u)(p0 1,p0 2). An informal sketch of the proof is as follows. From properties (C1) and (C2), we know that, setting v=MZ(u),thereexistss1∈S1(p1)such that v(p1,p2)=v(s1,p2) and v(p 1,p2)≤u(p 1,p2)for all p 1∈supp(s1),andthereexistss2∈S2(p2)such that v(p1,p2)=v(p1,s2)and v(p1,p 2)≥u(p1,p 2)for all p 2∈supp(s2). These properties define naturally stationary strategies for both designers that turn out to form an equilibrium. Assume that v(p1,p2)is the value of the continuation game starting at (p1,p2). From the point of view of Designer 1, (C1) means that it is possible to choose a splitting that preserves the expected continuation value. Hence, assume that v(p1,p2)<u (p1,p2); then Designer 1 can play non-revealingly without reducing the equilibrium continuation payoff. Intuitively, if v(p1,p2)<u (p1,p2), Designer 1 would be content if the game stopped, as Designer 1 would receive more than the value. If v(p1,p2)>u (p1,p2), then Designer 1 does not want the game to stop and chooses a splitting s1such that v(p 1,p2)≤u(p 1,p2)with probability one to reach a point where the realized payoff uwould potentially be greater than the value. The symmetry of (C1) and (C2) implies that v(p1,p2)can be enforced by both designers. Thus, this is the equilibrium payoff of this zero-sum game. Proof of Theorem 1.Letv=MZ(u), and define a strategy σ1of Designer 1 as follows. Given posteriors (p1,p2)∈P1×P2at stage n, Designer 1 chooses the nonrevealing splitting δp1if u(p1,p2)≥v(p1,p2), and otherwise chooses a splitting s1∈S1(p1) such that v(p1,p2)=v(s1,p2)and v(p 1,p2)≤u(p 1,p2),∀p 1∈supp(s1). According to Proposition 1, this strategy is well-defined. It has the property that for any strategy of Designer 2, u(pn+1 1,pn 2)≥v(pn+1 1,pn 2)almost surely. This implies that u(pN∗ 1,pN∗ 2)≥ v(pN∗ 1,pN∗ 2)almost surely: either N∗<+∞ and then pN∗−1 i=pN∗ i,orN∗=+∞and we consider the limit. Therefore, Eu(pN∗ 1,pN∗ 2)≥Ev(pN∗ 1,pN∗ 2).ByS2-convexity of v,E[v(pn+1 1,pn+1 2)|hn]≥E[v(pn+1 1,pn 2)|hn]=v(pn 1,pn 2)by construction of σ1.Itfollows that Ev(pn+1 1,pn+1 2)≥Ev(pn 1,pn 2), and by induction (and taking limit if N∗=+∞), Ev(pN∗ 1,pN∗ 2)≥Ev(pn 1,pn 2)≥v(p0 1,p0 2). Thus, there is a strategy of Designer 1 such that for any strategy of Designer 2, Eu(pN∗ 1,pN∗ 2)≥v(p0 1,p0 2). By symmetry, this is the value of the game. We have constructed a stationary equilibrium. Theoretical Economics 17 (2022) Long information design 899 It follows from the above proof that under the equilibrium strategies we have u(pN∗ 1,pN∗ 2)=v(pN∗ 1,pN∗ 2)almost surely. Thus, if the martingale stops, it must be at points where u(p1,p2)=v(p1,p2). Furthermore, if designers can always reach points where u(p1,p2)=v(p1,p2), the martingale actually stops after the first stage (N∗=2), after which the agent takes his decision. This is the case under the conditions below. Proposition 2. Assume that both sets Piare convex, that all splittings are admissible, and that there exists a continuous MZ function v=MZ(u). There is an equilibrium such that: –Ifu(p0 1,p0 2)<v (p0 1,p0 2), then Designer 1 plays revealing at the first stage, and both designers play nonrevealing at the second stage; –Ifu(p0 1,p0 2)>v (p0 1,p0 2), then Designer 2 plays revealing at the first stage, and both designers play nonrevealing at the second stage; –Ifu(p0 1,p0 2)=v(p0 1,p0 2), then both designers play nonrevealing at the first stage. This is a direct consequence of the following lemma. Lemma 4. Assume that both Piare convex and that all splittings are admissible, S1(p1)= (P1)∩S1(p1),S2(p2)=(P2)∩S2(p2). Then, for any (p1,p2)∈P1×P2, –ifu(p1,p2)≤v(p1,p2),thereexistss1∈S1(p1)such that v(p 1,p2)=u(p 1,p2), ∀p 1∈supp(s1), –ifu(p1,p2)≥v(p1,p2),thereexistss2∈S2(p2)such that v(p1,p 2)=u(p1,p 2), ∀p 2∈supp(s2). This lemma is a direct extension of a result in Heuer (1992) and in Oliu-Barton (2018). For the sake of completeness, the proof is recalled in the Appendix. It follows that in equilibrium both designers play nonrevealing and the martingale is constant if u(p1,p2)=v(p1,p2).Ifu(p1,p2)<v (p1,p2), Designer 1 splits to a point p 1such that u(p 1,p2)=v(p 1,p2)and the martingale is constant thereafter and if u(p1,p2)> v(p1,p2), Designer 2 splits to a point p 2such that u(p1,p 2)=v(p1,p 2)and the martingale is constant thereafter. Thus, if the sets of admissible posteriors are convex and all splittings are admissible, information disclosure lasts one stage at most, after which the agent takes his decision.10 In Section 3.4.2, we provide an example with finite sets of posteriors in which the number of disclosure stages is unbounded. Existence of MZ function A natural question arising from Theorem 1is whether an MZ function exists. To answer this, we introduce the following assumption, which strengthens continuity of splitting correspondences. A related condition of “nonexpansivity” appears in Renault and Venel (2017). 10A similar property also appears in Li and Norman (2021, Proposition 6) for nonzero sum sequential information design games with perfectly correlated states and a fixed deadline. 900 Koessler, Laclau, Renault, and Tomala Theoretical Economics 17 (2022) Assumption 1. The splitting correspondence of each Designer isatisfies the following. There exists θ∈(0, 1]such that ∀pi,p i∈Pi,∀si∈Si(pi),∀a,b≥0, ∃s i∈Sip i s.t. ∀f∈Di θ,af (si)−bf s i≤ api−bp i 1,(1) where Di θ={f:Pi→[−θ,θ],∀pi,p i∈Pi,∀a,b≥0, |af (pi)−bf (p i)|≤api−bp i1}. This condition should be thought of as one possible definition of a Lipschitz correspondence, similarly to the nonexpansivity condition in Renault and Venel (2017). The idea is that an existence result requires a fixed-point argument. Since the value of the game is a function of beliefs, we need to find a set of functions where the fixed-point argument will be applied. This set of functions must be compact, that is, not too large, but also stable by concavification and convexification, thus not too small. Consider for instance the special case where Designer 2 has a trivial correspondence with only noninformative policies. Then the solution would be the concavification by Designer 1. Thus, the value functions should at least lie in sets which are stable under concavification or convexification. As it turns out, Assumption 1can be reformulated as follows. Lemma 5. Assumption 1is equivalent to: For each Designer i,thereexistsθ∈(0, 1]such that for each f∈Di θ, the concavification and the convexification of fbelong to Di θ. This shows that the set of functions D1 θ(resp., D2 θ)ischosentobeclosedunderthe concavification (resp., convexification) operator. It is apparent in our proofs that the existence of the value relies on the set of functions closed under those operators; see, in particular, the proof of Proposition 7in the Appendix. Here are important cases where the assumption holds. Proposition 3. If for each i,Pi=(i)and all splittings are allowed, or Piis finite, then Assumption 1is satisfied. We now consider the interesting case of splitting correspondences induced by constraints on experiments. Suppose that Designer iis given a finite set of messages Mand a compact subset of experiments X⊆{x:i→(M)}, which contains a nonrevealing experiment. Consider all the splittings that Designer ican generate by repeatedly choosing experiments in X. For instance, if Xcontains only a nonrevealing experiment x0and a noisy experiment x1, the designer can run x1as many times as he wants, depending on the outcomes of previous trials. More generally, define an “auxiliary strategy” for Designer ias a sequence of measurable mappings σi=(σit ), where for each t≥1, σit :(X×M)t−1→(X). For each x∈ X,σit(x|x1,m1,,xt−1,mt−1)is the probability that the designer runs experiment x conditionally on previous experiments and message realizations x1,m1,,xt−1,mt−1. Apriorpiand an auxiliary strategy σiinduce a probability distribution over sequences of experiments and messages: conditional on state ωi,x1is selected according to σi1, Theoretical Economics 17 (2022) Long information design 901 then m1according to x1(·|ωi),thenx2according to σi2(x1,m1),thenm2according to x2(·|ωi), and so on. Denote by μT(pi,σi)the induced distribution of posteriors after T trials and SiT (pi)the set of all such distributions as σivaries. Finally, let Si(pi)be the weak-* closure of T≥1SiT (pi). It is easy to see that Si(pi)is a convex compact set and the correspondence Sisatisfies S2 i=Si. Proposition 4. The splitting correspondence generated by a set of experiments as above satisfies Assumption 1. We then have the following existence result. Theorem 2. Under Assumption 1, there exists a continuous MZ function of u, denoted by MZ(u), which is therefore the value of the long information design game G(p0 1,p0 2). The logic of the proof is the following. Introduce a “discounted game” where Designers 1 and 2 move simultaneously and in each stage, the game ends with exogenous probability 1 −δ, and the receiver takes an action, or the game continues to the next stage with probability δ<1. Let vδ(p0 1,p0 2)be the value of this game. The main argument is to show that the family of functions (vδ)δis equicontinuous and, therefore, admits a limit point vas δ→1. This uses the fact that function vδis the fixed point of an appropriate Bellmann equation. We then show that vsatisfies all the conditions ensuring that v=MZ(u). The proof follows the steps of the proofs of Propositions 2 and 3 in Laraki and Renault (2020), albeit using a different “continuity” condition (Assumption 1). Laraki and Renault (2020) use a “nonexpansivity” assumption, which guarantees also the existence of a continuous MZ function. Their assumption is satisfied if Pi= (i)and all splittings allowed or if Piis finite. However, the splitting correspondence generated by a set of experiments need not be nonexpansive. We provide a counterexample in Appendix A.2. 3.4 Examples In this section, we provide two simple examples in which we characterize the values and optimal strategies. In the first example, each designer can either fully reveal the state or reveal nothing, and illustrates that the values and equilibrium strategies differ in the simultaneous, sequential, and long information design games. In the second example, the sets of admissible posteriors are finite and all splittings on those sets are admissible, and the equilibrium martingale of posteriors is unbounded. 3.4.1 Illustrative example Consider the following situation. The designers are opposed lobbyists or NGO. The agent is a journalist who would like to write an article or talk about a lobbyist’s case in a TV show only if this lobbyist has disclosed relevant information about his case. The second lobbyist would like to appear in the journalist’s article or TV show only if the first lobbyist does, while the first lobbyist would like the reverse. Formally, for each i,leti={0, 1}, identify pi∈(i)with pi(1)∈[0, 1],andlet 902 Koessler, Laclau, Renault, and Tomala Theoretical Economics 17 (2022) p0 i=1 2. Suppose that each designer has only two available disclosure policies: nonrevealing or fully revealing, or equivalently that each designer can only use deterministic experiments. The possible posteriors are thus P1=P2={0, 1 2,1 }, and all splittings are available on those sets. For each pair of feasible posteriors, the agent takes some optimal action that induces the following payoff for Designer 1: u= p1=1010 p1=1/2101 p1=0010 p2=0p2=1/2p2=1 Designer 1 would like to fully reveal his own state when Designer 2 is silent at p2=1/2, and Designer 2 would like to reveal his own state if Designer 1 has already revealed. The payoff function uis neither S1-concave nor S2-convex. The concavification and convexification of uare given by the following: cavp1u= 010 111 010 vexp2u= 000 101 000 If designers can disclose information simultaneously at a single stage, then Designer 1 can guarantee an expected payoff of 1 2by revealing the state with probability 1 2and remaining silent with probability 1 2. Indeed, in this case the posterior belief of the agent on 1is p1=0 with probability 1 4,p1=1 with probability 1 4and p1=1 2with probability 1 2. Hence, the probability of obtaining the payoff of 1 is equal to 1 2for any disclosure policy of Designer 2. Similarly, Designer 2 can guarantee that Designer 1’s expected payoff is not higher than 1 2by revealing the state with probability 1 2. This is the splitting value of the model, that is, the equilibrium payoff of Designer 1 of the one-stage simultaneous move game. Suppose now that designers play sequentially once and that Designer 1 moves last. In that case, Designer 1 can guarantee a payoff of 1 by playing the opposite of what Designer 2 did (Designer 1 discloses if Designer 2 has not disclosed at stage 1, and does not disclose if Designer 2 has disclosed). This is the vex cav value. Similarly, if Designer 2 moves second, Designer 2 can guarantee that Designer 1’s payoff is 0 by playing the same way as Designer 1 (Designer 2 discloses if Designer 1 has disclosed before, and does not disclose otherwise). This is the cav vex value. What is the equilibrium if the long information design game? The game is not symmetric. Designer 2 can still apply the strategy above: when Designer 1 discloses, Designer 2 discloses right after, and does not disclose otherwise. Clearly, the resulting payoff is 0 regardless of what Designer 1 does. This is the Mertens–Zamir value of this Theoretical Economics 17 (2022) Long information design 903 example. Summarizing, we have the following:11 vexp2cavp1u= 000 111 000 cavp1vexp2u= 000 101 000 and SV(u)= 000 11 21 000 MZ(u)= 000 101 000 3.4.2 Unbounded disclosure stages As in the previous example, for each i,leti={0, 1} and identify pi∈(i)with pi(1)∈[0, 1].LetPi={0, 1/3, 2/3, 1}and assume that all splittings on Piare admissible. Consider the following “matching pennies” utility function for Designer 1: u= p1=10 0 1 1 p1=2/30 0 1 1 p1=1/31 1 0 0 p1=01 1 0 0 p2=0p2=1/3p2=2/3p2=1 Consider, for example, the situation in which (p1,p2)=(1, 2 3). If Designer 2 does not disclose any information, then the utility of Designer 1 is u(1, 2 3)=1. The optimal information policy of Designer 2 is to induce the posteriors p 2=1andp 2=1 3with the same probability and we have vexp2u(1, 2 3)=1 2u(1, 1 3)+1 2u(1, 1)=1 20+1 21=1 2.More generally, we have the following: vexp2u= 001 21 001 21 11 200 11 200 cavp1u= 0011 1 2 1 211 111 2 1 2 1100 cavp1vexp2u= 001 21 1 2 1 4 1 21 11 2 1 4 1 2 11 200 vexp2cavp1u= 001 21 1 2 1 2 3 41 13 4 1 2 1 2 11 200 11In Koessler et al. (2021), we present an extension of this example where P1=(1),P2=(2),all splittings are admissible, and SV(u)=MZ(u). 904 Koessler, Laclau, Renault, and Tomala Theoretical Economics 17 (2022) Using cavp1vexp2u≤MZ(u)≤vexp2cavp1uand the symmetries of the example, the MZ value function can be written as follows: MZ(u)= 001 21 1 2xy1 1yx1 2 11 200 where 1 4≤x≤1 2and 1 2≤y≤3 4. It is easy to see that the solution of the Mertens–Zamir system MZ(u)(p1,p2)=cavp1min(u,MZ(u))(p1,p2)=vexp2max(u,MZ(u))(p1,p2) gives x=1 3and y=2 3.12 From this solution, we immediately obtain the following equilibrium strategies: –Ifp1=2 3and p2∈{0, 1 3},orifp1=1 3and p2∈{2 3,1 }, then Designer 1 splits the belief p1to the two neighborhood posteriors with the same probability; otherwise, he discloses no information. –Ifp2=2 3and p1∈{2 3,1 },orifp2=1 3and p1∈{0, 1 3}, then Designer 2 splits the belief p2to the two neighborhood posteriors with the same probability; otherwise, he discloses no information. The equilibrium martingale of posteriors is represented by Figure 1. Note that, as in the “four frogs” example of Forges (1984,1990), the number of equilibrium disclosure stages in the long information design game is unbounded, but disclosure stops with probability one in finite time, at a profile of posterior beliefs (p1,p2)in {(0, 1),(0, 2 3),(0, 0),(1 3,0 ),(2 3,1 ),(1, 1),(1, 1 3),(1, 0)}, represented by a “∗”symbolin Figure 1. It is important to emphasize that an unbounded number of stages is required for this equilibrium, even though every designer is able to induce in a single stage all distributions of posteriors that he would be able to induce by iterating information policies Figure 1. Equilibrium martingale of posteriors in the example of Section 3.4.2. 12An algorithm for computing the solution more generally is provided in Koessler et al. (2021). Theoretical Economics 17 (2022) Long information design 905 in multiple stages; that is, S2 1(p1)=S1(p1)and S2 2(p2)=S2(p2)is satisfied in the example. However, Proposition 2, which guarantees that all the equilibrium information is disclosed in a single stage, does not apply because the sets of admissible posteriors P1 and P2are not convex. It is also easy to check that the one-stage splitting value is as follows: SV(u)= 001 21 1 2 1 2 1 21 11 2 1 2 1 2 11 200 Indeed, if a designer splits an interior belief to the two closest posteriors with the same probability when his payoff is different from his first best payoff then, whatever the belief induced by the other designer, the expected utility is 1 2. For example, when (p1,p2)=(1 3,2 3), if Designer 1 splits p1=1 3to p 1=2 3with probability 1 2and to p 1=0 with probability 1 2, then the expected utility of Designer 1 is 1 2, and if Designer 2 plays a nonrevealing strategy, then the expected utility of Designer 1 is at most 1 2.Hence, in these situations, the value of the game is SV(u)(p1,p2)=1 2. Otherwise, the value is 0 or 1. For example, when (p1,p2)=(2 3,1 ), Designer 2 cannot modify p2and Designer 1 gets his first best, so SV(u)( 2 3,1 )=1 and no information is disclosed. Similarly, when (p1,p2)=(1, 1 3), Designer 1 cannot modify p1and Designer 2 gets his first best, so SV(u)(1, 1 3)=0 and no information is disclosed. In this example, for every interior posteriors (p1,p2)we have cavp1vexp2u(p1,p2)<MZ(u)(p1,p2)= SV(u)(p1,p2)< vexp2cavp1u(p1,p2). 4. Competition in product demonstration In this section, we study the case of two designers acting as sellers of products with uncertain match value. The agent is a buyer who chooses the seller from whom to buy the product. The agent prefers to buy the product with the highest expected match. The static simultaneous-move game Gsim(p0 1,p0 2)is studied in Boleslavsky and Cotton (2015), who characterize the equilibrium and the value of the game. The model is as follows. The match value for the product of each seller can be either high (H)orlow(L): 1= 2={L,H}.Letp0 1∈(0, 1)be the prior probability that the state is Hfor Designer 1 and p0 2∈(0, 1)be the prior probability that the state is Hfor Designer 2. Denote by p1∈[0, 1] and p2∈[0, 1]the corresponding posteriors. The agent must choose either Designer 1 or Designer 2. The agent’s payoff is 1 if the chosen designer’s state is Hand is 0 otherwise. Hence, Designer 1 is chosen if p1>p 2; Designer 2 is chosen if p1<p 2, and we assume that the agent randomizes uniformly if p1=p2. Designer 1’s payoff is equal to 1 if that designer is chosen and is −1otherwise. Given the posteriors (p1,p2), the expected payoff of Designer 1 is thus as follows: u(p1,p2)=1{p1>p 2}−1{p1<p 2}. 906 Koessler, Laclau, Renault, and Tomala Theoretical Economics 17 (2022) Note that u(p1,p2)is discontinuous at p1=p2. The expected payoff of the agent is as follows: uA(p1,p2)=max{p1,p2}. This example can fit the following economic scenario. Designers are two competing firms (e.g., Canon and Nikon) that are about to release a new product or system (e.g., a new camera technology together with plans for lens development and compatibility). The agent is a representative consumer (e.g., a professional photographer) who plans to switch in the near future to one of these two new products. Ex ante, the agent and the firms do not know the match value for the products (it can be high (H)orlow(L) for each product). Suppose that the agent already owns a similar product from an older product generation and is ready to spend some time acquiring match information before making a decision. Firms can release public match information to consumers about their respective products through product demonstration (e.g., press events, reviews, trade fairs, product testing, or gradual announcements of a lens roadmap). When choosing how to disclose information, the firm does not know the consumer’s match value for the product. Additionally, it does not know in advance the public information feedback (e.g., the ratings of reviews) its product will receive. Finally, firms cannot significantly adjust theirs prices but are able to adjust their information policies to signals generated by their competitors.13 It follows that they have strictly opposite preferences, and both want to attract the consumer. Below, we characterize and compare the equilibrium strategies and values for the simultaneous, sequential and long information design games. We use those results to analyze the informativeness of firms’ strategies and the resulting welfare of the consumer. If we consider finite sets of admissible posteriors, the example fits all our assumptions. For instance, this represents situations in which the consumer reads reviews that apply some rating system (e.g., from one to five stars, or scores corresponding to relevant features of the product). To compare our results with the equilibria of the static simultaneous game Gsim(p0 1,p0 2)studied in Boleslavsky and Cotton (2015), we perform the equilibrium analysis assuming that all splittings of all possible posteriors are admissible. Note that if all splittings are admissible, designers’ utility functions are discontinuous on the diagonal p1=p2.14 To deal with this discontinuity, we extend some of the proofs of the previous section to this particular example.15 In the static simultaneousmove game and in the long information design game, we show that equilibria exist. The 2-stage sequential games admit ε-equilibria for all ε>0 but not exact equilibria. Before proceeding to equilibrium analysis, we first consider the fully revealing (FR) and non-revealing (NR) benchmarks. If information is fully revealed to the agent, then 13See, for example, Boleslavsky, Cotton, and Gurnani (2016) for more details on product demonstration and on price flexibility in a similar scenario with a single information designer. 14The designers’ utility functions are discontinuous at p1=p2for any tie-breaking rule adopted by the agent. 15An alternative approach would be to assume a continuous utility by letting the agent tremble and choose Designer 1 with probability that is a continuous function of p1−p2, increasing from 0 to 1 (e.g., following a logit rule). Theoretical Economics 17 (2022) Long information design 913 in Gentzkow and Kamenica (2017). Denote by =1=2the common set of states and assume that all information disclosure policies are available. In all versions of the information design game, full revelation is an equilibrium. Indeed, when one designer fully discloses the state, the other is indifferent and may fully disclose as well. Thus, the value of each version of the game is that obtained by full revelation. This value coincides with the cavp1vexp2and vexp2cavp1values, and thus also with the SV and MZ values. Indeed, all of those functions have to be both concave and convex with respect to the prior. Therefore, they must be linear and equal to ωμ(ω)u(z(ω),ω),wherez(ω)is the optimal action of the agent when the agent knows that the state is ω. 5.4 Unobserved messages Consider an alternative model in which posteriors are not observed at each stage. More precisely, designers only observe splittings chosen at each stage, until the game between the designers stops. Then the sequence of splittings chosen is drawn and posteriors are finally privately observed before the agent chooses an action. Contrary to the case where posteriors are publicly observed at each stage by the designers, all the versions of this game, one-stage simultaneous, two-stage sequential or long, have the same value and optimal strategies, which are the splitting value and the one-stage optimal strategies. In particular, when posteriors are privately observed by the agent, it does not matter for him how he would acquire information from competing designers. This observation directly follows from the fact that SV(u)(p1,p2)=max s1∈S1(p1)min s2∈S2(p2)u(s1,s2)=min s2∈S2(p2)max s1∈S1(p1)u(s1,s2). That is, in every version of the game with unobserved messages, Designer 1 can guarantee that his expected utility is at least SV(u)(p1,p2)by playing his one-stage equilibrium strategy in his first move, and Designer 2 can guarantee that the expected utility of Designer 1 is at most SV(u)(p1,p2)by playing his one-stage equilibrium strategy in his first move. Hence, SV(u)(p1,p2)is the value. It is worth noting that the previous observation does not apply if one considers more general utility functions. To see this, consider the following nonzero sum example, where the set of admissible posteriors is P1=P2={0, 1/2, 1}, all splittings on P1and P2 are admissible, the priors are p0 1=p0 2=1 2, and the utilities of the designers are given by p1=1 1,1 4,0 1,1 p1=1/2 0,4 3,3 0,4 p1=0 1,1 4,0 1,1 p2=0p2=1/2p2=1 The one-stage game has similarities with the prisoner dilemma: it is a dominant strategy for both designers to disclose information and, by backward induction, it is the unique equilibrium whatever the finite and fixed number of stages. However, it is easy to see that the infinite horizon game (whether the realized posteriors are observed or not) has a (subgame perfect) equilibrium sustaining no information revelation. Indeed, along 914 Koessler, Laclau, Renault, and Tomala Theoretical Economics 17 (2022) the equilibrium path both designers play nonrevealing, and they punish by disclosing information if they observe a deviation from the nonrevealing splitting. Appendix A.1 Proofs Proof of Lemma 1. We show that cavp1w(p1,p2)=max{w(s1,p2):s1∈S1(p1)}.Fixp2 and let h(p1):=max{w(s1,p2):s1∈S1(p1)}. Consider a function g:P1→R,S1-concave such that g(·)≥w(·,p2). This pointwise inequality implies that for any s1∈S1(p1), g(s1)≥w(s1,p2)and since gis S1-concave, g(p1)≥g(s1)≥w(s1,p2). Since this holds for any s1∈S1(p1), this implies g(p1)≥h(p1), and since this holds for any g(·)≥w(·,p2), we obtain cavp1w(p1,p2)≥h(p1). To obtain the converse inequality, since h(p1)≥w(p1,p2), it is enough to prove that his S1-concave. Hence, take s1∈S1(p1)and consider h(s1)=h(p 1)ds1(p 1). Define a measurable selection of S1,f:P1→(P1)such that f(p 1)∈argmax{u(s 1,p 1):s 1∈ S1(p 1)}. We have for each p 1∈P1that h(p 1)=w(f(p 1),p2)=w(p 1,p2)df (p 1|p 1). We then obtain the following: h(s1)= wp 1,p2df p 1|p 1ds1p 1=w(˜ p1,p2)d(f∗s1)( ˜ p1) =w(f∗s1,p2)≤h(p1), since f∗s1∈S1(p1)by assumption. We thus have cavp1w(p1,p2)=max{w(s1,p2):s1∈ S1(p1)}for all p1and p2. Proof of Lemma 2.(i)Continuity. From Lemma 1,vex p2u,cav p1u,cav p1vexp2u, vexp2cavp1uare continuous. The correspondence (p1,p2)→ S1(p1)×S2(p2)is both upper and lower hemicontinuous; hence, by the maximum theorem SV(u)is continuous. (ii) The function cavp1vexp2uis S2-convex.DenoteF(p1,p2)=vexp2u(p1,p2)and take s1∈S1(p1)such that cavp1vexp2u(p1,p2)=F(s1,p2)=Fp 1,p2ds1p 1. Since Fis S2-convex, F(p 1,p2)≤F(p 1,s2)for each s2∈S2(p2).Thus, cavp1vexp2u(p1,p2)=Fp 1,p2ds1p 1 ≤Fp 1,s2ds1p 1 ≤cavp1vexp2up 1,s2ds1p 1 ≤cavp1vexp2u(p1,s2), Theoretical Economics 17 (2022) Long information design 915 where the last inequality holds due to cavp1vexp2ubeing S1-concave. By symmetry, vexp2cavp1uis S1-concave. (iii) The function SV(u)is S1-concave.Fix (p1,p2)in P1×P2and s1∈S1(p1);we have to show that SV(u)(p1,p2)≥SV(u)(s1,p2).Letf:P1→(P1)beameasurable selection of S1such that for each p 1in p1,f(p 1)∈S1(p 1)is an optimal strategy of Designer 1 in game Gsim(p 1,p2). We have for each p 1in p1that ∀s2∈S2(p2), u(f(p 1),s2)≥SV(u)(p 1,p2). Taking expectation with respect to s1implies the following: ∀s2∈S2(p2),ufp 1,s2ds1p 1≥SV(u)p 1,p2ds1p 1=SV(u)(s1,p2). We get ∀s2∈S2(p2),u(f∗s1,s2)≥SV(u)(s1,p2). Since f∗s1∈S1(p1), we obtain that the value SV(u)(p1,p2)of Gsim(p1,p2)is at least SV(u)(s1,p2).Hence,SV(u)is S1-concave. (iv) For each (p1,p2)∈P1×P2,cavp1vexp2u(p1,p2)≤SV(u)(p1,p2).Itsuffices to show that there exists s1∈S1(p1)such that for all s2∈S2(p2),u(s1,s2)≥ cavp1vexp2u(p1,p2). Choose s1such that cavp1vexp2u(p1,p2)=vexp2u(s1,p2).Then ∀s2∈S2(p2),u(s1,s2)≥vexp2u(s1,p2)=cavp1vexp2u(p1,p2). The other inequalities are either trivial or deduced by symmetry. Proof of Lemma 3. To prove the inequality cavp1vexp2u(p1,p2)≤MZ(u)(p1,p2),observe that max(u,MZ(u)) ≥u;thus, MZ(u)=vexp2maxu,MZ(u)≥vexp2u. Since MZ(u)is S1-concave, this implies MZ(u)≥cavp1vexp2u. The other inequality is obtained by symmetry. Proof of Proposition 1. We first prove the following lemma that states a useful property of concavification and optimal splittings: at an optimal splitting s1such that cavp1w(p1)=w(s1),itmustbethatcav p1w(p 1)=w(p 1)on the support of s1.Thisimplies that cavp1w(s1)=cavp1wp 1ds1p 1=wp 1ds1p 1=cavp1w(p1). Therefore, cavp1wis “linear” on the support of S1. Lemma 6. Let w:P1→Rbe a continuous function. For each p1∈P1and each s1∈S1(p1) such that cavp1w(p1)=w(s1)=max{w(s 1):s 1∈S1(p1)}, we have the following: s1p 1∈P1:wp 1=cavp1wp 1=1. Proof. Assume the contrary that cavp1w(p1)=w(s1)and s1({p 1∈P1:cav p1w(p 1)> w(p 1)})>0. Then there exist ε,α>0suchthats1({p 1∈P1:cav p1w(p 1)≥w(p 1)+ε})= α>0. Let B={p 1∈P1:cav p1w(p 1)≥w(p 1)+ε}; define a measurable selection fof S1such that cavp1w(p 1)=w(f(p 1)) for all p 1∈Band f(p 1)=δp 1for all p 1/∈B,and 916 Koessler, Laclau, Renault, and Tomala Theoretical Economics 17 (2022) consider the splitting f∗s1. We have the following: w(f∗s1)=B cavp1wp 1ds1p 1+P1\B wp 1ds1p 1≥εα +wp 1ds1p 1. Since f∗s1∈S1(p1), this contradicts w(s1)=max{w(s 1):s 1∈S1(p1)}. To complete the proof of the proposition, we first show that if v=MZ(u), then vis S1-concave and S2-convex, and (C1) and (C2) hold. MZ(u)is S1-concave and S2-convex; by symmetry, it is enough to prove (C1). Suppose that v=cavp1min(u,v)and consider some (p1,p2)∈P1×P2. According to Lemma 6,thereexistss1∈S1(p1)such that v(p1,p2)=min(u,v)(s1,p2)and v(p 1,p2)=min(u,v)(p 1,p2),∀p 1∈supp(s1). It follows that v(p 1,p2)≤u(p 1,p2),∀p 1∈supp(s1)and v(p1,p2)=v(s1,p2), so (C1) holds. Second, we show that if vis S1-concave and S2-convex and (C1) and (C2) hold, then v=MZ(u). Suppose that vis S1-concave and S2-convex, and let w(p1,p2)= min(u,v)(p1,p2). Since w≤v,cav p1w≤cavp1v. According to (C1), there exists s1∈ S1(p1)such that v(p1,p2)=v(s1,p2)and w(p 1,p2)=v(p 1,p2)≤u(p 1,p2)for all p 1 in the support of S1.Hence,v(p1,p2)=v(s1,p2)=w(s1,p2)≤cavp1w(p1,p2).Thus, v=cavp1min(u,v)and by symmetry, v=vexp2max(u,v). For the proofs of the next four results, we let i=1. The proofs are similar for i=2. Proof of Lemma 4. Consider a continuous function w:P1→R. For each p1such that w(p1)<cavp1w(p1), there exists a splitting with finite support s1=mλmδp1msuch that cavp1w(p1)=mλmw(p1m)and w(p 1)<cavp1w(p 1)for each p 1in the relative interior of the convex hull of {p1m:m}. The reason is that (p1,cav p1w(p1))lies on a face of the convex hull hypograph of the function wif w(p1)<cavp1w(p1). One just has to obtain this point as a “minimal” convex combination of extreme points of the hypograph (minimal in the sense that the convex hull of its extreme points is minimal for inclusion) among the splittings s1such that cavp1w(p1)=w(s1).Fixp2and apply this logic to v(p1,p2)=cavp1min(u,v)(p1,p2).Weobtainv(p1,p2)=mλmmin(u,v)(p1m,p2), with v(p1m,p2)≤u(p1m,p2)for all mand min(u,v)(p 1,p2)<v (p 1,p2)for all p 1in the relative interior of the convex hull of {p1m:m}. By continuity, this implies that v(p1m,p2)=u(p1m,p2)for all m. Proof of Lemma 5.=⇒ Take fin D1 θ,p1,p 1in P1,a,b≥0. Also, let s1∈S1(p1)be such that f(s1)=cavp1f(p1). By Assumption 1,thereexistss 1∈S1(p 1)such that af (s1)− bf (s 1)≤ap1−bp 11.Thus,acavp1f(p1)−bcavp1f(p 1)≤af (s1)−bf (s 1)≤ap1− bp 11. Applying this logic to −fyields the result. ⇐= Fix p1,p 1in P1,s1∈S1(p1),a,b≥0. Define γ:S(p 1)×D1 θ→Rby γs 1,f=af (s1)−bf s 1− ap1−bp 1 1. For each f,thereexistss 1∈S(p 1)such that f(s 1)=cavp1f(p 1).Then γs 1,f≤acavp1f(p1)−bcavp1fp 1− ap1−bp 1 1≤0. Theoretical Economics 17 (2022) Long information design 917 From Sion’s minmax theorem, there exists s 1∈S1(p 1),∀f∈D1 θ,γ(s 1,f)≤0. Considering fand −fgives the result. Proof of Proposition 3.(i)P1=(1), all splittings admissible. Fix p1,p 1in (1), f∈D1 θ,a,b≥0. Consider s1=mλmδp1ma splitting of p1with finite support such cavp1f(p1)=f(s1). This splitting is induced by the experiment which sends message m with probability λω1 m=λmp1m(ω1) p1(ω1)in state ω1. Consider now the same experiment used at prior p 1, and denote by s 1=mλ mδp 1mthe splitting induced by that experiment at p 1.Wehavep 1=mλ mδp 1m,with λ m= ω1 λmp1m(ω1)p 1(ω1) p1(ω1)and p 1m(ω1)=p 1(ω1)λω1 m λ m . Then acavp1f(p1)−bcavp1fp 1≤af (s1)−bf s 1= m aλmf(p1m)−bλ mfp 1m ≤ m aλmp1m−bλ mp 1m 1 = m ω1aλmp1m(ω1)−bλ mp 1m(ω1) = m ω1ap1(ω1)λω1 m−bλω1 mp 1(ω1)= ap1−bp 1 1. (ii) P1is finite. Assume that P1contains at least two points. Define θ∈(0, 1]by θ=min αp1−(1−α)p 1 1,α∈[0, 1],p1,p 1∈P1,p1=p 1. Fix p1,p 1in P1,f∈D1 θ,a,b≥0. If p1=p 1,wehaveforanys1∈S1(p1),s 1∈S1(p 1), af (s1)−bf s 1≤(a+b)θand  ap1−bp 1 1≥(a+b)θ. If p1=p 1,givens1∈S1(p1),wehaveaf (s1)−bf (s1)≤|a−b|θ≤|a−b|p11, since θ≤1. Proof of Proposition 4. Lemma 7. Let Mbe a nonempty compact subset of an Euclidean space (set of messages), and let x:1→(M)be fixed. For p1in (1), let ϕ(p1)in ((1)) be the splitting of p1induced by x,thatis, ϕ(p1)=m∈M δν(m)dζ(m), where ζis the distribution of messages induced by p1and ν(m)is a conditional probability on 1given m. We have for any θ∈(0, 1]: ∀p1,p 1∈(1),∀f∈D1 θ,∀a,b≥0, af ϕ(p1)−bf ϕp 1≤ ap1−bp 1 1. 918 Koessler, Laclau, Renault, and Tomala Theoretical Economics 17 (2022) Proof. Define the probability λon Mby λ(B)=(1/|1|)ω1∈1x(B|ω1)for all Borel subsets Bof M. By the Radon–Nikodym theorem, for each ω1∈1,thereexistsa measurable gω1:M→R+such that dx(m|ω1)=gω1(m)dλ(m), that is, for each B, x(B|ω1)=m∈Bgω1(m)dλ(m). Fix now p1and p 1.Letζbe the distribution of minduced by p1:dζ(m)= g(m)dλ(m),withg(m)=ω1∈1p1(ω1)gω1(m). A conditional distribution of ω1given mis given by the measurable map ν:M→(1)such that for all ω1in 1and Borel B⊆M: p1(ω1)x(B|ω1)=m∈B ν(ω1|m)dζ(m). So for all (ω1,m),p1(ω1)gω1(m)=g(m)ν(ω1|m). Similarly, let ζbe the distribution of minduced by p 1:dζ(m)=g(m)dλ(m),withg(m)=ω1∈1p 1(ω1)gω1(m).Consider ν:M→(1)such that for all ω1in 1and Borel B⊆M: p 1(ω1)x(B|ω1)=m∈B ν(ω1|m)dζ(m). So for all (ω1,m),p 1(ω1)gω1(m)=g(m)ν(ω1|m). Fix fin D1 θ,a,b≥0. Since ϕ(p1)=m∈Mδν(m)dζ(m), fϕ(p1)=m∈M fν(m)dζ(m)=m∈M fν(m)g(m)dλ(m). Therefore, af ϕ(p1)−bf ϕp 1=m∈Maf ν(m)g(m)−bf ν(m)g(m)dλ(m), ≤m∈M ag(m)ν(m)−bν(m)g(m) 1dλ(m), =m∈M ω1∈1 |ap1(ω1)gω1(m)−bp 1(ω1)gω1(m)|dλ(m), = ω1∈1m∈M gω1(m)|ap1(ω1)−bp 1(ω1)|dλ(m), = ap1−bp 1 1. We now show that Assumption 1is satisfied for S1.Fixp1and p 1in P1,s1∈ S1(p1)and a sequence Tnand σ1nsuch that (μTn(p,σ1n))nconverges to s1.Wehave μTn(p 1,σ1n)∈S1(p 1)for each n.Lets 1be a limit point of the sequence (μTn(p 1,σ1n))n. Take f∈D1 θ,a,b≥0. The auxiliary strategy σ1ndefines an experiment z:1→((X× M)Tn)with compact set of messages. From Lemma 7, ∀n,af μTn(p1,σ1n)−bf μTnp 1,σ1n≤ ap1−bp 1 1. Theoretical Economics 17 (2022) Long information design 919 So af (s1)−bf s 1≤af (s1)−af μTn(p1,σ1n)+af μTn(p1,σ1n) −bf μTnp 1,σ1n+bf μTnp 1,σ1n−bf s 1 af (s1)−bf s 1≤af (s1)−af μTn(p,σ1n)+bf μTnp 1,σ1n −bf s 1+ ap1−bp 1 1. Passing to the limit as n→∞,wegetaf (s1)−bf (s 1)≤ap1−bp 11, concluding the proof. Proof of Theorem 2. First, note that uniqueness has been proved by Laraki and Renault (2020, Proposition 4), which implies the following. Lemma 8. Let u:P1×P2→Rbe a continuous function. There exists at most one continuous function v:P1×P2→R,whichisS1-concave S2-convex and satisfies (C1) and (C2) of Proposition 1for u. The reader is referred to Laraki and Renault (2020) for the proof. We focus then on existence. From Assumption 1,thereexistsθ∈(0, 1]such that condition (1) is satisfied for both designers. Define D θas the set of functions from P1×P2to [−θ,θ]such that ∀p1,p 1∈P1,∀p2,p 2∈P2,∀a,b≥0, af (p1,p2)−bf p 1,p2≤ ap1−bp 1 1and af (p1,p2)−bf p1,p 2≤ ap2−bp 2 1. Notice that any fin D θis 1-Lipschitz in each variable, therefore, D θis a set of equicontinuous functions from P1×P2to [−θ,θ]. The following proposition shows existence of MZ(u)for u∈D θ. Proposition 7. Assume u∈D θ. There exists a function v=MZ(u)∈D θwhich is S1concave, S2-convex and satisfies (C1) and (C2) of Proposition 1. Moreover, for u,uin D θ, MZ(u)−MZ(u)∞≤u−u∞. Proof. Proposition 7follows from a series of several lemmas and propositions, where u∈D θis assumed. For δ∈[0, 1)and f∈D θ, define (f):P1×P2→Rby (f)(p1,p2)=(1−δ)u(p1,p2)+δValS1(p1)×S2(p2)f(s1,s2), where ValS1(p1)×S2(p2)f(s1,s2)=max s1∈S1(p1)min s2∈S2(p2)f(s1,s2)=min s2∈S2(p2)max s1∈S1(p1)f(s1,s2). Lemma 9. For any fin D θ,(f)is well-defined and belongs to D θ.Also,has a unique fixed point in D θ,whichwedenotebyvδ. 920 Koessler, Laclau, Renault, and Tomala Theoretical Economics 17 (2022) Proof.Fixδand fin D θ. For every p1,p2,Val S1(p1)×S2(p2)f(s1,s2)is well-defined by Sion’s minmax theorem. So the function (f):P1×P2→Ris well-defined. We now show that (f)∈D θfor f∈D θ.Fixp1,p 1,p2,a≥0, b≥0, and we want to prove that a(f)(p1,p2)−b(f)(p 1,p2)≤ap1−bp 11. Since u∈D θ,wehavetoprovethatA≤ ap1−bp 11,where A=max s1∈S1(p1)min s2∈S2(p2)af (s1,s2)−max s 1∈S1(p 1) min s2∈S2(p2)bf s 1,s2. Let s1∈S1(p1)be optimal in maxs1∈S1(p1)mins2∈S2(p2)f(s1,s2). By Assumption 1,there exists s 1∈S1(p 1)s.t. for all s2∈(P2),bf (s 1,s2)≥af (s1,s2)−ap1−bp 11.So bVal S1(p 1)×S2(p2)f≥amins2∈S2(p2)f(s1,s2)−ap1−bp 11.Hence,A=mins2∈S2(p2)af (s1, s2)−bVal S1(p 1)×S2(p2)f≤ap1−bp 11and (f)∈D θ. The operator is a δ-contraction, so by the contracting fixed-point theorem, has a unique fixed-point vδ. The family of functions (vδ)δis equicontinuous since all belong to D θ.ByAscoli’s theorem, it admits a limit point v∈D θfor uniform convergence. Taking limit in the fixed-point equation as δ→1 implies that for all (p1,p2)∈P1×P2,v(p1,p2)= Val S1(p1)×S2(p2)v(s1,s2). We now have the following lemma. Lemma 10. Any such limit point vis S1-concave, S2-convex and satisfies (C1) and (C2) of Proposition 1for u. Thus, an MZ function MZ(u)exists. Proof. The proofs of Propositions 2 and 3 in Laraki and Renault (2020) show that any limit point vof (vδ)δis S1-concave and S2-convex and has the following property: for all (p1,p2)in P1×P2,thereexistss1∈S1(p1)such that v(p1,p2)=v(s1,p2)≤u(s1,p2)and there exists s2∈S2(p2)such that v(p1,p2)=v(p1,s2)≥u(p1,s2).Weshownowthatv satisfies (C1), (C2) being obtained by a symmetric argument. Fix (p0 1,p0 2)in P1×P2. Consider the correspondence S 1from P1to (P1)defined for p1in P1by S 1(p1)=s1∈S1(p1):vp1,p0 2=vs1,p0 2≤us1,p0 2. This correspondence admits measurable selections and any such selection fdefines a strategy for Designer 1: at each p1,playf(p1). Starting from p0 1,thestrategyfinduces a martingale (pn 1)n, which converges a.s. to some random variable p∞ 1with E(p∞ 1)= p0 1and E(v(p∞ 1,p0 2)) =v(p0 1,p0 2). Thus, the distribution of p∞ 1denoted s∞ 1belongs to S1(p0 1)and satisfies v(s∞ 1,p0 2)=v(p0 1,p0 2). The proof of Lemma 10 is concluded by the following claim. Claim 1. v(p∞ 1,p0 2)≤u(p∞ 1,p0 2)almost surely. Proof.Letg:P1→Rbe the continuous function given by g(p1)=v(p1,p0 2)− u(p1,p0 2). We want to show that g(p∞ 1)≤0 a.s. From the choice of f, for each p1in P1,g(f(p1)) =Ef(p1)(g)≤0. Since gis continuous, g(pn+1 1)−g(pn 1)→n→∞ 0 a.s., so Theoretical Economics 17 (2022) Long information design 921 by the dominated convergence theorem E((g(pn+1 1)−g(pn 1))2)→n→∞ 0. Define the random variable: Yn=Egpn+1 1−gpn 12|pn 1. We have then Yn≥0, Ynis bounded and E(Yn)→n→∞ 0, which imply that there exists a subsequence (Yϕ(n))nwhich converges to 0 almost surely. Moreover, since E[g(pn+1 1)− g(f(pn 1))|pn 1]=0, we have Yn=Egpn+1 1−gfpn 12|pn 1+Egfpn 1−gpn 12|pn 1 =Egpn+1 1−gfpn 12|pn 1+gfpn 1−gpn 12. Thus, Yϕ(n)tends to 0 and is the sum of two positive terms, so each term tends to 0. Thus, gfpϕ(n) 1−gpϕ(n) 12→n→∞ 0a.s. Since g(f(pϕ(n) 1)) ≤0 for all nand g(pϕ(n) 1)−→n→∞ g(p∞ 1),wegetg(p∞ 1)≤0a.s. This proves Lemma 10. We know then that any limit point of vδis the unique MZ function (Lemma 8). This implies that vδconverges to v=MZ(u)as δ→1. Since vδ(u)−vδ(u)∞≤u−u∞for all δ,wegetMZ(u)−MZ(u)∞≤u−u∞. This ends the proof of Proposition 7. To complete the proof of Theorem 2, we must extend the existence result to all continuous functions. The argument is that functions in D θ(or their multiples) are dense within continuous functions. Lemma 11. (i) The set L={lf ,l≥0, f∈D1 θ}is dense (for the uniform norm) in the set of continuous functions from P1to R. (ii) The set L={lf ,l≥0, f∈D θ}is dense (for the uniform norm) in the set of continuous functions from P1×P2to R. Proof. (i) First, observe that Lis a lattice. Take f,gin Land l,lsuch that f/l and g/lbelong to D1 θand let l =max{l,l}.Thenf l ,g l belong to D1 θand max(f,g)/l and min(f,g)/l as well. We get that max(f,g)and min(f,g)are in L. Second, for every p0 1=p1 1in P1and real numbers α,β,thereexistsf∈Lsuch that f(p0 1)=αand f(p1 1)=β. We know that there exists a linear mapping ϕon R1such that ϕ(p0 1)=αand ϕ(p1 1)=β. Because ϕis linear, there exists l≥0 such that for every p1,p 1 and a,b≥0, aϕ(p1)−bϕp 1=ϕap1−bp 1≤l ap1−bp 1 1. Thus, ϕrestricted to p1 1belongs to L. By the (lattice) Stone–Weierstrass theorem, Lis dense in continuous functions. 922 Koessler, Laclau, Renault, and Tomala Theoretical Economics 17 (2022) (ii) As in the previous point, Lis a lattice. It is thus enough to prove that for (p0 1,p0 2) and (p1 1,p1 2),with(p0 1,p0 2)=(p1 1,p1 2),inP1×P2,α,β∈R,thereexistsfin Lsuch that f(p0 1,p0 2)=αand f(p1 1,p1 2)=β. Assume p0 1=p1 1. From the previous point, we know that there exists gin Lsuch that g(p0 1)=αand g(p1 1)=β. Define f(p1,p2)=g(p1)for all p1,p2. To see that f∈L,considerp1,p 1,p2,p 2and a≥0, b≥0. We have af (p1,p2)−bf p 1,p2=ag(p1)−bgp 1≤l ap1−bp 1 1 since g∈L.Then af (p1,p2)−bf p1,p 2=g(p1)|a−b|=g(p1)·ap21− bp 2 1≤l ap2−bp 2 1 with l=maxp1|g(p1)|. We may now conclude the proof of Theorem 2. Consider a continuous payoff function u:P1×P2→R. For each n≥1, there exists unin Lsuch that u−un1≤1 nand ln>0suchthatun/ln∈D θ. By Lemma 10,MZ(un/ln)exists and we let vn=lnMZ(un/ln). This function is S1-concave, S2convex and satisfies (C1) and (C2) of Proposition 1 for un.Sovn=MZ(un). Fix now n,m,andsetl:=max{ln,lm}.Wehaveun/l and um/l are in D θ,vn=lMZ(un/l)and vm=lMZ(um/l). Also, we know that vn−vm1≤ lun/l −um/l1=un−um1. Since (un)nis a Cauchy sequence, so is (vn)n.Hence, (vn)nconverges to a continuous vwhich is S1-concave and S2-convex. By taking limits, vsatisfies (C1) and (C2) for u. Therefore, v=MZ(u)exists. This ends the proof of Theorem 2. Proof of Proposition 6. Consider the stationary strategy σ1of Designer 1 who plays nonrevealingly if v(p1,p2)≤u(p1,p2),ands1isgivenby(C1)otherwise.Considerany strategy σ2of Designer 2. The definition of σ1implies that for each n, Evpn+1 1,pn 2|pn 1,pn 2=vpn 1,pn 2. Since vis S2-convex, Evpn+1 1,pn+1 2|pn+1 1,pn 2≥vpn+1 1,pn 2. Taking expectation, for each n,E[v(pn+1 1,pn+1 2)] ≥E[v(pn+1 1,pn 2)] ≥E[v(pn 1,pn 2)] ≥ v(p0 1,p0 2)by induction. Denote X={(p1,p2)∈P1×P2:v(p1,p2)≤u(p1,p2)};byconstruction, (pn+1 1,pn 2)∈Xalmost surely for each n. Since by assumption Xis closed, (pN∗ 1,pN∗ 2)∈Xa.s., that is, u(pN∗ 1,pN∗ 2)≥v(pN∗ 1,pN∗ 2)a.s. With an abuse of notation, denote by (pn 1,pn 2)nthe martingale stopped at N∗,thatis, such that (pn 1,pn 2)=(pN∗ 1,pN∗ 2)for all n>N∗. Claim 2. limsupnv(pn 1,pn 2)≤v(p∞ 1,p∞ 2)a.s. Proof. Fix a realized play path and consider a converging subsequence of (v(pn 1,pn 2))n denoted by (v(pn 1∗,pn 2∗))n. We show that limnv(pn 1∗,pn 2∗)≤v(p∞ 1,p∞ 2).Thereare2cases.