Adversarial coordination and public information design
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Inostroza, Nicolás; Pavan, Alessandro Article Adversarial coordination and public information design Theoretical Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Inostroza, Nicolás; Pavan, Alessandro (2025) : Adversarial coordination and public information design, Theoretical Economics, ISSN 1555-7561, The Econometric Society, New Haven, CT, Vol. 20, Iss. 2, pp. 763-813, https://doi.org/10.3982/TE5768 This Version is available at: https://hdl.handle.net/10419/320298 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. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. 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 20 (2025), 763–813 1555-7561/20250763 Adversarial coordination and public information design Nicolas Inostroza Rotman School of Management, University of Toronto and FTG Alessandro Pava n Department of Economics, Northwestern University and CEPR We study flexible public information design in global games. In addition to receiving public information from the designer, agents are endowed with exogenous private information and must decide between two actions (invest and not invest), the profitability of which depends on unknown fundamentals and the agents’ aggregate action. The designer does not trust the agents to play favorably to her and evaluates any policy under the “worst-case scenario.” First, we show that the optimal policy removes any strategic uncertainty by inducing all agents to take the same action, but without permitting them to perfectly learn the fundamentals and/or the beliefs that rationalize other agents’ actions. Second, we identify conditions under which the optimal policy is a simple “pass/fail” test. Finally, we show that when the designer cares only about the probability the aggregate investment is successful, the optimal policy need not be monotone in fundamentals but then identify conditions on payoffs and exogenous beliefs under which the optimal policy is monotone. Keywords. Global games, adversarial coordination, Bayesian persuasion, robust public information design. JEL classification. D83, G28, G33. 1. Introduction Coordination plays a major role in many socioeconomic environments. The damages to society of miscoordination can be severe and often call for government intervention. Think of the possibility of default by major financial institutions in case investors run or refrain from rolling over their short-term positions. Such defaults can trigger a collapse in financial markets, with severe consequences for the real economy. Confronted with such prospects, governments and supervising authorities have incentives to intervene. Nicolas Inostroza: [email protected] Alessandro Pavan: [email protected] A previous version circulated under the title “Persuasion in Global Games with Application to Stress Testing.” For comments and useful suggestions, we thank the referees, Marios Angeletos, Gadi Barlevy, Eddie Dekel, Tommaso Denti, Laura Doval, Stephen Morris, Marciano Siniscalchi, Bruno Strulovici, Jean Tirole, Xavier Vives, Leifu Zhang, and seminar participants at various conferences and institutions where the paper was presented. Matteo Camboni provided excellent research assistance. Pavan acknowledges financial support from the NSF under the grant SES-1730483 and from Bocconi University where part of this research was conducted. The usual disclaimer applies. ©2025 The Authors. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at https://econtheory.org.https://doi.org/10.3982/TE5768
764 Inostroza and Pavan Theoretical Economics 20 (2025) These interventions often take the form of public information disclosures, such as stress testing or, more broadly, releases of information aimed at influencing market beliefs. In this paper, we study public information design in markets in which a large number of receivers (e.g., investors in financial markets) must choose whether to play an action favorable to the designer (e.g., pledging funds to a financial institution), or an “adversarial” action (e.g., refraining from pledging). A policymaker can flexibly design a policy disclosing information to market participants about relevant economic fundamentals. The analysis delivers results that are important for various situations in which coordination plays a major role, including bank runs, currency crises, and technology and standards adoption. In the context of stress testing, the policymaker may represent a supervising authority attempting to prevent a run against the banking sector (see, e.g., Henry and Kok (2013)andHomar, Kick, and Salleo (2016)). In the case of currency crises, the policymaker may represent a central bank attempting to dissuade speculators from short-selling the domestic currency by releasing information about the bank’s reserves and/or domestic economic fundamentals. In the case of technology adoption, the policymaker may represent the owners of an intellectual property trying to persuade heterogenous market users of the merits of a new product (Lerner and Tirole (2006)). Thebackboneofthemodelisaglobal game of regime change in which multiple agents must choose between “attacking” a status quo or “refraining from attacking,” and where the success of the attack depends on its aggregate size and on exogenous fundamentals. In addition to receiving public information from the designer, agents are endowed with exogenous private information. The designer does not trust the agents to play favorably to her and evaluates any policy of her choice under the “worst-case” scenario. That is, when multiple rationalizable strategy profiles are consistent with the information disclosed, the designer takes a “robust approach” by looking at the outcome that prevails when agents play according to the rationalizable profile least favorable to her.1 We assume the policymaker can flexibly design a policy that disseminates publicly information about relevant economic fundamentals. We use the model to address the following questions: (a) Are there benefits to preventing market participants from predicting each others’ actions and beliefs? (b) When are simple policies such as pass/fail tests optimal? (c) Are there merits to nonmonotone rules that induce the market to play favorably for intermediate fundamentals but not necessarily for stronger ones? Our first result establishes that, despite the fear of adversarial coordination, the optimal policy satisfies the “perfect coordination property.” In each state, it induces all market participants to take the same action, but without creating homogenous beliefs among market participants. In other words, the optimal policy completely removes any strategic uncertainty while preserving structural uncertainty. Given the public information disclosed, each receiver can perfectly predict the action of any other receiver, but not the beliefs that rationalize such actions. For example, an agent who is induced to invest must not be able to determine whether other agents invest because they know that 1Such a robust approach is motivated by the applications the analysis is meant for. For example, when concerned about runs to the banking sector, policymakers typically do not trust the market to play favorably.
Theoretical Economics 20 (2025) Adversarial coordination and information design 765 the fundamentals are so strong that the investment will always succeed, irrespective of the behavior of other agents (e.g., the bank will never default), or because they are confident that other agents will invest. The optimal policy leverages the heterogeneity of the agents’ primitive beliefs by making investing dominant for some agents based on their first-order beliefs, but only iteratively dominant for others based on their higher-order beliefs.2Under adversarial coordination, preserving uncertainty over beliefs is key to the minimization of the risk of an undesirable outcome such as a bank default, a currency collapse, or the failure of a new technology to take off. When the designer trusts the agents to follow her recommendations, the optimality of the perfect coordination property is straightforward and follows from arguments similar to those establishing the Revelation Principle (e.g., Myerson (1986)). This is not the case under adversarial coordination for information that facilitates perfect coordination may also favor the emergence of rationalizable profiles in which some of the agents play adversarially to the designer. Our second result shows that, when the economic fundamentals and the agents’ beliefs comove in the sense that states in which fundamentals are strong are also states in which most agents expect other agents to expect the fundamentals to be strong, and so on, then the optimal policy takes the form of a simple “pass/fail” test, with no further information disclosed to the market. It is known that, when the distribution from which the agents’ private signals are drawn is log-supermodular, or equivalently, satisfies the Monotone Likelihood Ratio Property— in short MLRP—all agents follow monotone (i.e., cut-off) strategies, no matter the public information. This is because, under MLRP, the agents’ “optimism ranking” is preserved under Bayesian updating. If agent jis more optimistic than agent ibefore the public announcement is made (formally, j’s beliefs dominate i’s beliefs according to the MLRP order), then this continues to be the case after any public announcement. When this is the case, disclosing information to the market in addition to whether or not the policymaker expects the agents’ investment to succeed when they play adversarially does not help. We also show that MLRP is key to the optimality of simple pass/fail policies. When the information the policymaker discloses can be used to change the ranking of the agents’ optimism, the policymaker can leverage the optimism reversal to spare more fundamentals from the undesirable outcome by disclosing information in addition to whether or not she expects the investment to succeed.3 In the context of stress testing, these results provide a foundation for the optimality of simple pass/fail policies. Importantly, optimal stress tests should be transparent,in the sense of facilitating coordination among investors, but should not generate consensus among market participants about the soundness of the financial institutions under scrutiny. 2The optimal policy does not ensure that investing is the unique rationalizable action based on firstorder beliefs for all agents. It relies on a contagion argument through higher-order beliefs to induce all agents to invest under the unique rationalizable profile. 3When, instead, the designer trusts her ability to coordinate the receivers on the course of action most favorable to her, optimal policies always take the form of action recommendations, and hence pass/fail policies are optimal, irrespective of the agents’ primitive beliefs. This is not the case under adversarial/robust design.
766 Inostroza and Pavan Theoretical Economics 20 (2025) Our third result is about the optimality of monotone pass/fail policies, that is, rules that grant a pass grade if and only if the exogenous fundamentals are above a given threshold. We show that the optimality of such rules is related to the extent to which the policymaker’s preferences for a favorable outcome (e.g., for avoiding a bank default) vary with the fundamentals. We identify precise conditions involving the policymaker’s preferences and the agents’ payoffs and exogenous beliefs under which monotone rules are optimal. Such conditions are fairly sharp in the sense that, when violated, one can identify instances in which nonmonotone rules strictly outperform monotone ones.4 The reason is that nonmonotone rules make it more difficult for the agents to commonly learn the fundamentals and hence permit the policymaker to give a pass grade to a larger set of fundamentals. When the policymaker’s preferences for the favorable outcome (i.e., for avoiding a bank default) do not vary much with the exogenous fundamentals (in particular, when they are constant), nonmonotone rules may be optimal. Organization The rest of the paper is organized as follows. Below, we wrap up the introduction with a brief review of the most pertinent literature. Section 2presents the model. Section 3contains all the results about properties of optimal policies (perfectcoordination, pass/fail, monotonicity). Section 4discusses how the results accommodate enrichments that are useful in applications (e.g., more general payoffs, as well as the possibility that the policymaker faces uncertainty about the outcome induced by her information dissemination). Section 5concludes. The Appendix contains all proofs with the exception of the proofs of Examples 2 and 3, which are in the Supplementary Appendix (available at https://econtheory.org/supp/5768/supplement.pdf). The manuscript Inostroza and Pavan (2024a) contains additional material. In particular, (a) it extends Theorem 1* in the main text (about the optimality of perfectly coordinating the market response) to a broader class of economies, (b) discusses the benefits of discriminatory disclosures, when the latter are feasible, and (c) expands the material in Section 4.3 discussing the role of the multiplicity of the receivers and their exogenous private information for the optimality of monotone rules. (Most) pertinent literature The paper is related to a few strands of the literature. The first one is the literature on adversarial coordination and unique implementation.See, among others, Segal (2003), Winter (2004), Sakovics and Steiner (2012), Frankel (2017), Halac, Kremer, and Winter (2020), and Halac, Lipnowski, and Rappoport (2021). These papers focus on the design of transfers. Instead, we focus on the design of public information in settings in which the receivers are endowed with exogenous private information. Li, Song, and Zhao (2023), and Morris, Oyama, and Takahashi (2024)consider the design of private information in binary supermodular games in which the receivers’ exogenous information is symmetric. Halac, Lipnowski, and Rappoport (2022)study unique implementation when the designer can use a combination of transfers and information provision. Goldstein and Huang (2016)andGalvão and Shalders (2022)also 4We also show that the conditions guaranteeing the optimality of monotone rules are more stringent when the policymaker faces multiple privately-informed receivers than when she faces either a single (possibly privately-informed) receiver, or multiple receivers who possess no exogenous private information.
Theoretical Economics 20 (2025) Adversarial coordination and information design 767 study public information design in settings in which the receivers possess exogenous private information. Goldstein and Huang (2016) restrict the policymaker to binary monotone rules, whereas Galvão and Shalders (2022) to partitional structures whereby when two states are pooled into the same cell, all in-between states are also pooled into the same cell. Related is also Alonso and Zachariadis (2023) who study the complementarity between private and public information. Relative to these works, our paper establishes three key results: (a) it proves that inducing all agents to take the same action is always optimal, despite the fear of adversarial coordination; (b) it shows why, in general, binary policies are suboptimal but then identifies sharp conditions under which such policies are optimal; (c) it shows why, in general, nonmonotone rules permit the policymaker to induce a favorable outcome over a larger set of fundamentals but then identifies sharp conditions under which optimal policies are monotone.5 The second strand is the literature on information design with multiple receivers. See, among others, Alonso and Camara (2016a), Arieli and Babichenko (2019), Bardhi and Guo (2018), Basak and Zhou (2020), Che and Hörner (2018), Doval and Ely (2020), Galperti and Perego (2023), Gick and Pausch (2012), Gitmez and Molavi (2022), Heese and Lauermann (2021), Laclau and Renou (2017), Mathevet, Perego, and Taneva (2020), Shimoji (2021), and Taneva (2019). The key contribution vis-a-vis this literature is in showing how the interaction between (a) adversarial coordination and (b) exogenous private information among the receivers shapes the optimal provision of public information.6 The third strand is the literature on global games with endogenous information.Angeletos, Hellwig, and Pavan (2006)andAngeletos and Pavan (2013) study signaling in global games. Angeletos and Werning (2006) investigate the role of prices as a vehicle for information aggregation. Angeletos, Hellwig, and Pavan (2007) consider a dynamic model in which agents learn from the accumulation of private information and from the (possibly noisy) observation of past outcomes. Cong, Grenadier, and Hu (2020)consider a dynamic setting similar to the one in Angeletos, Hellwig, and Pavan (2007) but allowing for policy interventions. Edmond (2013)andKyriazis and Lou (2024) consider propaganda in global games, in a setting in which the policymaker manipulates the agents’ private signals. Szkup and Trevino (2015), Yang (2015), Morris and Yang (2022), and Denti (2023) study the acquisition of private information in global games. Our paper contributes to this strand by identifying properties of flexible public information provision when (a) the sender can commit, and (b) the receivers play adversarially. Finally, the paper is related to the literature on stress testing.SeeGoldstein and Sapra (2014) for an overview of some of the early contributions. Bouvard, Chaigneau, and de Motta (2015) study a setting where a policymaker must choose between full transparency and full opacity but cannot commit to a disclosure policy. Williams (2017)and Goldstein and Leitner (2018) study the design of stress tests when the receivers do not possess exogenous private information. Orlov, Zryumov, and Skrzypacz (2023) study the joint design of stress tests and precautionary recapitalizations whereas Faria-e Castro, 5In particular, Example 3below shows that nonmonotone rules strictly outperform monotone ones in the same environment of Goldstein and Huang (2016). 6See Bergemann and Morris (2019) and Kamenica (2019) for an overview on information design.
768 Inostroza and Pavan Theoretical Economics 20 (2025) Martinez, and Philippon (2016)andGarcia and Panetti (2017) the joint design of stress tests and government bailouts. Inostroza (2023) studies regulatory disclosures with multiple audiences of investors who care about different aspects of a financial institution’s balance sheet. Alvarez and Barlevy (2021)andQuigley and Walther (2024) study the incentives of banks to disclose balance sheet (hard) information. Corona, Nan, and Gaoqing (2017) study how stress tests disclosures may favor banks’ coordinated risk taking in the spirit of Farhi and Tirole (2012). Morgan, Persitani, and Vanessa (2014), Flannery, Hirtleb, and Kovner (2017), and Petrella and Resti (2013) conduct an empirical analysis of the information provided by stress tests in the US and the EU. Our paper contributes to this literature along the following dimensions: (a) it shows that optimal stress tests should not create conformism in market participants’ beliefs about exogenous fundamentals but should be sufficiently transparent to eliminate any ambiguity about the market response to the tests; (b) it identifies conditions under which simple pass/fail policies are optimal; (c) it provides conditions for optimal tests to be monotone (see also Inostroza and Pavan (2024b) for a discussion of how the toughness of optimal stress tests relates to the type of securities issued by the banks). 2. Model Global games have been used to study the interaction between information and coordination in many socioeconomic environments, including bank runs, debt crises, currency attacks, investment in technologies with network externalities, technological spillovers, and political change. To ease the exposition, hereafter we describe the model and all the results in the context of a specific game in the spirit of Rochet and Vives (2004)inwhichtheagents are investors (e.g., fund managers, or unsecured bank depositors) deciding whether or not to pledge funds to one, or multiple financial institutions, and where these institutions default on their obligations when the size of the aggregate investment is not large enough.7The analysis, however, readily extends to many other global games. Players and actions A policymaker designs an information disclosure policy, for example, stress tests, call reports, publication of accounting standards, and disclosure of various macro and financial variables that are jointly responsible for the profitability of the agents’ decisions. The market is populated by a measure-one continuum of agents (the receivers) distributed uniformly over [0, 1]. Each agent may either take a “friendly” action, ai=1, or an “adversarial” action, ai=0. The friendly action is interpreted as the decision to invest (more generally, to “refrain from attacking” a status quo the policymaker wants to preserve). The adversarial action is interpreted as the decision to not invest (more generally, to “attack”). We denote by A≡1 0aidi∈[0, 1]thesizeofthe aggregate investment. 7Rochet and Vives (2004) consider a three-period economy à la Diamond and Dybvig (1983)butwith heterogenous investors, in which banks may fail early or late. As shown in that paper, the full model admits a reduced-form version similar to the one considered here.
Theoretical Economics 20 (2025) Adversarial coordination and information design 769 Fundamentals and exogenous information Consistently with the rest of the literature, we parameterize the relevant fundamentals by θ∈R. The fundamentals are exogenous to the policymaker’s choice of a disclosure policy. It is commonly believed (by the policymaker and the agents alike) that θis drawn from a distribution F, absolutely continuous over an interval ⫌[0, 1], with a smooth density fstrictly positive over . In addition, each agent i∈[0, 1]is endowed with private information summarized by a unidimensional statistic xi∈Rdrawn independently across agents given θfrom an absolutely continuous cumulative distribution function P(x|θ)with smooth density p(x|θ)strictly positive over an (open) interval θ≡(θ,¯ θ)containing θ,withθ,¯ θmonotone in θ,andwithp(x|θ)bounded over (x,θ). The bounds θ,¯ θcan be either finite or infinite. For example, when xi=θ+σεi,withεidrawn from a uniform distribution over [−1, +1],thenforanyθ,θ=θ−σand ¯ θ=θ+σ. When, instead, xi=θ+σεi,with εidrawn from a standard Normal distribution, then for any θ,θ=−∞and ¯ θ=+∞. Furthermore, in this latter case, P(x|θ)=((x−θ)/σ ),whereis the cumulative distribution function of the standard Normal distribution. We denote by x≡(xi)i∈[0,1]a profile of private signals and by X(θ)the collection of all x∈R[0,1]that are consistent with the fundamentals being equal to θ. As usual, we assume that any pair of signal profiles x,x∈X(θ)has the same cross-sectional distribution of signals, with the latter equal to P(x|θ). Regime outcome The fundamentals θparameterize the critical size of the aggregate investment that is necessary to avoid default (more generally, an undesirable regime change). If A>1−θ, short-term obligations are met and default is avoided. If, instead, A≤1−θ, default occurs. We denote by r=1 the event in which default is avoided and by r=0 the event in which default occurs.8 Dominance regions For any θ≤0, default occurs irrespective of the size of the aggregate investment, whereas for any θ>1 default is averted with certainty. For θ∈(0, 1], instead, whether or not default occurs is determined by the behavior of the market. Payoffs Each agent’s payoff differential between investing and not investing, u(θ,A), is equal to g(θ)>0 in case default is avoided, and b(θ)<0 otherwise. In turn, the policymaker’s payoff is equal to W(θ)in case default is avoided, and L(θ)in case of default, with W(θ)>L (θ)for all θ.WhenWand Lare invariant in θ, the policymaker’s objective reduces to minimizing the probability of default. The functions b,g,W,andLare all bounded. For any (θ,A)∈×[0, 1],thenlet u(θ,A)≡g(θ)1(A>1−θ)+b(θ)1(A≤1−θ), UP(θ,A)≡W(θ)1(A>1−θ)+L(θ)1(A≤1−θ) denote the payoffs of a representative agent and of the policymaker, respectively, when the fundamentals are θand the aggregate investment is A. 8The model assumes that, given Aand θ, the regime outcome is binary. The case in which default is “partial” is qualitatively similar, from a strategic standpoint, to the case where, given Aand θ,theregime outcome is stochastic and determined by variables that are not observable by the policymaker at the time of her public announcements (see the discussion in Section 4).
770 Inostroza and Pavan Theoretical Economics 20 (2025) Policy Let Sbe a compact Polish space defining the set of possible signal realizations. Apolicy =(S,π)consists of the set Salong with a measurable mapping π:→(S) specifying, for each θ, a probability distribution over the information disclosed to the market. Timing The sequence of events is the following: (i) The policymaker publicly announces the policy =(S,π)and commits to it.9 (ii) The fundamentals θare drawn from the distribution Fand the agents’ exogenous signals x∈X(θ)are drawn from the distribution P(x|θ). (iii) The public signal sis drawn from the distribution π(θ)and is publicly observed. (iv) Agents simultaneously choose whether or not to invest. (v) The regime outcome is determined (i.e., whether or not default occurred) and payoffs are realized. Adversarial coordination and robust information design The policymaker does not trust the market to follow her recommendations and play favorably to her (i.e., invest whenever θ>0).10 Instead, she adopts a robust/conservative approach. She evaluates any policy under the “worst-case” scenario, that is, she assumes that the market plays according to the rationalizable strategy profile that is most adversarial to her, among all those consistent with the policy . Definition 1. Given any policy ,themost aggressive rationalizable profile (MARP) consistent with is the strategy profile a≡(a i)i∈[0,1]that minimizes the policymaker’s ex ante expected payoff over all profiles surviving iterated deletion of interim strictly dominated strategies (henceforth IDISDS). In the IDISDS procedure leading to MARP, agents use Bayes rule to update their beliefs about the fundamentals θand the other agents’ exogenous information x∈X(θ) using the common prior F, the distribution of private signals P(x|θ), and the policy . Under MARP, given (x,s), each agent i∈[0, 1], after receiving exogenous information xfrom Nature and endogenous information sfrom the policymaker, refrains from investing whenever there exists at least one conjecture over (θ,A)consistent with the above Bayesian updating and supported by all other agents playing strategies surviving IDISDS, under which refraining from investing is a best response for the individual. Remarks. Hereafter, we confine attention to policies for which MARP exists.11 Because the game among the agents is supermodular (no matter the prior F,thedistribution Pfrom which the exogenous signals are drawn, and the policy ), the strategy 9See Leitner and Williams (2023) for a discussion of the commitment assumption in stress testing. 10If she did, a simple monotone policy revealing whether or not θ>0 would be optimal. 11Because the state is continuous, in principle, one can think of policies for which the agents’ common posteriors are not well-defined or, when combined with the agents’ exogenous information, are such that the agents’ hierarchies of beliefs are not well-defined, in which case MARP may not exist.
Theoretical Economics 20 (2025) Adversarial coordination and information design 777 occur. This means that the probability that each agent with information xLassigns to the event that default does not occur is at least equal to Pr[θ>1/3|xL,(1, ext )] =11/15, implying that it is optimal for the agent to invest. Next, consider the case in which fundamentals are intermediate, that is, θ∈ [5/6, 7/6]. In this case, the ranking of the agents’ optimism is reversed, with those agents observing the xLsignal assigning higher probability to higher states. In particular, because each agent with information xLassigns probability 2/3>cto θ≥1, any such agent finds it dominant to invest. Because, for any θ∈(5/6, 1),1/3 of the agents receives information xL, the minimal size of investment that each agent with signal equal to xHcanexpectatanyθ∈(5/6, 1)is equal to p(xL|θ)=1/3>1−θ, implying that even if all the less optimistic agents with signal xHrefrained from investing, default would not occur. But this means that investing is iteratively dominant for those agents receiving the xHsignal. Hence, the proposed policy spares any θ>0 from default. Because all agents invest when they observe a pass grade, no matter whether they learn that the fundamentals are extreme or intermediate, one may find it surprising that the policymaker needs to provide the extra information. This is a consequence of the policymaker not trusting the market to play favorably to her. The extra information is precisely what guarantees the uniqueness of the rationalizable action. As anticipated above, the benefits from disclosing information in addition to the pass (or fail) grade stem from the possibility to reverse the ranking of the agents’ optimism, which is possible only when the distribution p(x|θ)is not log-supermodular. In the example above, the most optimistic agents are those observing the xLsignals when the fundamentals are intermediate, whereas they are those observing the xHsignals when the fundamentals are extreme. The reversal in the agents’ optimism in turn permits the policymaker to guarantee that investing is the unique rationalizable action over a larger set of fundamentals (the entire set θ>0 in the example). The above example also illustrates the failure of the Revelation Principle when the policymaker is concerned with unique implementation (equivalently, when the market is expected to play according to MARP). It is well known that, in this case, confining attention to policies that take the form of action recommendations is with loss of generality. The contribution of Theorem 2is in showing that, notwithstanding such a qualification, the optimal policy does take the form of action recommendations in the special case in which beliefs comove with fundamentals according to MLRP. 3.3 Monotone rules We now turn to the optimality of policies that fail with certainty institutions with weak fundamentals and pass with certainty those with strong fundamentals. As anticipated in the Introduction, the optimality of such rules crucially depends on whether the policymaker’s preferences for avoiding default when fundamentals are large are strong enough to compensate for the possibility that nonmonotone rules may permit her to reduce the ex-ante probability of default (i.e., the possibility that default may occur over a set of fundamentals of smaller ex ante probability under a nonmonotone rule).
778 Inostroza and Pavan Theoretical Economics 20 (2025) In this subsection, we identify a condition relating the policymaker’s preferences to the agents’ exogenous beliefs and payoffs under which monotone rules are optimal. We show that the condition is fairly sharp in that, when violated, one can identify economies in which nonmonotone rules do strictly better than monotone ones. These economies include many of the examples considered in the literature, for example, Goldstein and Huang (2016). We assume hereafter that x∈R: uθ,1−P(x|θ)1(θ>0)p(x|θ)dF(θ)≤0= ∅.(1) When Condition (1) is violated, the expected payoff differential between investing and not investing is positive for any agent who is informed that fundamentals are nonnegative and who expects each other agent to invest (alternatively, not invest) when receiving a signal above (alternatively, below) hers. In this case, the information-design problem is uninteresting because the policymaker can save all θ>0 through a policy that announces whether or not θ>0. Then, let xmax ≡supx∈R: uθ,1−P(x|θ)1(θ>0)p(x|θ)dF(θ)≤0.(2) As we show in the Appendix,xmax is an upper bound for the set of cut-offs characterizing the strategies consistent with MARP across all disclosure policies satisfying the perfect coordination property. For any x,letΘ(x)≡{θ∈:x∈θ}denote the set of fundamentals that, given the distribution P(·|θ)from which the agents’ signals are drawn, are consistent with private information x. Condition M. The following properties hold: (i) infΘ(xmax )≤0; (ii) for any θ0,θ1∈[0, 1],withθ0<θ 1,andx≤xmax such that (a) θ1≤P(x|θ1)and (b) x∈θ0, UP(θ1,1 )−UP(θ1,0 ) UP(θ0,1 )−UP(θ0,0 )>p(x|θ1)b(θ1) p(x|θ0)b(θ0).(3) Property (i) in Condition Msays that the lower bound of the support of the beliefs of an agent with signal xmax,wherexmax is the threshold defined in (2), is nonpositive and, therefore, that according to this agent there is a positive probability that default is unavoidable, no matter the aggregate investment. Clearly, this property trivially holds when, for any θ, the agents’ signals are drawn from a distribution whose support is large enough (and hence, a fortiori, when the noise in the agents’ signals is drawn from a distribution with unbounded support, e.g., a Normal distribution). Property (ii) of Condition Msays that the value the policymaker assigns to avoiding default increases with the underlying fundamentals at a large enough rate. Specifically, the property requires that the benefit that the policymaker derives from changing the
Theoretical Economics 20 (2025) Adversarial coordination and information design 779 agents’ behavior (inducing all agents to invest starting from a situation in which no agent invests) must increase with the fundamentals at a sufficiently high rate, with the critical rate determined by a combination of the agents’ payoffs in case of default and beliefs. Theorem 3. Suppose that p(x|θ)is log-supermodular and Condition Mholds. Given any regular policy satisfying the perfect-coordination property, there exists a regular deterministic binary monotone policy ˆ θ=({0, 1},πˆ θ)that also satisfies the perfectcoordination property and such that, when the agents play according to MARP under both and ˆ θ, the policymaker’s ex ante expected payoff is weakly higher under ˆ θthan under .15 When Condition Mholds, the choice of the optimal policy reduces to the choice of the smallest threshold ˆ θsuch that, when agents commonly learn that θ> ˆ θ,underthe unique rationalizable profile, all agents invest irrespective of their exogenous private information. For this to be the case, it must be that, ∞ ˆ θu(θ,1−P(x|θ))p(x|θ)dF(θ)>0, for any x∈R. The above problem, however, does not have a formal solution, due to the lack of upper semicontinuity of the policymaker’s payoff in ˆ θ. Notwithstanding these complications, hereafter we follow the pertinent literature and refer to the “optimal monotone policy” as the one defined as follows. For any θ∈(0, 1),letx∗(θ)be the critical signal threshold such that, when agents follow a cut-off strategy with threshold x∗(θ), default occurs if and only if the fundamentals are below θ.16 Let θ∗≡infˆ θ≥0:∞ ˆ θ u˜ θ,1−Px∗(θ)|˜ θpx∗(θ)|˜ θdF(˜ θ)≥0forallθ∈[ˆ θ,1 )(4) be the lowest truncation point ˆ θsuch that, when the policy reveals that fundamentals are above ˆ θ, then for any possible default threshold θ∈[ˆ θ,1 ), if default were to occur for fundamentals below θand not for fundamentals above θ, then the marginal agent with signal x∗(θ)would find it optimal to invest. Hereafter, we assume that θ∗is well-defined, whichisalwaysthecasewhen 17 θ## ≡supθ∈(0, 1): u˜ θ,1−Px∗(θ)|˜ θpx∗(θ)|˜ θdF(˜ θ)≤0<1. 15The policy ˆ θis such that there exists a threshold ˆ θ∈[0, 1]such that, for any θ≤ˆ θ,πˆ θ(θ)assigns probability one to s=0, whereas for any θ> ˆ θ,πˆ θ(θ)assigns probability one to s=1. 16For any θ∈(0, 1),thethresholdx∗(θ)is implicitly defined by P(x∗(θ)|θ)=θ. When the noise in the agents’ signals is bounded, the definition of x∗(θ)can be extended to θ=0 and θ=1. When the noise is unbounded, abusing notation, one can extend the definition to θ=0 and θ=1 by letting x∗(0)=−∞and x∗(1)=+∞. 17For any ˆ θ∈(θ##,1 ), and any θ∈[ˆ θ,1 ), 0< u˜ θ,1−Px∗(θ)|˜ θpx∗(θ)|˜ θdF(˜ θ)<∞ ˆ θ u˜ θ,1−Px∗(θ)|˜ θpx∗(θ)|˜ θdF(˜ θ). Hence, when θ## <1, θ∗is well-defined.
780 Inostroza and Pavan Theoretical Economics 20 (2025) The optimal monotone policy is the one with cut-off ˆ θ=θ∗.18 The previous literature (e.g., Goldstein and Huang (2016)) characterized the threshold θ∗by restricting attention to monotone rules. The contribution of Theorem 3is in identifying the conditions under which such rules are optimal. Importantly, these conditions are not met in the works that restrict attention to monotone rules. As the examples below suggest, in those settings, the policymaker can strictly increase her payoff through a nonmonotone rule. As we show in the Appendix, Property (i) in Condition Mguarantees that, starting from the optimal monotone policy (the one with cut-off θ∗), one cannot perturb the policy by assigning a pass grade also to a small interval of fundamentals [θ,θ],with0≤ θ<θ <θ ∗, while guaranteeing that investing remains the unique rationalizable action when the policymaker announces a pass grade (i.e., when the signal s=1isdisclosed). This property trivially holds when the noise in the agents’ signals is large (and hence, a fortiori, when noise is unbounded), but plays a key role when the noise is drawn from a bounded interval of small size (see Example 2below for an illustration). Property (ii) of Condition Min turn guarantees that the higher payoff the policymaker obtains, under the new policy, from avoiding default when fundamentals are stronger compensates for the possibility that, from an ex ante perspective, the probability of default may be larger under monotone policies than under nonmonotone ones (see Example 3for an illustration of why nonmonotone rules may permit the policymaker to avoid default over a set of fundamentals of larger ex ante probability). As anticipated above, Condition Mis fairly sharp in the sense that, when violated, one can identify economies in which the optimal policy is nonmonotone. We provide two such examples below. Example 2illustrates the role of Property (i) in Condition M, whereas Example 3illustrates the role of Property (ii) in Condition M. These examples also illustrate why nonmonotone rules, in general, may reduce the set of fundamentals over which default happens. Let θMS ∈(0, 1)be implicitly defined by the unique solution to 1 0 uθMS,AdA=0. (5) The threshold θMS corresponds to the value of the fundamentals at which an agent who knows θand holds Laplacian beliefs with respect to the aggregate investment is indifferent between investing and not investing.19 Importantly, θMS is independent of the initial common prior Fand of the distribution of the agents’ signals. 18The reason why this is an abuse is that, under the monotone policy with cut-off θ∗,inthecontinuation game that starts after the policymaker announces s=1, there exists a rationalizable profile in which some of the agents refrain from investing. However, there exists a monotone policy with cut-off ˆ θarbitrarily close to the threshold θ∗such that, after the policymaker announces s=1 (equivalently, that θ≥ˆ θ), the unique rationalizable profile features all agents investing. Because the policymaker’s payoff under the latter policy is arbitrarily close to the one she obtains when all agents invest for θ>θ ∗and refrain from investing when θ≤θ∗, the abuse appears justified. 19This means that the agent believes that aggregate investment is uniformly distributed over [0, 1].See Morris and Shin (2006).
Theoretical Economics 20 (2025) Adversarial coordination and information design 781 Figure 2. Suboptimality of deterministic binary monotone policies. Example 2. Suppose that there exist scalars g,b∈R,withg>0>b,suchthat,forany θ,g(θ)=g,andb(θ)=b. Assume that θis drawn from a uniform distribution with support [−K,1+K],forsomeK∈R++. Finally, assume that the agents’ exogenous signals are given by xi=θ+σi,withσ∈R++ and with each idrawn independently across agents from a uniform distribution over [−1, 1],withσ<K/2. Let θ∗ σbe the thresholddefinedin(4), applied to the primitives described in this example.20 There exists σ#∈(0, K/2)such that (a) inf Θ(x∗ σ#(θMS)) >0, and (b) for all σ∈(0, σ#), starting from the optimal monotone policy with cut-off θ∗ σ, there exists a deterministic nonmonotone policy satisfying the perfect-coordination property and permitting the policymaker to avoid default over a set of fundamentals of strictly larger probability measure than the optimal monotone policy. ♦ The proof is in the Supplementary Appendix. Here, we sketch the key arguments. To fix ideas, let g=1−cand b=−c,withc∈(0, 1),asinExample1,andrecallthat, under such a payoff specification, investing is optimal when the probability of default is no greater than 1 −c, whereas not investing is optimal when such a probability exceeds 1−c. For any binary policy =({0, 1},π), and any threshold θ∈[0, 1]such that (x∗ σ(θ),1 ) are mutually consistent under ,let V σ(θ)≡U σx∗ σ(θ),1|x∗ σ(θ), denote the payoff of the marginal agent with signal x∗ σ(θ), after the policy announces that s=1, where U σis the function defined after Theorem 2. Now, for any ˆ θ∈,letˆ θ=({0, 1},πˆ θ)be the deterministic, binary, monotone rule with cut-off ˆ θ. Note that the absence of any public disclosure is equivalent to a monotone policy with cut-off ˆ θ=min=−Kand that, under such a policy, default occurs if and only if θ≤θMS =c. 20Hereafter, the subscript σin θ∗ σand x∗ σis meant to highlight that these thresholds are those for the economy in which the noise in the agents’ exogenous private signals is scaled by σ.
782 Inostroza and Pavan Theoretical Economics 20 (2025) A necessary and sufficient condition for all agents to invest under MARP consistent with the policy ˆ θ, after hearing that s=1, is that, for any possible default threshold θ> ˆ θ,Vˆ θ σ(θ)>0. The lowest fundamental in the support of x∗ σ(θ)’s beliefs is x∗ σ(θ)−σ. Hence, when x∗ σ(θ)−σ>ˆ θ, the marginal agent with signal x∗ σ(θ)already knows from his private information that fundamentals are above ˆ θ. Because, in the absence of any public disclosure, the payoff of the marginal agent is strictly negative for all θ<θ MS,this implies that the cut-off θ∗ σfor the optimal monotone rule is θ∗ σ=x∗ σ(θMS)−σ. Now to see that the optimal monotone policy is improvable, assume that σis small so that x∗ σ(θMS)−σ>0. Next, pick γ,δ>0 small and let θ ≡x∗ σ(θMS −δ)−σand θ≡θ −γ,withθ>0. Consider a binary policy γ,δ=({0, 1},πγ,d), that in addition to announcing a pass grade s=1 when fundamentals are above θ∗ σ(as the optimal monotone rule does) also announces s=1whenθ∈[θ,θ].LetVγ,δ σ(θ)be the payoff of the marginal agent with signal x∗ σ(θ)under the new rule γ,δ, after the policymaker announces that s=1. This payoff is represented in Figure 2along with the payoff Vθ∗ σ σ(θ) under the optimal monotone rule. Provided that γand δare small, Vγ,δ σ(θ)≥0for all θfor which (x∗ σ(θ),1 )are mutually consistent under γ,δ,withVγ,δ σ(θ)=0ifand only if θ=θMS. Starting from γ,δ, one can then further perturb the policy γ,δby giving a fail grade to banks with fundamentals in [θ∗ σ,θ∗ σ+ε],withε>0 small. The new policy ˜ so constructed is such that V˜ σ(θ)>0forallθfor which (x∗ σ(θ),1 )are mutually consistent under ˜ , meaning that, when the policymaker announces that s=1, investing is the unique rationalizable action for all agents. The policy ˜ thus satisfies the perfect-coordination property and guarantees that default occurs over a set of fundamentals of strictly smaller probability under Fthan the optimal monotone policy θ∗ σ=({0, 1},πθ∗ σ). The reason why the nonmonotone policy ˜ constructed in the proof of Example 2 guarantees that default occurs over a smaller set of fundamentals than the optimal monotone policy is that agents receiving signals around θMS are highly sensitive to the grade the policy gives to institutions with fundamentals around θMS but not so much so to the grade given to fundamentals far from θMS. In the above example with bounded noise, an agent receiving a signal x∗ σ(θMS)is not sensitive at all to the grade the policy gives to fundamentals below x∗ σ(θMS)−σbecause his private signal informs him that the fundamentals are above x∗ σ(θMS)−σ. Hence, while it is impossible to amend the optimal monotone policy (the one with cut-off θ∗ σ=x∗ σ(θMS)−σ) by giving a pass grade also to fundamentals slightly below θ∗ σwithout inducing some of the agents to refrain from investing, it is possible to amend the optimal monotone policy by extending the pass grade to an interval [θ,θ]of fundamentals sufficiently “far away” from θ∗ σ, while continuing to induce all agents to invest under MARP. The reason why such improvements are not feasible under Condition Min Theorem 3is that Property (i) in Condition Mimplies that x∗ σ(θMS)−σ<0, thus making the above construction unfeasible.21 Interestingly, when θ∈[θ,θ], the assumption of bounded support of the agents’ beliefs 21Under Property (i), the marginal agent with signal x∗ σ(θMS )does not rule out any fundamental in (0, θMS ). Hence, any perturbation of the optimal monotone policy passing fundamentals to the left of θMS induces the agent to refrain from investing.
Theoretical Economics 20 (2025) Adversarial coordination and information design 783 implies that a positive-measure set of agents know with certainty that θ∈[θ,θ]and yet, under the unique rationalizable profile, all agents invest; this is because, by design, the policy ˜ constructed in Example 2guarantees that, when θ∈[θ,θ], such an event is not commonly learned. The next example considers an economy in which the noise in the agents’ exogenous signals is drawn from a distribution with an unbounded support (in which case, Property (i) in Condition Mtrivially holds), but Property (ii) is violated. Given any binary, deterministic policy =({0, 1},π)(i.e., any policy such that, for any θ,π(θ)is a degenerate Dirac distribution assigning probability 1 either to s=1or to s=0), let D={(θi,¯ θi]:i=1, ,N}denote the partition of (0, θMS]induced by π, with N∈N,θ1=0, and θN=θMS.22 Let d∈Ddenote a generic cell of the partition Dand, for any θ∈(0, θMS],denotebyd(θ)∈Dthe cell that contains θ. Finally, let M()≡maxi=1,,N|¯ θi−θi|denote the mesh of D, that is, the Lebesgue measure of the cell of Dof maximal Lebesgue measure. Example 3below shows that, when the noise in the agents’ information is small, any deterministic binary policy of large mesh can be improved upon by a non-monotone deterministic binary policy with a smaller mesh. This property in turn implies that optimal policies are highly nonmonotone. Example 3. Suppose that θis drawn from an improper uniform prior over Rand that the agents’ signals are given by xi=θ+σεi,withεidrawn from a standard Normal distribution.23 Further assume that there exist scalars g,b,W,L∈R,withg>0>band W>L, such that, for any θ,g(θ)=g,b(θ)=b,W(θ)=Wand L(θ)=L.Thereexists a scalar ¯σ>0 and a function E:(0, ¯σ]→R+,withlimσ→0+E(σ)=0, such that, for any σ∈(0, ¯σ], in the game in which the noise in the agents’ information is scaled by σ, the following is true: given any deterministic binary policy =({0, 1},π)satisfying the perfect-coordination property and such that M()>E(σ), there exists another deterministic binary policy ∗with M(∗)<E(σ)that also satisfies the perfect-coordination property and such that the ex ante probability of default under ∗is strictly smaller than under .♦ See the Supplementary Appendix for a detailed proof of the result. Here, we discuss the main ideas. Nonmonotone policies permit the policymaker to avoid default over a larger set of fundamentals by making it difficult for the agents to commonly learn the fundamentals when the latter are between 0 and θMS and the policymaker announces a pass grade. Intuitively, if the policymaker assigned a pass grade to an interval (θ,θ]⊂(0, θMS]of large Lebesgue measure, when σis small and θ∈(θ,θ],most agents would receive private signals xi∈(θ,θ]. No matter the grade assigned to fundamentals outside the interval (θ,θ], in the continuation game that starts after the policymaker announces a pass grade, most agents with signals xi∈(θ,θ]would then 22That is, either (a) π(θ)=0forallθ∈∪ i=2k,k≤N(θi,¯ θi]and π(θ)=1forallθ∈∪ i=2k−1,k≤N(θi,¯ θi],or(b) π(θ)=1forallθ∈∪ i=2k,k≤N(θi,¯ θi]and π(θ)=0forallθ∈∪ i=2k−1,k≤N(θi,¯ θi]. 23The improperness of the prior simplifies the exposition but is not important. The agents’ hierarchies of beliefs are still well-defined.
784 Inostroza and Pavan Theoretical Economics 20 (2025) assign high probability to the joint event that θ∈(θ,θ], that other agents assign high probability to θ∈(θ,θ], and so on. When this is the case, it is rationalizable for such agents to refrain from investing. Hence, when σis small, the only way the policymaker can guarantee that, when θ∈(0, θMS], the agents invest after hearing a pass grade is by dividing the set (0, θMS]into a collection of disjoint intervals, each of small Lebesgue measure. This guarantees that the support of each agent’s posterior beliefs after a pass grade is announced is not connected. Connectedness of the supports facilitates rationalizable profiles where some agents refrain from investing. Next, suppose that the intervals (θi,¯ θi]⊂(0, θMS],i=1, ,N, receiving a pass grade are far apart, implying that the policymaker fails an interval (θ,θ]⊂(0, θMS]of large Lebesgue measure (note that this is indeed the case under the optimal monotone deterministic rule with cutoff θ∗ σ,whereθ∗ σis the threshold defined in (4).24 The detailed derivations in the Supplementary Appendix then show that, starting from , the policymaker could assign a pass grade to fundamentals in the middle of [θ,θ]and a fail grade to some fundamentals to the right of θ, in such a way that (a) investing continues to be the unique rationalizable action for all agents after hearing a pass grade, and (b) the set of fundamentals receiving a pass grade under the new policy is strictly larger than under the original one. Furthermore, the construction sketched above can be iterated until one arrives at a new policy with a mesh smaller than E(σ)under which default occurs over a set of fundamentals of strictly smaller measure than under the original policy. When the benefit W(θ)−L(θ)of avoiding default is constant in θ, as in the example above, the new policy thus yields the policymaker a strictly higher payoff than the original one. Finally, one can show that, when σis small, a pass grade can be given to all θ> θMS +ε,withε>0 small, while guaranteeing that all agents invest after the policymaker announces the pass grade s=1.25 The above properties thus also imply that, if the policymaker is restricted to deterministic policies (arguably, the most relevant case in practice), when the precision of the agents’ exogenous information is large, the optimal policy is highly nonmonotone over (0, θMS)and announces a pass grade when fundamentals are above θMS. 4. Extensions We first introduce a few enrichments in Section 4.1, then establish the analog of the three theorems above for these richer economies in Section 4.2, and then conclude in Section 4.3 discussing the role of the multiplicity of the receivers and their exogenous private information. 24The subscript simply highlights the dependence of the cutoff θ∗ σon σ. 25Formally, for any ε>0, there exists σ(ε)such that, for any σ<σ (ε), given any pass/fail policy satisfying PCP, there exists another pass/fail policy also satisfying PCP that agrees with on any θ<θ MS and gives a pass grade to any θ≥θMS +ε.
Theoretical Economics 20 (2025) Adversarial coordination and information design 785 4.1 Generalizations The fundamentals are given by (θ,z),withθdrawn from according to the absolutely continuous cdf F,andwithzdrawn from [z,z]according to Qθ(z),withthecdfQθ(z) weakly decreasing in θ,foranyz.26 The variable θcontinues to parameterize the maximal information the policymaker can collect about the fundamentals. The additional variable zparameterizes risk that the agents and the policymaker face at the time of the disclosure (e.g., macroeconomic variables that are only imperfectly correlated with the fundamentals). As in the baseline model, conditional on θ, the private signals x=(xi)i∈[0,1]are i.i.d. draws from an (absolutely continuous) cumulative distribution function P(x|θ), with associated density p(x|θ)strictly positive and bounded over the interval θ∈R. There exists a function R:×[0, 1]×[z,z]→Rsuch that, given any (θ,A,z), default occurs (i.e., r=0) if, and only if, R(θ,A,z)≤0. The function Ris continuous and strictly increasing in (θ,z,A).Forany (θ,A), the probability of avoiding default is thus given by r(θ,A)≡P[R(θ,A,z)>0|θ,A]. There exist functions ˆ W,ˆ L:×[0, 1]×[z,z]→Rsuch that, given any (θ,A,z),the policymaker’s payoff is equal to ˆ UP(θ,A,z)=ˆ W(θ,A,z)1R(θ,A,z)>0+ˆ L(θ,A,z)1R(θ,A,z)≤0.(6) Hence, ˆ W(θ,A,z)is the policymaker’s payoff in case default is avoided, whereas ˆ L(θ,A,z)is her payoff in case of default. Likewise, there exist functions ˆ g,ˆ b:×[0, 1]× [z,z]→Rsuch that, given any (θ,A,z), the agents’ payoff differential between investing and not investing is equal to ˆ u(θ,A,z)=ˆ g(θ,A,z)1R(θ,A,z)>0+ˆ b(θ,A,z)1R(θ,A,z)≤0,(7) with ˆ g(θ,A,z)>0>ˆ b(θ,A,z),forany(θ,A,z).Forany(θ,A),t henlet g(θ,A)≡E1R(θ,A,z)>0ˆ g(θ,A,z)|θ,A r(θ,A)and b(θ,A)≡E1R(θ,A,z)≤0ˆ b(θ,A,z)|θ,A 1−r(θ,A) denote the agents’ expected payoff differential in case of no default and in case of default, respectively. Likewise, for any (θ,A),let W(θ,A)≡E1R(θ,A,z)>0ˆ W(θ,A,z)|θ,A r(θ,A)and L(θ,A)≡E1R(θ,A,z)≤0ˆ L(θ,A,z)|θ,A 1−r(θ,A) 26All the results extend to the case where Qθ(z)has unbounded support. Note that Qθ(z)is not required to be absolutely continuous in z(in fact, it is not absolutely continuous in the baseline model, where the distribution has a mass point of 1 at z=0).
786 Inostroza and Pavan Theoretical Economics 20 (2025) denote the policymaker’s expected payoff, again in case of no default and default, respectively. The agents’ and the policymaker’s expected payoffs can then be conveniently expressed as a function of θand Aonly, by letting u(θ,A)≡r(θ,A)g(θ,A)+1−r(θ,A)b(θ,A)and UP(θ,A)≡r(θ,A)W(θ,A)+1−r(θ,A)L(θ,A). Hereafter, we assume that |u(θ,A)|is bounded and that there exist θ,θ∈R,withθ < θ,suchthat(a)u(θ,1 )<0forallθ≤θ,(b)u(θ,0 )>0forallθ>θ,and(c)u(θ,1 )> 0>u (θ,0 )for all θ∈(θ,θ].Thethresholdsθand θdefine the “critical region” (θ,θ] where the sign of the agents’ payoff differential depends on the response of the market.27 We also assume that both u(θ,A)and UP(θ,A)are nondecreasing in Aand such that UP(θ,1 )>UP(θ,0 )for all θ∈(θ,θ].28 4.2 Results We identify conditions under which Theorems 1–3extend to these richer economies. 4.2.1 Perfect-coordination property Given any distribution G∈ over , say that G is “regular” if, when the common posterior over is Gand, for any θ, agents receive private signals according to P(·|θ), MARP is well-defined. Then, for any regular G,any θ,letA(θ;G)denote the aggregate investment at θwhen agents play according to MARP, under the common posterior G. Condition PC. For any distribution τ∈()over posterior beliefs consistent with the common prior F(i.e., such that Gτ(dG)=F), the following condition holds: 1uθ,A(θ;G)>0UP(θ,1 )+1uθ,A(θ;G)≤0UP(θ,0 )G(dθ)τ(dG) ≥UPθ,A(θ;G)G(dθ)τ(dG). 27The critical region can also be defined in terms of the regime outcome. That is, let θ,¯ θ∈R,with θ<¯ θ, be defined by R(θ,1,z)=R(¯ θ,0,z)=0. Note that default occurs with certainty when θ<θ and never occurs when θ> ¯ θ, no matter (A,z). Because the agents’ payoff differential is strictly negative (alternatively, strictly positive) when there is default (alternatively, when there is no default), (θ,θ]⊆(θ,θ]. All the results below hold also under this alternative definition. The reason for defining the critical region as done above is that it permits us to weaken some of the assumptions by requiring that they hold over a smaller set of fundamentals. Clearly, the two definitions coincide when the regime outcome is a deterministic function of (θ,A), as in the baseline model. 28That u(θ,A)is monotone in Aimplies that the continuation game remains supermodular. That UP(θ,A)is nondecreasing in Aimplies that, for any , MARP continues to coincide with the “smallest” rationalizable profile, that is, the one involving the smallest measure of agents investing. Finally, that for any θin the critical region, the policymaker strictly prefers that all agents invest to no agent investing guarantees that, when the optimal policy has a pass/fail structure, it is obtained by maximizing the probability that a pass grade is given when fundamentals are in the critical range.
Theoretical Economics 20 (2025) Adversarial coordination and information design 793 The results are worth extending in a few directions. The analysis assumes that the policymaker is Bayesian and knows the distribution from which the agents’ exogenous private information is drawn. While this is a natural starting point, in future work it would be interesting to investigate how the structure of the optimal policy is affected by the policymaker’s uncertainty about the agents’ information sources.34 Motivated by the applications the analysis is meant for (most notably, stress testing), we have confined attention to nondiscriminatory disclosures. In future work, it would be interesting to extend the analysis to settings in which agents are endowed with exogenous private information (as assumed here) but the designer can disclose different information to different agents (discriminatory policies). The analysis in the present paper is static. Many applications of interest are dynamic, with agents coordinating on multiple attacks and/or learning over time (for the role of dynamics in global games, see, among others, Angeletos, Hellwig, and Pavan (2007)). In future work, it would be interesting to consider dynamic extensions and investigate how the timing of information disclosures is affected by the agents’ behavior in previous periods.35 Finally, the analysis is conducted by assuming that the maximal information that the designer can collect about the fundamentals (in the paper, θ) is exogenous. In future work, it would be interesting to accommodate for the possibility that part of this information is endogenous. For example, in stress testing, the policymaker may solicit information from the same banks that are under scrutiny. This creates an interesting screening +persuasion problem in the spirit of the literature on privacy in sequential contacting (see, e.g., Calzolari and Pavan (2006a,b), and Dworczak (2020)).36 Appendix Proof of Theorem 1*. Given any regular policy =(S,π)and any n∈N,letT (n)be the set of strategies surviving nrounds of iterated deletion of interim strictly dominated strategies (IDISDS), with T (0)denoting the entire set of strategy profiles a=(ai(·))i∈[0,1], where for any i∈[0, 1],ai(x,s)denotes the probability agent iinvests, given (x,s). Let a (n)≡(a (n),i(·))i∈[0,1]∈T (n)denote the most aggressive profile surviving nrounds of IDISDS (i.e., the profile in T (n)that is most adversarial to the policymaker, in the sense that it minimizes the policymaker’s ex ante payoff).The profiles (a (n))n∈Ncan be constructed inductively as follows. The profile a (0)≡(a (0),i(·))i∈[0,1]prescribes that all 34See Dworczak and Pavan (2022) for a notion of robustness in information design that accounts for this type of ambiguity. 35For models of dynamic persuasion, see, among others, Ely (2017) and Basak and Zhou (2022). 36Calzolari and Pavan (2006a) considers an auction setting in which the sender is the initial owner of a good and where the different receivers are privately-informed bidders in an upstream market who then resell in a downstream market. Calzolari and Pavan (2006b) studies information design in a model of sequential contracting with multiple principals, where upstream principals play the role of senders persuading downstream principals (the receivers). Dworczak (2020) contains a general analysis of persuasion in mechanism-design environments with aftermarkets in which senders restrict attention to cut-off mechanisms.
794 Inostroza and Pavan Theoretical Economics 20 (2025) agents refrain from investing, irrespective of (x,s).Next,letU i(x,s;a)denote the payoff differential between investing and not investing for agent ireceiving information (x,s)when, under , all other agents follow the strategy in a.Thena (n),i(x,s)=0if U i(x,s;a (n−1))≤0anda (n),i(x,s)=1ifU i(x,s;a (n−1))>0. MARP consistent with is the profile (a i(·))i∈[0,1]given by a i(·)=limn→∞a (n),i(·),alli∈[0, 1]. Next, observe that, for any n, there exists a function a (n)(·)such that a (n),i(·)=a (n)(·) for all i∈[0, 1]. With an abuse of notation, hereafter we thus denote by athe common strategy that all agents follow under MARP consistent with .Foranyθand s∈supp(π(θ)), aggregate investment under MARP consistent with given (θ,s)is thus thesameforanyx,x∈X(θ)and is given by A(θ,s)≡a(x,s)dP(x|θ). Next, consider the policy +=(S+,π+),S+≡S×{0, 1}, that, for each θ,drawsthe public signal sfrom the same distribution π(θ)∈(S)as the original policy ,andthen, for each sit draws, it also announces the sign of the agents’ payoff differential at (θ,s), when agents play according to MARP consistent with the original policy . That is, for any θand any s∈supp(π(θ)), the new policy +thus announces (s,1(u(θ,A(θ,s)) > 0)). In the baseline model of Section 2,thesignofu(θ,A(θ,s)) is uniquely determined by the regime outcome r(θ,s). In that environment, for any θ,andanys∈supp(π(θ)), the new policy +thus announces (s,r(θ,s)). Define T+ (n)and a+ (n)analogously to T (n)and a (n)above, but with respect to +. The proof is in three steps. Steps 1 and 2 show that any agent iwho, given (x,s), finds it dominant (alternatively, iteratively dominant) to invest under , also finds it dominant (alternatively, iteratively dominant) to invest under +when receiving information (x,(s,1 )). Step 3 uses the above property to establish that, because the game is supermodular and a+is “less aggressive” than a(meaning that any agent who, given (x,s), invests under aalso invests under a+when receiving information (x,(s,1 )), then, under a+, all agents invest (alternatively, refrain from investing) when receiving information (s,1 )(alternatively, (s,0 )). Step 1. We prove that {(x,s):U i(x,s;a)>0∀a}⊆{(x,s):U+ i(x,(s,1 );a)>0∀a}, for all i∈[0, 1].Thatis,anyagentiwho, under , finds it dominant to invest, given information (x,s), also finds it dominant to invest under +when receiving information (x,(s,1 )). First, note that the supermodularity of the game implies that {(x,s):U i(x,s;a)> 0∀a}={(x,s):U i(x,s;a (0))>0}and {(x,s):U+ i(x,(s,1 );a)>0∀a}={(x,s):U+ i(x, (s,1 );a+ (0))>0}. Now let i(x,s)denote the distribution over describing the beliefs of agent i∈ [0, 1]when receiving information (x,s)∈R×Sunder ,and+ i(x,(s,1 )) the corresponding beliefs under +, when receiving information (x,(s,1 )) under +. Bayesian updating implies that + idθ|x,(s,1 )=1uθ,A(θ,s)>0 i(1|x,s) i(dθ|x,s), (11) where i(1|x,s)≡{θ∈:u(θ,A(θ,s))>0} i(dθ|x,s)is the total probability an agent with information (x,s)assigns, under , to fundamentals for which u(θ,A(θ,s)) >0.
Theoretical Economics 20 (2025) Adversarial coordination and information design 795 Next, observe that, for any i∈[0, 1]and (x,s)∈R×Ssuch that U ix,s;a (0)=θ u(θ,0 ) i(dθ|x,s)>0, (12) we also have that U+ ix,(s,1 );a+ (0) i(1|x,s)=θ u(θ,0 )1uθ,A(θ,s)>0 i(dθ|x,s) ≥θ u(θ,0 ) i(dθ|xi,s)=U ix,s;a (0)>0. The first equality follows from the fact that, under a (0), no agent invests, along with the property of posterior beliefs in (11). The first inequality follows from the monotonicity of u(θ,A)in Aalong with the fact A(θ,s)≥0, which together imply that u(θ,0 )≤0for any θfor which u(θ,A(θ,s)) ≤0. The second equality follows from the definition of U i(x,s;a (0)). Finally, the second inequality follows from (12). Thus, any agent for whom investing was dominant after receiving information (x,s) under , continues to find it dominant to invest after receiving information (x,(s,1 )) under +. Step 2. Next, take any n>1. Assume that, for any 1 ≤k≤n−1, any i∈[0, 1], (x,s):U i(x,s;a)>0∀a∈T (k−1)⊆(x,s):U+ ix,(s,1 );a>0, ∀a∈T+ (k−1). (13) Arguments similar to those establishing the result in Step 1 above imply that (x,s):U i(x,s;a)>0∀a∈T (n−1)⊆(x,s):U+ ix,(s,1 );a>0, ∀a∈T+ (n−1). (14) Intuitively, the result follows from the following three properties: (a) because the game is supermodular, {(x,s):U i(x,s;a)>0∀a∈T (n−1)}={(x,s):U i(x,s;a (n−1))>0},where recall that a (n−1)is the most aggressive profile surviving n−1 rounds of IDISDS (clearly, the same property holds for +); (b) a+ (n−1)is “less aggressive” than a (n−1),inthesense that any agent who, given (x,s), invests under a (n−1)also invests under +when receiving information (x,(s,1 )); and (c) the extra information that θis such u(θ,A(θ,s)) >0 removes from the support of the agents’ posterior beliefs states in which the payoff differential from investing is nonpositive under a, and hence also under a (n−1)(recall that a (n−1)is more aggressive that a, meaning that any agent who, given (x,s), invests under a (n−1), also invests under awhen receiving the same information (x,s)). Step 3. Equipped with the results in Steps 1 and 2 above, we now prove that, for all θ∈and s∈supp(π(θ)) such that u(θ,A(θ,s)) >0, a+(x,(s,1 )) ≡limn→∞ a+ (n)(x,(s, 1)) =1forallx. This follows directly from the fact that, as shown above, a(x,s)=1⇒ a+(x,(s,1 )) =1. The announcement that θis such that u(θ,A(θ,s)) >0 thus reveals to each agent that, when all other agents play according to MARP consistent with the new policy +, the payoff differential from investing is strictly positive. Any agent ireceiving information (s,1 )under +thus necessarily invests, no matter x. Under the new policy +, all agents thus invest when they learn that θis such that u(θ,A(θ,s)) >0.
796 Inostroza and Pavan Theoretical Economics 20 (2025) That they all refrain from investing when they learn that θis such that u(θ,A(θ,s)) ≤0 follows from the fact that such an announcement makes it common certainty that θ≤θ. We conclude that the new policy +satisfies the perfect-coordination property. That, when the agents play according to MARP, for any θ, no agent is worse off (and some agents are strictly better off) under +than under follows from the fact that, for all s∈supp(π(θ)), the following are true: (1) when (θ,s)is such that u(θ,A(θ,s))>0, all agents who are not investing under (thus obtaining an expected payoff of zero) invest under +(obtaining an expected payoff u(θ,1 )>0), and all agents who are investing under continue to invest but obtain a larger payoff u(θ,1 )>u (θ,A(θ,s)) because of the monotonicity of u(θ,A)in A; (2) when, instead, (θ,s)is such that u(θ,A(θ,s)) ≤0, all agents who are not investing under (thus obtaining an expected payoff of zero) continue not to invest under +, whereas all agents who are investing under (obtaining a negative payoff) now refrain from investing thus obtaining a payoff of zero. Next, suppose that, under MARP consistent with ,foranyθand s∈supp(π(θ)), the regime outcome is a deterministic function of (θ,s). Then, for any (θ,s),thesignof u(θ,A(θ,s))is determined by the regime outcome (it is strictly positive when r(θ,s)= 1, i.e., when there is no default, and it is weakly negative when r(θ,s)=0, i.e., when there is default). Because the regime outcome is monotone in A, by inducing all agents to invest when u(θ,A(θ,s)) >0 and not to invest when u(θ,A(θ,s)) ≤0, the policy +induces the same regime outcome as . To see that the policymaker is better off under +than under , for any set of signals S⊆S,anyθ,letπ+(S,1|θ)(alternatively, π+(S,0|θ)) denote the probability that the policy π+selects signals (s,1 )(alternatively, (s,0 ))withs∈S.Thenlet+(S,1 )≡ θπ+(S,1|θ)dF(θ)(alternatively, +(S,0 )≡θπ+(S,1|θ)dF(θ)) denote the ex ante probability of announcements (s,1 )(alternatively, (s,0 ))withs∈S, under the policy +. Finally, for any S⊆S,let(S)≡π(S|θ)dF(θ)denote the ex ante probability the policy selects signals in S. Condition PC implies that S1uθ,A(θ,s)>0UP(θ,1 )+1uθ,A(θ,s)≤0UP(θ,0 )(dθ|s)(ds) ≥SUPθ,A(θ,s)(dθ|s)(ds). Hence, the policymaker is better off under +than under . The result in the theorem then follows by taking ∗=+. Proof of Theorem 2*. The proof is in 2 steps. Step 1 shows that, when p(x|θ)is logsupermodular and Condition FB holds, then under any regular policy, MARP is in cut-off strategies. Step 2 then leverages the result in Step 1 to show that, starting from any policy that satisfies the perfect-coordination property, one can construct a binary policy ∗ that also satisfies the perfect-coordination property and such that, for any θ,theprobability that each agent invests under ∗is the same as under , which implies the result in the theorem.
Theoretical Economics 20 (2025) Adversarial coordination and information design 797 Step 1. Fix an arbitrary policy =(S,π)and, for any pair (x,s)∈R×S,let (θ|x,s)represent the endogenous posterior beliefs over of each agent receiving exogenous information xand endogenous information s.Next,letU(x,s|k)≡u(θ,1− P(k|θ))(dθ|x,s)denote the expected payoff differential of an agent with information (x,s), when all other agents follow a cut-off strategy with cut-off k(i.e., they invest if their private signal exceeds kand refrain from investing if it is below k). Lemma 1. Suppose that p(x|θ)is log-supermodular and that Condition FB holds. Given any policy =(S,π), for any s∈S,thereexistsξ;s∈Rsuch that MARP consistent with is given by the strategy profile a≡(a i)i∈[0,1]such that, for any s∈S,x∈R,i∈[0, 1], a i(x,s)=1(x>ξ ;s)with ξ;s≡sup{x:U(x,s|x)≤0}if {x:U(x,s|x)≤0}= ∅,and ξ;s≡−∞otherwise. Moreover, the strategy profile ais a BNE of the continuation game that starts with the announcement of the policy . Proof of Lemma 1. Fix the policy =(S,π).Foranys∈S,letξ;s (1)≡sup{x: limk→∞ U(x,s|k)≤0}. Given the public signal s, it is dominant for any agent with private signal xexceeding ξ;s 1to invest. Next, recall that, for any n∈N,T (n)denotes the set of strategy profiles that survive the first nrounds of iterated deletion of interim strictly dominated strategies (IDISDS), and a (n)≡(a (n),i)i∈[0,1]the most aggressive profile in T (n). Observe that the profile a (1)is given by a (1),i(x,s)=1(x>ξ ;s (1))for all (x,s)∈R×S, and all i∈[0, 1], and minimizes the policymaker’s payoff not just in expectation but for any (θ,s). This follows from the fact that, when nobody else invests, the expected payoff differential u(θ,0 )(dθ|x,s)between investing and not investing crosses 0 only once and from below at x=ξ;s (1). The single-crossing property of u(θ,0 )(dθ|x,s)in turn is a consequence of the fact that u(θ,0 )crosses 0 only once from below at θ=θ(as implied by Condition FB and the definition of θ) along with Property SCB below. Property SCB. Suppose that the function h:R→Rcrosses 0only once from below at θ=θ0(i.e., h(θ)≤0for all θ≤θ0and h(θ)≥0for all θ>θ 0). Let q:R2→R+be a log-supermodular function and suppose that, for any θ, there is an open interval θ= (θ,¯ θ)⊂Rcontaining θsuch that q(x,θ)>0for all x∈θand q(x,θ)=0for (almost) all x∈R\θ, with the bounds θ,¯ θnondecreasing in θ. Choose any (Lebesgue) measurable subset ⊆Rcontaining θ0and, for any x∈R, let (x;)≡h(θ)q(x,θ)dθ. Suppose there exists x∈θ0such that (x;)=0. Then, necessarily, (x;)≥0for all x∈θ0with x>x ,and(x;)≤0for all x∈θ0with x<x ,withbothinequalities strict if (a) {θ∈:h(θ)= 0}has strict positive Lebesgue measure, (b) qis strictly log-supermodular over R2.37 Proof of Property SCB.Foranyx∈R,letx≡{θ∈:x∈θ}. The monotonicity of θin θimplies that xis monotone in xin the strong-order sense. Pick any x∈θ0with x>x .Thatxand xbelong to θ0implies that θ0∈x∩x.Next,observethat x;=x h(θ)qx,θdθ 37That qis strictly log-supermodular over R2also implies that q(x,θ)>0forall(x,θ)∈R2.
798 Inostroza and Pavan Theoretical Economics 20 (2025) =x∩x h(θ)qx,θdθ+x\x h(θ)qx,θdθ =x∩x∩(−∞,θ0) h(θ)qx,θqx,θ qx,θdθ +x∩x∩(θ0,∞) h(θ)qx,θqx,θ qx,θdθ +x\x h(θ)qx,θdθ ≥qx,θ0 qx,θ0x∩x∩(−∞,θ0) h(θ)qx,θdθ+x∩x∩(θ0,∞) h(θ)qx,θdθ +x\x h(θ)qx,θdθ ≥qx,θ0 qx,θ0x; =0 +x\x h(θ)qx,θdθ≥0. The first equality follows from the fact that q(x,θ)=0 for almost all θ∈\x.The second equality follows from the fact that xcan be partitioned into x∩xand x\x. The third equality follows from noting that q(x,θ)>0forallθ∈x.The first inequality follows from the monotonicity of q(x,θ)/q(x,θ)over x∩xas a consequence of qbeing log-supermodular, along with the fact that θ0∈x∩xand the assumption that hcrosses 0 once from below at θ=θ0. The second inequality follows from the fact that, for any θ∈(x\x)∩(−∞,θ0),h(θ)≤0, along with the fact that x∩(θ0,+∞)=x∩x∩(θ0,∞), with the last property following from noting that the sets xare ranked in the strong-order sense. The last inequality follows from the observation that, for any θ∈x\x,h(θ)≥0, which in turn is a consequence of (i) the monotonicity of the sets xin x, (ii) the assumption that hcrosses 0 only once from below at θ=θ0, and (iii) the assumption that θ0∈x∩x. Similar arguments imply that, for x<x ,(x;)≤0. The same arguments also imply that, when (a) {θ∈:h(θ)= 0}has strict positive Lebesgue measure and (b) qis strictly log-supermodular over R2,then(x;)<0forallx<x and (x;)>0forall x>x . This completes the proof of Property SCB. The facts that (a) the continuation game is supermodular, (b) the density p(x|θ)is log-supermodular, and (c) when agents follow monotone strategies, the regime outcome is monotone in θimply that, for any s∈S, there exists a unique sequence (ξ;s (n))n∈Nsuch that, for any n≥1, a (n)is such that a (n),i(x,s)=1(x>ξ ;s (n))for all iand all (x,s)∈R×S, with each ξ;s (1)as defined above, and with all other cut-offs ξ;s (n),n>1, s∈S,defined inductively by ξ;s (n)≡sup{x:U(x,s|ξ;s (n−1))≤0}. Indeed, Condition FB together with Property SCB jointly imply that U(x,s|ξ;s (n−1))=u(θ,1−P(ξ;s (n−1)|θ))(dθ|x,s)
Theoretical Economics 20 (2025) Adversarial coordination and information design 799 crosses zero once from below in xand, therefore, U(x,s|ξ;s (n−1))>0 if, and only if, x>ξ ;s (n). Let T≡∩ ∞ n=1T ndenote the set of strategy profiles that survive IDISDS under . The most aggressive strategy profile in Tis then given by a i(x,s)≡1(x>ξ ;s)for all iand all (x,s)∈R×S,where,foranys∈S,ξ;s≡limn→∞ ξ;s (n). The sequence (ξ;s (n))nis monotone and its limit is given by ξ;s=sup{x:U(x,s|x)≤0}if {x:U(x,s|x)≤0}= ∅, and ξ;s≡−∞otherwise. This establishes the first part of the lemma. That the profile ais a BNE for the continuation game that starts with the announcement of the policy follows from the fact that, given any s∈S, when all agents follow a cut-off strategy with cutoff ξ;s, the best response for each agent i∈[0, 1]is to invest for xi>ξ ;sand to refrain from investing for xi<ξ ;s. This completes the proof of the lemma. Step 2. Now take any regular policy =(S,π)satisfying the perfect-coordination property. Given the result in Theorem 1, without loss of generality, assume that = (S,π)is such that S={0, 1}׈ S, for some measurable set ˆ S, and is such that (a) when the policy discloses any signal s=(ˆ s,1 ), all agents invest and default does not happen, whereas (b) when the policy discloses any signal s=(ˆ s,0 ), all agents refrain from investing and default happens. Equipped with the result in Lemma 1, we show that, starting from =(S,π),onecan construct a binary policy ∗=({0, 1},π∗)also satisfying the perfect-coordination property and such that the probability of default under ∗isthesameasunder. The policy ∗=({0, 1},π∗)is such that, for any θ,π∗(1|θ)=ˆ Sπ(d(ˆ s,1 )|θ). That is, for each θ,the binary policy ∗recommends to invest with the same total probability as the original policy discloses signals leading all agents to invest.38 We now show that, under ∗, when the policy announces that s=1, the unique rationalizable action for each agent is to invest. To see this, for any (x,1 )that are mutually consistent given ∗,letU∗(x,1|k)denote the expected payoff differential for any agent with private signal x, when the policy ∗announces s=1, and all other agents follow a cut-off strategy with cut-off k.39 From the law of iterated expectations, we have that U∗(x,1|k)=ˆ S Ux,(ˆ s,1 )|kς(dˆ s|x,1 )(15) where ς(·|x,1 )is the probability measure over ˆ Sobtained by conditioning on the event (x,1 ),under. For any signal s=(ˆ s,1 )in the range of π, MARP consistent with is such that a i(x,(ˆ s,1 )) =1allx∈R,andalli, meaning that investing is the unique rationalizable action after announces s=(ˆ s,1 ). Lemma 1in turn implies that, for all s=(ˆ s,1 ) in the range of π,ˆ s∈ˆ S,allk∈R,U(k,(ˆ s,1 )|k)>0. From (15), we then have that, for all all k∈R,U∗(k,1|k)>0. In turn, this implies that, given the new policy ∗,whens=1 is disclosed, under MARP consistent with ∗, all agents invest, that is, a∗ i(x,1 )=1allx, all i∈[0, 1]. It is also easy to see that, when the policy ∗discloses the signal s=0, it becomes common certainty among the agents that θ≤θ. Hence, under MARP consistent 38ˆ Sπ(d(ˆ s,1 )|θ)represents the total probability that the measure π(θ)assigns to signal (ˆ s,1 ). 39Recall that (x,1 )are mutually consistent under ∗if p∗(x,1 )≡p(x|θ)π∗(1|θ)dF(θ)>0.
800 Inostroza and Pavan Theoretical Economics 20 (2025) with ∗, after s=0 is disclosed, all agents refrain from investing, irrespective of their private signals. The new policy ∗so constructed thus (a) satisfies the perfect-coordination property, and (b) is such that, for any θ, the probability of default under ∗is the same as under . Proof of Theorem 3*. The conditions in the theorem imply that Theorems 1* and 2* hold. Thus, assume that the policy =(S,π)(a) is a regular (possibly stochastic) “pass/fail”policy (i.e., S={0, 1},withπ(1|θ)=1−π(0|θ)denoting the probability that signal s=1 is disclosed when the fundamentals are θ), (b) is such that π(1|θ)=0for all θ≤θand π(1|θ)=1forallθ>θ, and (c) satisfies the perfect-coordination property. Theorems 1* and 2* imply that, if does not satisfy these properties, there exists another policy that satisfies these properties and yields the policymaker a payoff weakly higher than . The proof then follows from applying the arguments below to instead of . Suppose that is such that there exists no ˆ θsuch that π(1|θ)=0forF-almost all θ≤ˆ θand π(1|θ)=1forF-almost all θ> ˆ θ.40 We establish the result by showing that there exists a deterministic monotone policy ˆ θ=({0, 1},πˆ θ)satisfying the perfectcoordination property that yields the policymaker a payoff strictly higher than . Recall that, for the policy to satisfy the perfect-coordination property, it must be that, when the policy discloses the signal s=1, U(x,1|x)>0forallxsuch that (x,1 ) are mutually consistent, where U(x,1|x)is the expected payoff differential of an agent with signal xwho hears that s=1 and who expects all other agents to follow a cut-off strategy with threshold x. Let Gdenote the set of policies =(S,π)that, in addition to properties (a) and (b) above, are such that U(x,1|x)≥0forallxsuch that (x,1 )are mutually consistent.41 For any ∈G,letUP[]denote the policymaker’s ex ante expected payoff when, under , agents invest after hearing that s=1 and refrain from investing after hearing that s=0. Denote by argmax˜ ∈GUP[˜ ]the set of policies that maximize the policymaker’s payoff over G.42 Step 1 below shows that any ∈argmax˜ ∈GUP[˜ ]is such that π(1|θ)=0forFalmost all θ≤θ∗and π(1|θ)=1forF-almost all θ>θ ∗,withθ∗as defined in (4). Step 2 then shows that the policymaker’s payoff under the optimal monotone policy θ∗=({0, 1},πθ∗)with cut-off θ∗can be approximated arbitrarily well by a deterministic monotone policy ˆ θ=({0, 1},πˆ θ)∈Gthat satisfies the perfect-coordination property, thus establishing the theorem. Step 1. Given any policy ,let X≡x:(x,1 )-mutually consistent and U(x,1|x)=0. 40If this not the case, then the deterministic monotone policy ˆ θ=({0, 1},πˆ θ)with cut-off ˆ θalso satisfies the perfect-coordination property and yields the policymaker the same payoff as , in which case the result trivially holds. 41As explained in the main text, some policies in Gneed not satisfy the perfect-coordination property, namely those for which there exists x,with(x,1 )mutually consistent, such that U(x,1|x)=0. 42That argmax˜ ∈GUP[˜ ]= ∅ follows from the compactness of Gand the upper hemicontinuity of UP.
Theoretical Economics 20 (2025) Adversarial coordination and information design 801 Take any policy ∈Gfor which there exists no ˆ θsuch that π(1|θ)=0forF-almost all θ≤ˆ θand π(1|θ)=1forF-almost all θ> ˆ θ. Clearly, if X=∅, there exists another policy ∈Gthat yields the policymaker a payoff strictly higher than .43 Thus, assume that X= ∅,andlet ¯ x≡supX.ClaimAbelow shows that the set {θ∈Θ(¯ x):π(1|θ)< 1}has strict positive F-measure. Claim Bshows that, given any ∈Gfor which the posterior beliefs of the marginal agent with signal ¯ xdiffer from those obtained by Bayes’ rule conditioning on the event that fundamentals are above some threshold ˆ θ,there exists another policy ∈Gthat yields the policymaker a payoff strictly higher than . Finally, Claim Cshows that, under the properties in Condition M*, the only policies ∈ Gthat generate posterior beliefs for the marginal agents with signal ¯ xequal to those obtained from Bayes’ rule by conditioning on the event that fundamentals are above some threshold ˆ θare such that π(1|θ)=0forF-almost all θ≤θ∗and π(1|θ)=1for F-almost all θ>θ ∗. Jointly, the three claims thus establish the result that any policy ∈argmax˜ ∈GUP[˜ ],issuchthatπ(1|θ)=0forF-almost all θ≤θ∗and π(1|θ)=1for F-almost all θ>θ ∗. Given any x,letθ0(x)be the fundamental threshold below which the agents’ expected payoff differential is negative and above which it is positive, when all agents follow a cut-off strategy with cut-off x. Because Condition FB holds, θ0(x)is well-defined.44 For any policy =({0, 1},π)∈G,letp(x,1 )≡+∞ −∞ π(1|θ)p(x|θ)dF(θ)denote the joint probability density of the exogenous signal xand the endogenous signal s=1. Claim A. For any =({0, 1},π)∈Gsuch that X= ∅,{θ∈Θ(¯ x):π(1|θ)<1}has strict positive F-measure. Proof of Claim A. Suppose, by contradiction, that π(1|θ)=1forF-almost all θ∈ Θ(¯ x). Property (i*) in Condition M* then implies that ¯ x>x max,wherexmax is defined as in (8). In fact, if this was not the case, the monotonicity of Θ(·)would imply that infΘ(¯ x)≤infΘ(xmax )<θ .Thatπ(1|θ)=1forF-almost all θ∈Θ(¯ x)would then imply that π(1|θ)=1 for a set of fundamentals θ<θof strict positive F-measure, which is inconsistent with the assumption that ∈G. Thus, necessarily, ¯ x>x max. Now suppose that inf Θ(¯ x)≥θ.Thatπ(1|θ)=1forF-almost all θ∈Θ(¯ x)means that, from the perspective of an agent with signal ¯ x, the information conveyed by the announcement that s=1underis the same as under the monotone deterministic policy θ=({0, 1},πθ)with cut-off ˆ θ=θ.Asaresult,U(¯ x,1|¯ x)=Uθ(¯ x,1|¯ x). Because ¯ x>x max, and because, by definition of xmax,Uθ(x,1|x)>0forallx>x max,itmustbe that U(¯ x,1|¯ x)>0, which contradicts the assumption that ¯ x∈X. Hence,itmustbe that infΘ(¯ x)<θ . As explained above, however, this is inconsistent with the assumption that ∈G. 43In fact, because there exists no such a ˆ θ, there must exists a set (θ,θ )⊆[θ,¯ θ]of F-positive measure over which π(1|θ)<1. The policy can then be obtained from by increasing π(1|θ)over such a set. Provided the increase is small, ∈G.BecauseUP(θ,1 )>U P(θ,0 )over [θ,¯ θ], the policymaker’s payoff under is strictly higher than under . 44When the regime outcome is a function of Aand θonly, as in the baseline model, θ0(x)coincides with the threshold below which default occurs and above which it does not occur when agents follow a cut-off strategy with cut-off x.
802 Inostroza and Pavan Theoretical Economics 20 (2025) Next, for any =({0, 1},π)∈G,let θH≡supθ∈:∃δ>0s.t.π1|θ<1forF-almost all θ∈[θ−δ,θ). The result in Claim Aabove implies that θHis such that θH>inf Θ(¯ x). Claim B. Take any =({0, 1},π)∈Gsuch that X= ∅. Suppose that θ∈(θ,θH):π(1|θ)>0has strict positive F-measure. (16) Then there exists another policy ∈Gthat yields the policymaker a payoff strictly higher than . Claim Bessentially says that, if ∈Gis not a deterministic monotone rule, and there exists a ¯ xsuch that U(¯ x,1|¯ x)=0, then it is improvable. Proof of Claim B. The proof below distinguishes two cases. Case 1:θ<θ 0(¯ x)≤θH. Consider the policy ,δ=({0, 1},π,δ)defined by π,δ(1| θ)=π(1|θ)for all θ≤θ0(¯ x+δ),withδ>0 small so that θ0(¯ x+δ)<θ H,andπ,δ(1|θ)= min{π(1|θ)+,1 }for all θ>θ 0(¯ x+δ),with>0 also small. To see that, when and δ are small, ,δ∈G, note that by definition of θ0(·),foranyx,andanyθ>θ 0(x),u(θ,1− P(x|θ)) >0. This property, together with the monotonicity of θ0(·), jointly imply that, for any x≤¯ x+δ, ∞ −∞ uθ,1−P(x|θ)π(1|θ)1θ≤θ0(¯ x+δ) +minπ(1|θ)+,1 1θ>θ 0(¯ x+δ)p(x|θ)dF(θ) ≥∞ −∞ uθ,1−P(x|θ)π(1|θ)p(x|θ)dF(θ). (17) To see what justifies the inequality, observe that u(θ,1−P(¯ x+δ|θ)) >0forθ>θ 0(¯ x+δ), by definition of θ0(·). Because, for any θ,u(θ,1−P(x|θ)) is decreasing in x,we then have that, for any x≤¯ x+δ,u(θ,1−P(x|θ)) >0forallθ>θ 0(¯ x+δ). Because ∈G, the right-hand side of (17) is nonnegative.45 Hence, for any x≤¯ x+δsuch that (x,1 )are mutually consistent under ,δ, because the left-hand side of (17)is equal to U,δ(x,1|x)p,δ(x,1 )and because, for such x,p,δ(x,1 )>0, we have that U,δ(x,1|x)≥0. That U,δ(x,1|x)≥0alsoforallx> ¯ x+δsuch that (x,1 )are mutually consistent under ,δfollows from the fact that, by definition of ¯ x,foranyx≥¯ x+δ, the function J(x)≡+∞ −∞ u(θ,1−P(x|θ))π(1|θ)p(x|θ)dF(θ)is bounded away from 0, along with the fact that, for any δ>0, the function family (J,δ(·))whose elements J,δ(·)are given by J,δ(x)≡+∞ −∞ u(θ,1−P(x|θ))π,δ(1|θ)p(x|θ)dF(θ)is continuous in in the sup-norm in a neighborhood of 0.46 Because the new policy ,δ∈Gis such that 45Either (x,1 )are not mutually consistent under , in which case the right-hand side of (17)iszero,or they are mutually consistent, in which case the right-hand side of (17)isequaltoU(x,1|x)p(x,1 ),which is nonnegative because p(x,1 )>0 and U(x,1|x)≥0. 46That is, ∀k>0, ∃>0sothat∀0<<,|J,δ(x)−J(x)|≤k,∀x≥¯ x+δ.
Theoretical Economics 20 (2025) Adversarial coordination and information design 809 UθH(¯ x,1|¯ x)=0. Property (i*) of Condition M* then implies that infΘ(¯ x)<θ .Hence, for x=¯ x, the inequality in (23) is strict, which in turn implies that Uθ∗ (¯ x,1|¯ x)<0, contradicting the assumption that θ∗∈G. Therefore, it must be that θH≤θ∗. However, by definition of θ∗,ifθH<θ ∗, there exists an xsuch that (a) UθH(x,1|x)<0, and (b) (x,1 ) are mutually consistent under θH. Because, for all such x,U(x,1|x)is well-defined (i.e., (x,1 )are mutually consistent also under )andUθH(x,1|x)=U(x,1|x),wethus have that U(x,1|x)<0, which contradicts the assumption that ∈G.Hence,θH=θ∗. This completes the proof of Claim C. Step 2. Step 1 implies that arg max˜ ∈GUP[˜ ]= ∅ and that any ∗=({0, 1},π)with ∗∈argmax˜ ∈GUP[˜ ]is such that π(1|θ)=0forF-almost all θ≤θ∗and π(1|θ)=1for F-almost all θ>θ ∗. The result in the theorem then follows from observing that, given any ∗∈argmax˜ ∈GUP[˜ ], there exists a nearby deterministic monotone policy ˆ θ∈G with cut-off ˆ θ=θ∗+˜ε,for˜ε>0 small, such that ˆ θsatisfies the perfect-coordination property (i.e., Uˆ θ(x,1|x)>0allxsuch that (x,1 )are mutually consistent under ˆ θ).50 The continuity of UP[ˆ θ]in ˆ θthen implies that, for ˜ε>0 small, UP[ˆ θ]>UP[],thus establishing the result in the theorem. References Alonso, Ricardo and Odilon Camara (2016a), “Persuading voters.” American Economic Review, 106, 3590–3605. [0767] Alonso, Ricardo and Konstantinos E. Zachariadis (2023), “Persuading large investors.” Available at SSRN 3781499. [0767] Alvarez, Fernando and Gadi Barlevy (2021), “Mandatory disclosure and financial contagion.” Journal of Economic Theory, 194, 105237. [0768] Angeletos, George-Marios, Christian Hellwig, and Alessandro Pavan (2006), “Signaling in a global game: Coordination and policy traps.” Journal of Political Economy, 114, 452–484. [0767] Angeletos, George-Marios, Christian Hellwig, and Alessandro Pavan (2007), “Dynamic global games of regime change: Learning, multiplicity, and the timing of attacks.” Econometrica, 75, 711–756. [0767,0793] Angeletos, George-Marios and Alessandro Pavan (2013), “Selection-free predictions in global games with endogenous information and multiple equilibria.” Theoretical Economics, 8, 883–938. [0767] Angeletos, George-Marios and Iván Werning (2006), “Crises and prices: Information aggregation, multiplicity, and volatility.” American Economic Review, 96, 1720–1736. [0767] Arieli, Itai and Yakov Babichenko (2019), “Private Bayesian persuasion.” Journal of Economic Theory, 182, 185–217. [0767] 50The arguments are the same as those used in the proof of Claim Cfor the case θH>θ ∗.
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Theoretical Economics 20 (2025) Adversarial coordination and information design 813 Orlov, Dmitry, Pavel Zryumov, and Andrzej Skrzypacz (2023), “The design of macroprudential stress tests.” The Review of Financial Studies, 36, 4460–4501. [0767] Petrella, Giovanni and Andrea Resti (2013), “Supervisors as information producers: Do stress tests reduce bank opaqueness?” Journal of Banking and Finance, 37, 5406–5420. [0768] Quigley, Daniel and Ansgar Walther (2024), “Inside and outside information.” The Journal of Finance. [0768] Rochet, Jean-Charles and X. Vives (2004), “Coordination failures and the lender of last resort: Was bagethor right after all?” Journal of the European Economic Association,2, 1116–1147. [0768] Sakovics, Jozsef and Jakub Steiner (2012), “Who matters in coordination problems?” American Economic Review, 102, 3439–3461. [0766] Segal, Ilya (2003), “Coordination and discrimination in contracting with externalities: Divide and conquer?” Journal of Economic Theory, 113, 147–181. [0766] Shimoji, Makoto (2021), “Bayesian persuasion in unlinked games.” International Journal of Game Theory, 1–31. [0767] Szkup, Michal and Isabel Trevino (2015), “Information acquisition in global games of regime change.” Journal of Economic Theory, 160, 387–428. [0767] Taneva, Ina (2019), “Information design.” American Economic Journal: Microeconomics, 11, 151–185. [0767] Williams, Basil (2017), “Stress tests and bank portfolio choice.” Manuscript, Imperial College. [0767] Winter, Eyal (2004), “Incentives and discrimination.” American Economic Review, 94, 764–773. [0766] Yang, Ming (2015), “Coordination with flexible information acquisition.” Journal of Economic Theory, 158, 721–738. [0767] Co-editor Simon Board handled this manuscript. Manuscript received 7 July, 2023; final version accepted 18 August, 2024; available online 9 September, 2024.