Adversarial Coordination and Public Information Design

Adversarial Coordination

and Public Information Design

∗

Nicolas Inostroza

University of Toronto

Alessandro Pavan

Northwestern University and CEPR

July 11, 2024

Abstract

We study ﬂexible public information design in global games. In addition to receiving

public information from the designer, agents are endowed with exogenous private infor-

mation and must decide between two actions (invest and not invest), the proﬁtability

of which depends on unknown fundamentals and the agents’ aggregate action. The de-

signer 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’ ac-

tions. Second, we identify conditions under which the optimal policy takes the form of

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.

JEL classiﬁcation: D83, G28, G33.

Keywords: Global Games, Adversarial Coordination, Bayesian Persuasion, Robust Pub-

lic Information Design.

∗

Emails: [email protected]to.ca, alepavan@northwestern.edu. A previous version circu-

lated under the title “Persuasion in Global Games with Application to Stress Testing.” For comments and

useful suggestions, we thank Marios Angeletos, Gadi Barlevy, Eddie Dekel, Tommaso Denti, Laura Doval,

Stephen Morris, Marciano Siniscalchi, Bruno Strulovici, Jean Tirole, Xavier Vives, Leifu Zhang, and sem-

inar 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.

1 Introduction

Coordination plays a major role in many socio-economic environments. The damages to

society of mis-coordination 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. 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 policy maker 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, technology and standards adoption. In the context of stress testing,

the policy maker may represent a supervising authority attempting to prevent a run against

the banking sector (see, for example, Henry and Christoﬀer (2013) and Homar et al. (2016)).

In the case of currency crises, the policy maker 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 policy maker 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)).

The backbone of the model is a global game of regime change in which multiple agents

must choose between “attacking” a status quo or “refraining from doing so,” 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.

We assume the policy maker can ﬂexibly design a policy that disseminates publicly infor-

mation about relevant economic fundamentals. We use the model to address the following

questions: (a) Are there beneﬁts to preventing market participants from predicting each oth-

ers’ actions and beliefs? (b) When are simple policies such as pass/fail tests optimal? (c) Are

there merits to non-monotone 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 par-

ticipants 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 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.

Under adversarial coordination, preserving uncertainty over be-

liefs 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 in-

formation that facilitates perfect coordination may also favor the emergence of rationalizable

Such a robust approach is motivated by the applications the analysis is meant for. For example, when

concerned about runs to the banking sector, policy makers typically do not trust the market to play favorably.

The optimal policy does not ensure that investing is the unique rationalizable action based on ﬁrst-order

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.

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

co-move 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 j is more optimistic than agent i before 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 policy maker 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

policy maker discloses can be used to change the ranking of the agents’ optimism, the policy

maker 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.

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.

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 policy maker’s pref-

erences for a favorable outcome (e.g., for avoiding a bank default) vary with the fundamentals.

We identify precise conditions involving the policy maker’s preferences and the agents’ payoﬀs

When, 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.

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 non-monotone

rules strictly outperform monotone ones.

The reason is that non-monotone rules make it

more diﬃcult for the agents to commonly learn the fundamentals and hence permit the policy

maker to give a pass grade to a larger set of fundamentals. When the policy maker prefer-

ences 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), non-monotone rules may be

optimal.

Organization. The rest of the paper is organized as follows. Below, we wrap up the in-

troduction with a brief review of the most pertinent literature. Section 2 presents the model.

Section 3 contains all the results about properties of optimal policies (perfect-coordination,

pass/fail, monotonicity). Section 4 discusses how the results accommodate enrichments that

are useful in applications (e.g., more general payoﬀs, as well as the possibility that the pol-

icy maker faces uncertainty about the outcome induced by her information dissemination).

Section 5 concludes. The Appendix contains all proofs with the exception of the proofs of

Examples 2 and 3 which are in the Online Supplement Inostroza and Pavan (2024b). The

latter also expands the material in Subsection 4.3 discussing the role of the multiplicity of the

receivers and their exogenous private information for the optimality of monotone rules. 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 mar-

ket response) to a broader class of economies, and (b) discusses the beneﬁts of discriminatory

disclosures, when the latter are feasible.

(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

We also show that the conditions guaranteeing the optimality of monotone rules are more stringent when

the policy maker 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.

(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) and

Galv˜ao and Shalders (2022) also study public information design in settings in which the

receivers possess exogenous private information. Goldstein and Huang (2016) restrict the

policy maker to binary monotone rules, whereas Galv˜ao 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 sub-optimal but then identiﬁes sharp conditions under which such

policies are optimal; (c) it shows why, in general, non-monotone rules permit the policy maker

to induce a favorable outcome over a larger set of fundamentals but then identiﬁes sharp

conditions under which optimal policies are monotone.

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 H¨orner (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.

The third strand is the literature on global games with endogenous information. Angeletos,

Hellwig and Pavan (2006) and Angeletos and Pavan (2013) study signaling in global games.

Angeletos and Werning (2006) investigate the role of prices as a vehicle for information ag-

gregation. Angeletos, Hellwig and Pavan (2007) consider a dynamic model in which agents

In particular, Example 3 below shows that non-monotone rules strictly outperform monotone ones in the

same environment of Goldstein and Huang (2016).

See Bergemann and Morris (2019) and Kamenica (2019) for an overview on information design.

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) and Kyriazis and Lou (2024) consider propaganda in global games, in a setting in

which the policy maker manipulates the agents’ private signals. Szkup and Trevino (2015),

Yang (2015), Morris and Yang (2022), and Denti (2023) study the acquisition of private in-

formation 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. See Goldstein and Sapra

(2014) for an overview of some of the early contributions. Bouvard, Chaigneau and Motta

(2015) study a setting where a policy maker 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, Martinez and Philippon (2016) and

Garcia 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) and

Quigley 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 (2024c) 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 socio-economic 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) in which the agents 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.

The analysis, however, readily

extends to many other global games.

Players and Actions. A policy maker designs an information disclosure policy, e.g.,

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, a

= 1, or an “adversar-

ial” action, a

= 0. The friendly action is interpreted as the decision to invest (more generally,

to "refrain from attacking” a status quo the policy maker wants to preserve). The adversarial

action is interpreted as the decision to not invest (more generally, to “attack”). We denote

by A ≡

di ∈ [0, 1] the size of the aggregate investment.

Fundamentals and Exogenous Information. Consistently with the rest of the litera-

ture, we parameterize the relevant fundamentals by θ ∈ R. The fundamentals are exogenous

to the policy maker’s choice of a disclosure policy. It is commonly believed (by the policy

maker and the agents alike) that θ is drawn from a distribution F , absolutely continuous over

an interval Θ ⫌ [0, 1], with a smooth density f strictly positive over Θ. In addition, each

agent i ∈ [0, 1] is endowed with private information summarized by a uni-dimensional statistic

∈ R drawn independently across agents given θ from an absolutely continuous cumulative

distribution function P (x|θ) with smooth density p(x|θ) strictly positive over an (open) inter-

Rochet and Vives (2004) consider a three-period economy a’ la Diamond and Dybvig (1983) but with

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.

val ϱ

≡ (ϱ

, ¯ϱ

) containing θ, with ϱ

, ¯ϱ

monotone in θ, and with p(x|θ) bounded over (x, θ).

The bounds ϱ

, ¯ϱ

can be either ﬁnite or inﬁnite. For example, when x

= θ + σε

, with ε

drawn from a uniform distribution over [−1, +1], then, for any θ, ϱ

= θ − σ and ¯ϱ

= θ + σ.

When, instead, x

= θ +σε

, with ε

drawn from a standard Normal distribution, then, for any

θ, ϱ

= −∞ and ¯ϱ

= +∞. Furthermore, in this latter case, P (x|θ) = Φ((x − θ)/σ), where

Φ is the cumulative distribution function of the standard Normal distribution. We denote

by x ≡ (x

)

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 θ parameterize 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.

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 policy

maker’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. When W and L are invariant in θ, the policy maker’s objective reduces

to minimizing the probability of default. The functions b, g, W , and L are all bounded. For

any (θ, A) ∈ Θ × [0, 1], then let

u (θ, A) ≡ g(θ)1(A > 1 − θ) + b(θ)1(A ≤ 1 − θ),

(θ, A) ≡ W (θ)1(A > 1 − θ) + L(θ)1(A ≤ 1 − θ)

The model assumes that, given A and 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 A and θ, the regime

outcome is stochastic and determined by variables that are not observable by the policy maker at the time of

her public announcements (see the discussion in Section 4).

denote the payoﬀs of a representative agent and of the policy maker, respectively, when the

fundamentals are θ and the aggregate investment is A.

Policy. Let S be a compact Polish space deﬁning the set of possible signal realizations.

A policy Γ = (S, π) consists of the set S along with a measurable mapping π : Θ → ∆(S)

specifying, for each θ, a probability distribution over the information disclosed to the market.

Timing. The sequence of events is the following:

1. The policy maker publicly announces the policy Γ = (S, π) and commits to it.

2. The fundamentals θ are drawn from the distribution F and the agents’ exogenous signals

x ∈ X(θ) are drawn from the distribution P (x|θ).

3. The public signal s is drawn from the distribution π(θ) and is publicly observed.

4. Agents simultaneously choose whether or not to invest.

5. The regime outcome is determined (i.e., whether or not default occurred) and payoﬀs

are realized.

Adversarial Coordination and Robust Information Design. The policy maker does

not trust the market to follow her recommendations and play favorably to her (i.e., invest

whenever θ > 0).

Instead, she adopts a robust/conservative approach. She evaluates any

policy Γ under the “worst-case” scenario, i.e., 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 Γ.

Deﬁnition 1. Given any policy Γ, the most aggressive rationalizable proﬁle (MARP)

consistent with Γ is the strategy proﬁle a

≡ (a

)

i∈[0,1]

that minimizes the policy maker’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 θ and the other agents’ exogenous information x ∈ X(θ) using the

common prior F , the distribution of private signals P (x|θ), and the policy Γ. Under MARP,

See Leitner and Williams (2023) for a discussion of the commitment assumption in stress testing.

If she did, a simple monotone policy revealing whether or not θ > 0 would be optimal.

given (x, s), each agent i ∈ [0, 1], after receiving exogenous information x from Nature and

endogenous information s from the policy maker, 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 Γ for which MARP exists.

Because

the game among the agents is supermodular (no matter the prior F , the distribution P from

which the exogenous signals are drawn, and the policy Γ), the strategy proﬁle a

coincides with

the “smallest” Bayes-Nash equilibrium (BNE) of the continuation game among the agents, and

minimizes the policy maker’s payoﬀ state by state, and not just in expectation. The reason

why we consider MARP is that, in general, without imposing speciﬁc assumptions on F , P ,

and Γ, the only way the “smallest” BNE can be identiﬁed is by the iterated deletion of interim

dominated strategies. In standard global games, the “smallest” BNE is typically identiﬁed

by assuming the agents’ signals are drawn from a distribution P satisfying the monotone

likelihood property (MLRP), which is also used to guarantee equilibrium uniqueness. Here,

we allow for arbitrary policies Γ, and do not require that, given Γ, the continuation equilibrium

be unique.

Furthermore, given a policy Γ = (S, π), when describing the agents’ behavior, we do

not distinguish between pairs (x, s) that are mutually consistent given Γ (meaning that the

joint density of (x, s) is positive, i.e.,

θ:s∈supp(π(θ))

p(x|θ)dF (θ) > 0) and those that are not.

Because the policy maker commits to the policy Γ, the abuse is legitimate and permits us to

ease the exposition. Any claim about the optimality of the agents’ behavior, however, should

be interpreted to apply to pairs (x, s) that are mutually consistent given Γ.

3 Properties of optimal policies

We now introduce and discuss three key properties of optimal policies.

Because the state is continuous, in principle, one can think of policies Γ 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.

3.1 Perfect-coordination property

Deﬁnition 2. A policy Γ = (S, π) satisﬁes the perfect-coordination property (PCP) if,

for any θ ∈ Θ, any exogenous information x ∈ X(θ), any public announcement s ∈ supp(π(θ)),

and any pair of individuals i, j ∈ [0, 1], a

, s) = a

, s), where a

= (a

)

i∈[0,1]

is the most

aggressive rationalizable proﬁle (MARP) consistent with the policy Γ.

A disclosure policy thus has the perfect-coordination property if it coordinates all market

participants on the same action, after any information it discloses. For any θ ∈ Θ, any

s ∈ supp(π(θ)), let r

(θ, s) ∈ {0, 1} denote the regime outcome that prevails when agents

play according to a

, that is, r

(θ, s) = 1 (alternatively, r

(θ, s) = 0) means that default

does not occur (alternatively, occurs) when, given (θ, s), market participants play according

to MARP consistent with Γ. That the agents’ signals are drawn independently from P (x|θ),

conditional on θ, implies that the cross-sectional distribution of signals is pinned down by

P (x|θ), and hence the regime outcome (that is, whether default occurs or not) is the same

across any pair of signal proﬁles x, x

′

∈ X(θ) and thus depends only on Γ, θ, and s. Hereafter,

we say that the policy Γ is regular if MARP under Γ is well-deﬁned and the regime outcome

under a

is measurable in (θ, s).

Theorem 1. Given any regular policy Γ, there exists another regular policy Γ

∗

satisfying the

perfect-coordination property (PCP) and such that, when the agents play according to MARP

under both Γ and Γ

∗

, for any θ, (a) the probability of default under Γ

∗

is the same as under Γ,

(b) the transition from Γ to Γ

∗

leads to a Pareto improvement (the policy maker is indiﬀerent,

no agent is worse oﬀ, and some agents are strictly better oﬀ).

The policy Γ

∗

is obtained from the original policy Γ by disclosing, for each θ, in addition to

the information s ∈ supp(π(θ)) disclosed by the original policy Γ, a second piece of information

that reveals to the market the regime outcome r

(θ, s) ∈ {0, 1} that prevails at (θ, s) when

agents play according to MARP consistent with the original policy Γ, a

That, under the new policy Γ

∗

, it is rationalizable for all agents to invest when the pol-

icy discloses the information



s, r

(θ, s)



= (s, 1), and to refrain from investing when the

policy discloses the information



s, r

(θ, s)



= (s, 0), is fairly straight-forward. In fact, the

announcement of (s, 1) (alternatively, of (s, 0)) makes it common certainty among the agents

that θ > 0 (alternatively, that θ ≤ 1).

The reason why the result is not obvious is that the designer does not content herself with

one rationalizable proﬁle delivering the desired outcome; she is concerned with the possibility

of adversarial coordination and, as a result, when she recommends to all the agents to invest,

she must guarantee that investing is the unique rationalizable action for each agent, irrespec-

tive of his exogenous signal x. The proof in the Appendix shows that, when the additional

information is r

(θ, s), this is indeed the case.

To ﬁx ideas, consider ﬁrst the case where, under the original policy Γ, the regime outcome

(θ, s) is monotone in θ. The announcement that r

(θ, s) = 1 makes it common certainty

among the agents that θ >

θ(s), for some threshold

θ(s). In this case, all agents revise their

ﬁrst-order beliefs about θ upward when receiving the additional information that r

(θ, s) = 1.

That each agent is more optimistic about the strength of the fundamentals, however, does

not guarantee that, under MARP consistent with the new policy Γ

∗

, more agents invest than

under the original policy Γ. In fact, the new piece of information changes not only the agents’

ﬁrst-order beliefs about θ but also their higher-order beliefs and the latter matter for the

determination of the most-aggressive rationalizable proﬁle. More generally, r

(θ, s) need not

be monotone in θ. This is because MARP under the original policy Γ need not entail strategies

that are monotone in x. As a result, in general, the announcement that r

(θ, s) = 1 need not

trigger an upward revision of the agents’ beliefs.

The result in Theorem 1 follows instead from the game being supermodular along with the

fact that Bayesian updating preserves the likelihood ratio of any two states that are consistent

with no default under the original policy Γ. Formally, for any s ∈ supp(π(Θ)), any pair of

states θ

′

and θ

′′

such that (a) s ∈ supp π(θ

′

) ∩ supp π(θ

′′

), and (b) r

(θ

′

, s) = r

(θ

′′

, s) = 1,

the likelihood ratio of such two states under Γ

∗

is the same as under the original policy Γ.

This property implies that the posterior beliefs (over Θ) of each agent with private signal x

who, under the new policy Γ

∗

, receives information (s, 1), are a “truncation” of the posterior

beliefs the same agent would have had under the original policy Γ after receiving information

s. The truncation eliminates from the support of the agent’s original beliefs states θ at which,

under MARP consistent with the original policy Γ there would have been default and hence

the agent’s payoﬀ diﬀerential from investing would have been negative. Because the game is

supermodular, under any policy Γ, MARP is less aggressive than the most aggressive strategy

proﬁle surviving n − 1 rounds of IDISDS (in the sense that any agent who invests under the

latter proﬁle does so also under MARP, but the opposite is not necessarily true). This means

that the extra information r

(θ, s) = 1 also removes from the support of each agent’s beliefs

states θ at which the payoﬀ diﬀerential from investing is negative under the most aggressive

proﬁle surviving n − 1 rounds of IDISDS under Γ. Hence, at any stage n of the IDISDS

procedure, the truncation makes each agent more willing to invest. That is, any agent who

would have invested after hearing s under the original policy Γ, also invests after hearing (s, 1)

under the new policy Γ

∗

. Because this is true for any n, it is also true in the limit as n goes to

inﬁnity. In other words, after the new policy Γ

∗

announces (s, 1), each agent learns that his

payoﬀ diﬀerential from investing when all other agents play according to MARP consistent

with the new policy Γ

∗

is strictly positive. Hence, after the new policy announces (s, 1), each

agent’s unique rationalizable action is to invest, irrespective of her private information x.

When, instead, the new policy Γ

∗

announces (s, 0), each agent learns that the state θ is

among those at which there would have been default under MARP consistent with the original

policy Γ (that is, r

(θ, s) = 0). The announcement thus makes it common certainty among

the agents that θ ≤ 1. It is then immediate that, under MARP consistent with the new policy

∗

, all agents refrain from investing.

The policy Γ

∗

thus completely removes any strategic uncertainty. Indeed, when



s, r

(θ, s)



(s, 1) (alternatively,



s, r

(θ, s)



= (s, 0)) is announced, each agent knows that, under MARP

consistent with the new policy Γ

∗

, all other agents invest (alternatively, refrain from invest-

ing), irrespective of their exogenous private information. Importantly, while the policy Γ

∗

removes any strategic uncertainty, it preserves structural uncertainty, that is, heterogeneity

in the agents’ ﬁrst and higher-order beliefs about θ. As explained in the Introduction, it is

essential that agents who invest are uncertain as to whether other agents invest because they

ﬁnd it dominant to do so, or because when they count on other agents investing, they ﬁnd it

iteratively dominant to do so, which requires heterogeneity in posterior beliefs.

That the policy maker is indiﬀerent between Γ and Γ

∗

is a direct implication of the fact

that, for any θ, her payoﬀ depends on A only through the probability of default, which is

the same across the two policies. That, for any θ, no agent is worse oﬀ (and some agents

are strictly better oﬀ) follows from the fact that, under Γ

∗

, all agents refrain from investing

(alternatively, invest) in case of default (alternatively, no default), whereas this is not the case

under Γ.

When it comes to disclosures in ﬁnancial markets, Theorem 1 implies that optimal policies

should combine the announcement of a pass/fail result (captured by r ∈ {0, 1}) with the

disclosure of additional information (captured by s) whose role is to guarantee that, when a

pass grade is given, the extra information s the agents receive from the policy maker makes

investing the unique rationalizable action. This structure appears broadly consistent with

common practice. The theorem, however, says more. It indicates that optimal disclosure poli-

cies should be transparent about market responses but not in the sense of creating conformism

in beliefs about fundamentals. Rather, they should leave no room to ambiguity as to whether

or not default will be averted when a pass grade is announced. Preserving heterogeneity in

beliefs about fundamentals is key to minimizing the probability of default.

3.2 Pass/Fail

Our next result provides a foundation for policies that take a simple pass/fail form; it identiﬁes

a key property of the agents’ beliefs under which such policies are optimal.

Theorem 2. Suppose that p(x|θ) is log-supermodular. Then, given any regular policy Γ sat-

isfying the perfect-coordination property, there exists a regular binary policy Γ

∗

= ({0, 1}, π

∗

)

that also satisﬁes the perfect-coordination property and such that, when agents play according

to MARP under both Γ and Γ

∗

, for any θ, the probability of default and the payoﬀs (for each

agent and the policy maker) are the same under Γ

∗

and Γ.

As anticipated in the Introduction, the log-supermodularity of p(x|θ) (equivalently, the

assumption that the distribution p(x|θ) from which the agents’ private signals are drawn

satisﬁes the monotone likelihood ratio property – in short, MLRP)) implies that the policy

maker cannot reverse the ranking in the agents’ optimism through public announcements.

Whenever agent j is more optimistic than agent i (in the monotone-likelihood-ratio order)

based on her exogenous private information x

, she continues to be more optimistic after

hearing the policy maker’s announcement, irrespectively of the shape of the policy Γ. In turn,

The property that p(x|θ) is log-supermodular means that, for any x

′

, x

′′

∈ R, with x

′

< x

′′

, and any

′

, θ

′′

∈ Θ, with θ

′′

> θ

′

, then p(x

′′

|θ

′′

)p(x

′

|θ

′

) ≥ p(x

′′

|θ

′

)p(x

′

|θ

′′

this implies that MARP is always in monotone strategies, and hence that the policy maker

does not beneﬁt from disclosing any information beyond the fate of the regime r

(θ, s).

To see this more formally, take any policy Γ = (S, π) satisfying the perfect coordination

property. Given the result in Theorem 1, without loss of optimality, assume that Γ = (S, π)

is such that S = {0, 1} × S, for some Polish space S, and that, under MARP consistent with

Γ, when the policy maker discloses any signal (s, r

(θ, s)) = (s, 1), investing is the unique

rationalizable action for each agent, irrespective of their exogenous private information. Given

the policy Γ, let U

(x, (s, 1)|k) denote the expected payoﬀ diﬀerential of an agent with ex-

ogenous private information x who receives public information (s, r

(θ, s)) = (s, 1) and who

expects all other agents to invest if and only if their exogenous signal exceeds a cut-oﬀ k. No

matter the shape of the policy Γ, when p(x|θ) is log-supermodular, then MARP associated

with the policy Γ is in monotone (i.e., cut-oﬀ) strategies. Hence, each agent’s expected payoﬀ

diﬀerential when all other agents play according to MARP can be written as U

(x, (s, 1)|k)

for some k that depends on s. That the original policy Γ satisﬁes the perfect-coordination

policy in turn implies that, for any s and k such that (k, (s, 1)) are mutually consistent,

(k, (s, 1)|k) > 0. That is, the expected payoﬀ diﬀerential of any agent whose private signal

x coincides with the cutoﬀ k must be strictly positive. If this were not the case, the contin-

uation game would also admit a rationalizable proﬁle (in fact, a continuation equilibrium) in

which some of the agents refrain from investing, thereby contradicting the fact that investing

irrespectively of x is the unique rationalizable proﬁle following the announcement of (s, 1).

Now consider a policy Γ

∗

that, for any θ, draws the signal (s, 1) (alternatively, (s, 0)) from

the distribution π(θ) of the original policy Γ = (S, π) but conceals the information s and

only discloses r = 1 (alternatively, r = 0). By the law of iterated expectations, for all k with

(k, (s, 1)) mutually consistent, because U

(k, (s, 1)|k) > 0 then U

∗

(k, 1|k) > 0. This implies

that the new policy Γ

∗

also satisﬁes the perfect-coordination property. The policy maker can

thus drop the additional signals s from the original policy Γ and still guarantee that after r = 1

is announced, investing is the unique rationalizable action for all agents. That the probability

of default and the payoﬀs (for each agent and the policy maker) are the same under Γ and

∗

then follows directly from the fact that, for any θ, the probability that each agent invests

This means that the set θ ∈ Θ such that (a) k ∈ ϱ

and (b) (s, 1) ∈ supp(π(θ)) has strictly positive

measure under F .

is the same under the two policies, along with the fact that signals are payoﬀ-irrelevant when

ﬁxing the agents’ behavior.

The inability to change the ranking in the agents’ beliefs through public announcements

is key to the optimality of simple pass/fail policies, as the next example shows.

Example 1. Suppose that θ is drawn from a uniform distribution over [−1, 2]. Given θ, each

agent i ∈ [0, 1] receives an exogenous signal x

∈ {x

, x

}, drawn independently across agents

from a Bernoulli distribution with probability

p(x

|θ) =







2/3 if θ ∈ (0, 1/3) ∪ [2/3, 5/6) ∪ [1, 7/6) ∪ [4/3, 5/3)

1/3 if θ ∈ [1/3, 2/3) ∪ [5/6, 1) ∪ [7/6, 4/3) ∪ [5/3, 2).

The value of p(x

|θ) for θ ∈ [−1, 0] plays no role in this example, so it can be taken arbitrarily.

Suppose that agents’ payoﬀs are such that g(θ) = 1 − c and b(θ) = −c, for all θ, with

c ∈ (1/2, 8/15). There exits a deterministic policy that satisﬁes PCP and guarantees that

default does not occur for θ > 0, whereas no pass/fail policy can guarantee that default does

not occur for all θ > 0.

Proof of Example 1. Figure 1 illustrates the signal structure considered in Example 1.

The dash line depicts the probability of signal x

whereas the solid line the complementary

probability of signal x

, as a function of θ.

Note that the agents’ posterior beliefs under the signal structure of Example 1 can be

ranked according to FOSD, but not according to MLRP. Each agent observing x

has pos-

terior beliefs about θ that dominate those of each agent observing x

in the FOSD order.

Nonetheless, the ratio p(x

|θ)/p(x

|θ) is not increasing in θ over the entire domain, mean-

ing that p(x|θ) is not log-supermodular and hence posteriors cannot be ranked according to

MLRP. Also note that, under the payoﬀ speciﬁcation in the example, investing is optimal

for an agent assigning probability to default no greater than 1 − c, whereas not investing is

optimal if such a probability is at least 1 − c.

The example features signals drawn from a distribution with ﬁnite support. This property, however, is

not essential. Conclusions similar to those in the example obtain when the agents’ signals are drawn from a

continuous distribution. We thank Tommaso Denti for suggesting a similar example with ﬁnite signals and

Leifu Zhang for suggesting an example with continuous signals.

Figure 1: Sub-optimality of simple pass/tail tests

To see that there exists no pass/fail policy guaranteeing that default does not occur for

all θ > 0, note that, by virtue of Theorem 1, if such a policy existed, there would also exist

a binary policy satisfying PCP and such that π(1|θ) = 0 for all θ ≤ 0 and π(1|θ) = 1 for

all θ > 0, with π(1|θ) denoting the probability that the policy discloses signal 1 when the

fundamentals are θ. Under such a policy, after hearing that s = 1, no matter the private

signal x, each agent assigns probability 1/2 to θ ∈ [0, 1] and probability 1/2 to θ ∈ [1, 2].

Because c > 1/2, each agent expecting all other agents to refrain from investing (and hence

default to occur for all θ ∈ [0, 1]) then ﬁnds it optimal to do the same. Hence, under MARP

consistent with the above policy, after the signal s = 1 is announced, all agents refrain from

investing, meaning that the above policy fails to spare types θ ∈ [0, 1] from default, when the

agents play adversarially.

To see that, instead, the policy maker can avoid default for all θ > 0 using a richer

policy, consider the policy Γ = (S, π), with S = {0, (1, mid) , (1, ext)} that, in addition to

publicly announcing a pass grade, also announces whether the fundamentals are extreme (i.e.,

θ ∈ (0, 5/6) ∪ (7/6, 2]), or intermediate (i.e., θ ∈ [5/6, 7/6]). Formally, for any θ ∈ [−1, 0],

π (0|θ) = 1, meaning that the policy maker assigns a failing grade. For any θ ∈ [5/6, 7/6],

instead, π(1, mid|θ) = 1, meaning that the the policy maker announces a pass grade and

that fundamentals are intermediate. Finally, for any θ ∈ (0, 5/6) ∪ (7/6, 2], π(1, ext|θ) = 1,

meaning that the policy maker announces a pass grade and that fundamentals are extreme.

See Figure 1 for a graphical representation.

Under such a policy, investing is the unique rationalizable action for any agent observing

a pass grade, no matter whether the agent also learns that the fundamentals are intermediate

or extreme.

To see this, consider ﬁrst the case in which the fundamentals are extreme, i.e., θ ∈ (0, 5/6)∪

(7/6, 2]. All agents with exogenous information x

ﬁnd it dominant to invest when hearing

s = (1, ext). In fact, even if all other agents refrained from investing, the probability that

each agent with signal x

assigns to θ > 1 (and hence to the event that there is no default

) is Pr



θ > 1|x

, ext



= 8/15 > c, making it dominant to invest. As a consequence of this

property, each agent with exogenous private information x

ﬁnds it iteratively dominant to

invest. This is because, for any θ ∈ [1/3, 5/6], even if all agents with exogenous information

equal to x

refrained from investing, the aggregate investment from those individuals with

information x

would suﬃce for default not to occur. This means that the probability that

each agent with information x

assigns to the event that default does not occur is at least

equal to Pr



θ > 1/3|x

, (1, ext)



= 11/15, implying that it is optimal for the agent to invest.

Next, consider the case in which fundamentals are intermediate, i.e., θ ∈ [5/6, 7/6]. In

this case, the ranking of the agents’ optimism is reversed, with those agents observing the x

signal assigning higher probability to higher states. In particular, because each agent with

information x

assigns probability 2/3 > c to θ ≥ 1, any such agent ﬁnds it dominant to

invest. Because, for any θ ∈ (5/6, 1), 1/3 of the agents receives information x

, the minimal

size of investment that each agent with signal equal to x

can expect at any θ ∈ (5/6, 1)

is equal to p(x

|θ) = 1/3 > 1 − θ, implying that even if all the less optimistic agents with

signal x

refrained from investing, default would not occur. But this means that investing is

iteratively dominant for those agents receiving the x

signal.

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 policy maker needs to provide

the extra information. This is a consequence of the policy maker 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 x

-signals when the fundamentals are

intermediate, whereas they are those observing the x

-signals when the fundamentals are

extreme. The reversal in the agents’ optimism in turn permits the policy maker 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 policy

maker 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 2 is in showing that, notwithstanding such a qualiﬁcation, the optimal policy does

take the form of action recommendations in the special case in which beliefs co-move 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 policy maker’s

preferences for avoiding default when fundamentals are large are strong enough to compensate

for the possibility that non-monotone 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 non-monotone rule).

In this subsection, we identify a condition relating the policy maker’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 non-

monotone rules do strictly better than monotone ones. These economies include many of the

examples considered in the literature, e.g., Goldstein and Huang (2016).

We assume hereafter that



x ∈ R :

u (θ, 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 non-negative 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 policy maker can save all θ > 0 through a policy that announces whether or not

θ > 0. Then, let

max

≡ sup



x ∈ R :

u (θ, 1 − P (x|θ)) 1(θ > 0)p (x|θ) dF (θ) ≤ 0



. (2)

As we show in the Appendix, x

max

is an upper bound for the set of cut-oﬀs characterizing

the strategies consistent with MARP across all disclosure policies Γ satisfying the perfect

coordination property.

For any x, let Θ(x) ≡ {θ ∈ Θ : x ∈ ϱ

} denote the set of fundamentals that, given the

distribution P (·|θ) from which the agents’ signals are drawn, are consistent with private in-

formation x.

Condition M. The following properties hold:

1. inf Θ(x

max

) ≤ 0;

2. for any θ

, θ

∈ [0, 1], with θ

< θ

, and x ≤ x

max

such that (a) θ

≤ P (x|θ

) and (b)

x ∈ ϱ

(θ

, 1) − U

(θ

, 0)

(θ

, 1) − U

(θ

, 0)

p (x|θ

) b (θ

)

p (x|θ

) b (θ

)

. (3)

Property (1) in Condition M says that the lower bound of the support of the beliefs of an agent

with signal x

max

, where x

max

is the threshold deﬁned in (2), is non-positive 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 (2) of Condition M says that the value the policy maker assigns to avoiding

default increases with the underlying fundamentals at a large enough rate. Speciﬁcally, the

property requires that the beneﬁt that the policy maker derives from changing the 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 M holds. Given any

regular policy Γ satisfying the perfect-coordination property, there exists a regular deterministic

binary monotone policy Γ

= ({0, 1}, π

) that also satisﬁes the perfect-coordination property

and such that, when the agents play according to MARP under both Γ and Γ

, the policy

maker’s ex-ante expected payoﬀ is weakly higher under Γ

than under Γ.

When Condition M holds, the choice of the optimal policy reduces to the choice of the

smallest threshold

θ such that, when agents commonly learn that θ >

θ, under the unique

rationalizable proﬁle, all agents invest irrespective of their exogenous private information. For

this to be the case, it must be that, for any x ∈ R,

∞

u(θ, 1 − P (x|θ))p(x|θ)dF (θ) > 0.

The above problem, however, does not have a formal solution, due to the lack of upper-

semicontinuity of the policy maker’s payoﬀ in

θ. Notwithstanding these complications, here-

after we follow the pertinent literature and refer to the “optimal monotone policy” as the one

deﬁned as follows. For any θ ∈ (0, 1), let x

∗

(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.

Let

∗

≡ inf



θ ≥ 0 :

∞



θ, 1 − P



∗

(θ)|





∗

(θ)|



dF (

θ) ≥ 0 for all θ ∈

θ, 1





(4)

be the lowest truncation point

θ such that, when the policy reveals that fundamentals are above

The policy Γ

is such that there exists a threshold

θ ∈ [0, 1] such that, for any θ ≤

θ, π

(θ) assigns

probability one to s = 0, whereas for any θ >

θ, π

(θ) assigns probability one to s = 1.

For any θ ∈ (0, 1), the threshold x

∗

(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

∗

(1) = +∞.

θ, then for any possible default threshold θ ∈

θ, 1



, if default were to occur for fundamentals

below θ and 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, which is always the

case when

≡ sup



θ ∈ (0, 1) :



θ, 1 − P



∗

(θ)|





∗

(θ)|



dF (

θ) ≤ 0



< 1.

The optimal monotone policy is the one with cut-oﬀ

θ = θ

∗

The previous literature (e.g., Goldstein and Huang (2016)) characterized the threshold θ

∗

by restricting attention to monotone rules. The contribution of Theorem 3 is 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 policy maker can strictly increase her payoﬀ through a non-monotone rule.

As we show in the Appendix, Property (1) in Condition M guarantees 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

′′

], with 0 ≤ θ

′

< θ

′′

< θ

∗

while guaranteeing that investing remains the unique rationalizable action when the policy

maker announces a pass grade (i.e., when the signal s = 1 is disclosed). 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 2 below for an illustration).

Property (2) of Condition M in turn guarantees that the higher payoﬀ the policy maker

obtains, under the new policy, from avoiding default when fundamentals are stronger com-

For any

θ ∈ (θ

, 1), and any θ ∈

θ, 1



0 <

∞

−∞



θ, 1 − P



∗

(θ)|





∗

(θ)|



dF (

θ) <

∞



θ, 1 − P



∗

(θ)|





∗

(θ)|



dF (

θ).

Hence, when θ

< 1, θ

∗

is well-deﬁned.

The reason why this is an abuse is that, under the monotone policy with cut-oﬀ θ

∗

, in the continuation

game that starts after the policy maker 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

θ arbitrarily close

to the threshold θ

∗

such that, after the policy maker announces s = 1 (equivalently, that θ ≥

θ), the unique

rationalizable proﬁle features all agents investing. Because the policy maker’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.

pensates for the possibility that, from an ex-ante perspective, the probability of default may

be larger under monotone policies than under non-monotone ones (see Example 3 for an illus-

tration of why non-monotone rules may permit the policy maker to avoid default over a set

of fundamentals of larger ex-ante probability).

As anticipated above, Condition M is fairly sharp in the sense that, when violated, one

can identify economies in which the optimal policy is non-monotone. We provide two such

examples below. Example 2 illustrates the role of Property (1) in Condition M, whereas

Example 3 illustrates the role of Property (2) in Condition M. These examples also illustrate

why non-monotone rules, in general, may reduce the set of fundamentals over which default

happens.

Let θ

∈ (0, 1) be implicitly deﬁned by the unique solution to

u(θ

, A)dA = 0. (5)

The threshold θ

corresponds to the value of the fundamentals at which an agent who knows

θ and holds Laplacian beliefs with respect to the aggregate investment is indiﬀerent between

investing and not investing.

Importantly, θ

is independent of the initial common prior F

and of the distribution of the agents’ signals.

Example 2. Suppose that there exist scalars g, b ∈ R, with g > 0 > b, such that, for any θ,

g(θ) = g, and b(θ) = b. Assume that θ is drawn from a uniform distribution with support

[−K, 1 + K], for some K ∈ R

. Finally, assume that the agents’ exogenous signals are

given by x

= θ + σϵ

, with σ ∈ R

and with each ϵ

drawn independently across agents

from a uniform distribution over [−1, 1], with σ < K/2. Let θ

∗

be the threshold deﬁned

in (4), applied to the primitives described in this example.

There exists σ

∈ (0, K/2)

such that (a) inf Θ(x

∗

(θ

)) > 0, and (b) for all σ ∈ (0, σ

), starting from the optimal

monotone policy with cut-oﬀ θ

∗

, there exists a deterministic non-monotone policy satisfying

the perfect-coordination property and permitting the policy maker to avoid default over a set

of fundamentals of strictly larger probability measure than the optimal monotone policy.

This means that the agent believes that aggregate investment is uniformly distributed over [0, 1]. See

Morris and Shin (2006).

Hereafter, the subscript σ in θ

∗

and x

∗

is 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.

∗

(θ)

γ,δ

(θ)

′

2σ+1

∗

1 − c

′

′′

− δ

Figure 2: Sub-optimality of deterministic binary monotone policies.

The proof is in the Online Supplement, Inostroza and Pavan (2024b). Here we sketch the

key arguments. To ﬁx ideas, let g = 1 − c and b = −c, with c ∈ (0, 1), as in Example 1,

and recall that, 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 θ ∈ [0, 1], let x

∗

(θ) be the critical signal threshold such that, when all agents

invest for x > x

∗

(θ) and refrain from investing for x < x

∗

(θ), default occurs if and only if the

fundamentals are below θ. For any binary policy Γ = ({0, 1}, π), and any threshold θ ∈ [0, 1]

such that (x

∗

(θ), 1) are mutually consistent under Γ, let

(θ) ≡ U

∗

(θ), 1|x

∗

(θ)) ,

denote the payoﬀ of the marginal agent with signal x

∗

(θ), after the policy Γ announces that

s = 1, where U

is the function deﬁned after Theorem 2.

Now, for any

θ ∈ Θ, let Γ

= ({0, 1}, π

) be the deterministic, binary, monotone rule with

cut-oﬀ

θ. Note that the absence of any public disclosure is equivalent to a monotone policy

with cut-oﬀ

θ = min Θ = −K and that, under such a policy, default occurs if and only if

θ ≤ θ

= c.

A necessary and suﬃcient condition for all agents to invest under MARP consistent with

the policy Γ

, after hearing that s = 1, is that, for any possible default threshold θ >

θ,

(θ) > 0. The lowest fundamental in the support of x

∗

(θ)’s beliefs is x

∗

(θ) − σ. Hence,

when x

∗

(θ) − σ >

θ, the marginal agent with signal x

∗

(θ) already knows from his private

information that fundamentals are above

θ and thus learns nothing from the announcement

that s = 1. Because, in the absence of any public disclosure, the payoﬀ of the marginal agent

is strictly negative for all θ < θ

, this implies that the cut-oﬀ θ

∗

for the optimal monotone

rule is θ

∗

= x

∗

(θ

) − σ.

Now to see that the optimal monotone policy is improvable, assume that σ is small so that

∗





− σ > 0. Next, pick γ, δ > 0 small and let θ

′′

≡ x

∗

(θ

− δ) − σ and θ

′

≡ θ

′′

− γ,

with θ

′

> 0. Consider a binary policy Γ

γ,δ

= ({0, 1}, π

γ,d

) that, in addition to announcing a

pass grade s = 1 when fundamentals are above θ

∗

(as the optimal monotone rule does) also

announces s = 1 when θ ∈ [θ

′

, θ

′′

]. Let V

γ,δ

(θ) be the payoﬀ of the marginal agent with

signal x

∗

(θ) under the new rule Γ

γ,δ

, after the policy maker announces that s = 1. This

payoﬀ is represented in Figure 2 along with the payoﬀ V

∗

(θ) under the optimal monotone

rule. Provided that γ and δ are small, V

γ,δ

(θ) ≥ 0 for all θ for which (x

∗

(θ), 1) are mutually

consistent under Γ

γ,δ

, with V

γ,δ

(θ) = 0 if and only if θ = θ

. Starting from Γ

γ,δ

, one can

then further perturb the policy Γ

γ,δ

by giving a fail grade to banks with fundamentals in

[θ

∗

, θ

∗

+ ε], with ε > 0 small. The new policy

Γ so constructed is such that V

(θ) > 0 for all

θ for which (x

∗

(θ), 1) are mutually consistent under

Γ, meaning that, when the policy maker

announces that s = 1, investing is the unique rationalizable action for all agents. The policy

Γ thus satisﬁes the perfect-coordination property and guarantees that default occurs over a

set of fundamentals of strictly smaller probability under F than the optimal monotone policy

∗

= ({0, 1}, π

∗

). □

The reason why the non-monotone policy

Γ constructed in the proof of Example 2 guaran-

tees that default occurs over a smaller set of fundamentals than the optimal monotone policy

(the one with threshold θ

∗

) is that agents receiving signals around θ

are highly sensitive to

the grade the policy gives to institutions with fundamentals around θ

but not so much so

to the grade given to fundamentals far from θ

. In the above example with bounded noise,

an agent receiving a signal x

∗

(θ

) is not sensitive at all to the grade the policy gives to fun-

damentals below x

∗

(θ

) − σ because his private signal informs him that the fundamentals

are above x

∗

(θ

) − σ. Hence, while it is impossible to amend the optimal monotone policy

(the one with cut-oﬀ θ

∗

= x

∗

(θ

) − σ) by giving a pass grade also to fundamentals slightly

below θ

∗

without 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 fundamen-

tals suﬃciently “far away” from θ

∗

, while continuing to induce all agents to invest under

MARP. The reason why such improvements are not feasible under Condition M in Theorem

3 is that Property (1) in Condition M implies that x

∗

(θ

) − σ < 0, thus making the above

construction unfeasible.

Interestingly, when θ ∈ [θ

′

, θ

′′

], the assumption of bounded support

of the agents’ beliefs 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

Γ constructed in Example 2 guarantees 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 (1)

in Condition M trivially holds), but Property (2) is violated.

Given any binary, deterministic policy Γ = ({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 = 1 or

to s = 0), let D



(θ

] : i = 1, ..., N



denote the partition of



0, θ



induced by π,

with N ∈ N, θ

= 0, and θ

= θ

Let d ∈ D

denote a generic cell of the partition

and, for any θ ∈ (0, θ

], denote by d

(θ) ∈ D

the cell that contains θ. Finally, let

M (Γ) ≡ max

i=1,...,N

− θ

| denote the mesh of D

, that is, the Lebesgue measure of the cell

of D

of maximal Lebesgue measure.

Example 3 below shows that, when the noise in the agents’ information is unbounded, but

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 non-monotone.

Under Property (1), the marginal agent with signal x

∗

(θ

) does not rule out any fundamental in (0, θ

Hence, any perturbation of the optimal monotone policy passing fundamentals to the left of θ

induces the

agent to refrain from investing.

That is, either (a) π(θ) = 0 for all θ ∈ ∪

i=2k,k≤N





and π(θ) = 1 for all θ ∈ ∪

i=2k−1,k≤N





, or (b)

π(θ) = 1 for all θ ∈ ∪

i=2k,k≤N





and π(θ) = 0 for all θ ∈ ∪

i=2k−1,k≤N





Example 3. Suppose that θ is drawn from an improper uniform prior over R and that the

agents’ signals are given by x

= θ +σε

, with ε

drawn from a standard Normal distribution.

Further assume that there exist scalars g, b, W, L ∈ R, with g > 0 > b and W > L, such that,

for any θ, g(θ) = g, b(θ) = b, W (θ) = W and L(θ) = L. There exists a scalar ¯σ > 0

and a function E : (0, ¯σ] → R

, with lim

σ→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 Γ = ({0, 1}, π) satisfying the perfect-coordination

property and such that M (Γ) > E(σ), there exists another deterministic binary policy Γ

∗

with M (Γ

∗

) < E(σ) that also satisﬁes the perfect-coordination property and such that the

ex-ante probability of default under Γ

∗

is strictly smaller than under Γ.

See the Online Supplement, Inostroza and Pavan (2024b), for a detailed proof of the

result. Here we discuss the main ideas. Non-monotone policies permit the policy maker

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 θ

and the policy

maker announces a pass grade. Intuitively, if the policy maker assigned a pass grade to

an interval (θ

′

, θ

′′

] ⊂ (0, θ

] of large Lebesgue measure, when σ is small and θ ∈ (θ

′

, θ

′′

most agents would receive private signals x

∈ (θ

′

, θ

′′

]. No matter the grade assigned to

fundamentals outside the interval (θ

′

, θ

′′

], in the continuation game that starts after the policy

maker announces a pass grade, most agents with signals x

∈ (θ

′

, θ

′′

] would then 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 σ is small, the only way the policy maker can guarantee that,

when θ ∈ (0, θ

], the agents invest after hearing a pass grade is by dividing the set (0, θ

]

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





⊂



0, θ



, i = 1, ..., N, receiving a pass grade are

far apart, implying that the policy maker fails an interval (θ

′

, θ

′′

] ⊂ (0, θ

] of large Lebesgue

The improperness of the prior simpliﬁes the exposition but is not important. The agents’ hierarchies of

beliefs are still well-deﬁned.

measure (note that this is indeed the case under the optimal monotone deterministic rule with

cutoﬀ θ

∗

, where θ

∗

is the threshold deﬁned in (4).

The detailed derivations in the Online

Supplement Inostroza and Pavan (2024b) then show that, starting from Γ, the policy maker

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 till 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 policy maker

a strictly higher payoﬀ than the original one.

Finally, one can show that, when σ is small, a pass grade can be given to all θ > θ

+ ε,

with ε > 0 small, while guaranteeing that all agents invest after the policy maker announces

the pass grade s = 1.

The above properties thus also imply that, if the policy maker is restricted to determin-

istic policies (arguably, the most relevant case in practice), when the precision of the agents’

exogenous information is large, the optimal policy is highly non-monotone over (0, θ

) and

announces a pass grade when fundamentals are above θ

. □

4 Extensions

We ﬁrst introduce a few enrichments in Subsection 4.1, then establish the analog of the

three theorems above for these richer economies in Subsection 4.2, and then conclude in

Subsection 4.3 discussing the role of the multiplicity of the receivers and their exogenous

private information.

The subscript simply highlights the dependence of the cutoﬀ θ

∗

on σ.

Formally, for any ε > 0, there exists σ(ε) such that, for any σ < σ(ε), given any pass/fail policy Γ

satisfying PCP, there exists another pass/fail policy Γ

′

also satisfying PCP that agrees with Γ on any θ < θ

and gives a pass grade to any θ ≥ θ

+ ε.

4.1 Generalizations

The fundamentals are given by (θ, z), with θ drawn from Θ according to the absolutely con-

tinuous cdf F , and with z drawn from [z, ¯z] according to Q

(z), with the cdf Q

(z) weakly

decreasing in θ, for any z.

The variable θ continues to parameterize the maximal information the policy maker can

collect about the fundamentals. The additional variable z parameterizes risk that the agents

and the policy maker 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 = (x

)

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 ϱ

∈ R.

There exists a function R : Θ × [0, 1] × [z, ¯z] → R such that, given any (θ, A, z), default

occurs (i.e., r = 0) if, and only if, R(θ, A, z) ≤ 0. The function R is continuous and strictly

increasing in (θ, z, A). For any (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

W ,

L : Θ×[0, 1]×[z, ¯z] → R such that, given any (θ, A, z), the policy

maker’s payoﬀ is equal to

(θ, A, z) =

W (θ, A, z)1 (R(θ, A, z) > 0) +

L(θ, A, z)1 (R(θ, A, z) ≤ 0) . (6)

Hence,

W (θ, A, z) is the policy maker’s payoﬀ in case default is avoided, whereas

L(θ, A, z) is

her payoﬀ in case of default. Likewise, there exist functions ˆg,

b : Θ × [0, 1] × [z, ¯z] → R such

that, given any (θ, A, z), the agents’ payoﬀ diﬀerential between investing and not investing is

equal to

ˆu(θ, A, z) = ˆg(θ, A, z)1 (R(θ, A, z) > 0) +

b(θ, A, z)1 (R(θ, A, z) ≤ 0) , (7)

All the results extend to the case where Q

(z) has unbounded support. Note that Q

(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).

with ˆg(θ, A, z) > 0 >

b(θ, A, z), for any (θ, A, z). For any (θ, A), then let

g(θ, A) ≡

E [1 (R(θ, A, z) > 0) ˆg(θ, A, z)|θ, A]

r(θ, A)

and b(θ, A) ≡

1 (R(θ, A, z) ≤ 0)

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) ≡

1 (R(θ, A, z) > 0)

W (θ, A, z)|θ, A

r(θ, A)

and L(θ, A) ≡

1 (R(θ, A, z) ≤ 0)

L(θ, A, z)|θ, A

1 − r(θ, A)

denote the policy maker’s expected payoﬀ, again in case of no default and default, respectively.

The agents’ and the policy maker’s expected payoﬀs can then be conveniently expressed

as a function of θ and A only, by letting

u(θ, A) ≡ r(θ, A)g(θ, A) + (1 − r(θ, A))b(θ, A) and U

(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, with θ < θ, such

that (a) u(θ, 1) < 0 for all θ ≤ θ, (b) u(θ, 0) > 0 for all θ > θ and (c) u(θ, 1) > 0 > u(θ, 0)

for all θ ∈ (θ, θ]. The thresholds θ and θ deﬁne the “critical region” (θ, θ] where the sign of

the agents’ payoﬀ diﬀerential depends on the response of the market.

We also assume that

both u(θ, A) and U

(θ, A) are non-decreasing in A and such that U

(θ, 1) > U

(θ, 0) for all

θ ∈ (θ, θ].

4.2 Results

We now identify conditions under which Theorems 1-3 extend to these richer economies.

The critical region can also be deﬁned in terms of the regime outcome. That is, let θ

′

∈ R, with

′

, be deﬁned by R(θ

′

, 1, z) = R(

′

, 0, z) = 0. Note that default occurs with certainty when θ < θ

′

and never occurs when θ >

′

, no matter (A, z). One could then let the critical region be deﬁned by (θ

′

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 of the critical region. The reason for deﬁning the critical region in terms of the sign of the agents’

payoﬀ diﬀerential 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.

That u(θ, A) is monotone in A implies that the continuation game remains supermodular. That U

(θ, A)

is non-decreasing in A implies that, for any Γ, MARP continues to coincide with the “smallest” rationalizable

proﬁle, that is, the one involving the smallest measure of agents investing. Finally, that, for any θ in the

critical region, the policy maker 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.

4.2.1 Perfect-coordination property

Given any distribution G ∈ ∆Θ over Θ, say that G is “regular” if, when the common posterior

over Θ is G and, for any θ, agents receive private signals according to p(·|θ), MARP is well

deﬁned. Then, for any regular G, any θ, let A(θ; G) denote the aggregate investment at θ

when agents play according to MARP, under the common posterior G.

Condition PC. For any distribution τ ∈ ∆∆(Θ) over posterior beliefs consistent with the

common prior F (i.e., such that

Gτ(dG) = F ), the following condition holds:





1 (u(θ, A(θ; G)) > 0) U

(θ, 1) + 1 (u(θ, A(θ; G)) ≤ 0) U

(θ, 0)



G (dθ)



τ(dG)

≥



(θ, A(θ; G))G (dθ)



τ(dG).

To appreciate the meaning of Condition PC, suppose that the policy maker, through her

disclosure policy Γ, generates a distribution τ over common posteriors G over Θ, and that,

for any G, agents play according to MARP. Now suppose that, in each state θ, the policy

maker also informs the agents of the sign of their expected payoﬀ diﬀerential u(θ, A(θ; G))

under MARP consistent with G. Finally, suppose that, after each posterior G is generated,

the additional information induces all agents to invest when they learn that u(θ, A(θ; G)) > 0

and not to invest when they learn that u(θ, A(θ; G)) ≤ 0. Then, the additional information

makes the agents better oﬀ. Condition PC says that the policy maker is also weakly better

oﬀ. In other words, the condition requires that the policy maker’s and the agents’ payoﬀs be

not too misaligned. Condition PC trivially holds when the policy maker faces no aggregate

uncertainty (i.e., when each distribution Q

over [z, ¯z] is degenerate), W is weakly increasing

in A and L is invariant in A, as in the baseline model. For example, in case of stress testing,

the condition says that the policy maker prefers more agents to invest in case the bank under

examination avoids default, but is indiﬀerent as to how many investors pull their money out

of the bank when the latter defaults. More generally, Condition PC accommodates for the

possibility that both W and L depend on A, possibly non-monotonically, provided that, on

average, the loss to the policy maker from having no agent invest in states θ in which the

agents’ expected payoﬀ diﬀerential (under MARP given the induced common posterior G) is

negative is more than compensated by the beneﬁt from having all agents invest in states θ in

which the diﬀerential is positive. The average is over both the induced posteriors G and the

fundamentals θ.

As in the baseline model, let A

(θ, s) denote the aggregate size of investment at θ under

MARP consistent with Γ, when the policy discloses s.

Theorem 1*. Given any regular policy Γ = (S, π), there exists another regular policy Γ

∗

satisfying the perfect-coordination property and such that, when, under both Γ and Γ

∗

agents

play according to MARP, the following are true: (1) for any θ, no agent is worse oﬀ under Γ

∗

than under Γ, and some agents are strictly better oﬀ; (2) if, for any θ and s ∈ supp(π(θ)), the

regime outcome is deterministic (i.e., r(θ, A

(θ, s)) ∈ {0, 1}), then, for any θ, the probability

of default under Γ

∗

is the same as under Γ; (3) when Condition PC holds, the policy maker

is better oﬀ under Γ

∗

than under Γ.

Theorem 1* extends Theorem 1 to the richer class of economies under consideration, in

which the regime outcome is determined by additional variables that are not observable by

the policy maker, and where both the policy maker’s and the agents’ payoﬀs depend on the

aggregate investment A beyond its eﬀect on the regime outcome. The policy Γ

∗

in the theorem

is obtained from the original policy Γ by disclosing, for each θ, in addition to the information

s ∈ supp(π(θ)) disclosed by the original policy Γ, a second piece of information that reveals

to the market whether, at (θ, s), under MARP consistent with the original policy Γ, the

agents’ expected payoﬀ diﬀerential is positive or negative. Note in particular that because

the sign of the payoﬀ diﬀerential in the baseline model is given by the regime outcome, this

additional piece of information, in the baseline model, coincides with the regime outcome

(θ, s) ∈ {0, 1}.

In Inostroza and Pavan (2024a), we show that the perfect-coordination property is fairly

general and extends to a class of economies even richer than the one introduced in Subsection

4.1 in which (a) the agents’ prior beliefs need not be consistent with a common prior, nor be

generated by signals drawn independently across agents, conditionally on θ, (b) the number

of agents is arbitrary (in particular, ﬁnitely many agents), (c) payoﬀs can be heterogenous

across agents, (d) agents have a level-K degree of sophistication, (e) the policy maker may

possess imperfect information about the payoﬀ state and/or the agents’ beliefs, (f) the policy

maker may engage in ﬂexible discriminatory disclosures and disclose diﬀerent information to

diﬀerent agents. The key property is the possibility for the policy maker to have access to

information that is a suﬃcient statistic of the agents’ information when predicting the sign

of the agents’ payoﬀ diﬀerential under MARP. This property holds when, for example, the

correlation in the agents’ exogenous beliefs originates in public signals the policy maker has

access to.

4.2.2 Pass/Fail Policies

Condition FB. For any x, u(θ, 1 − P (x|θ)) ≥ 0 (alternatively, u(θ, 1 − P (x|θ)) ≤ 0) implies

that u(θ

′′

, 1 − P (x|θ

′′

)) > 0 for all θ

′′

> θ (alternatively, u(θ

′

, 1 − P (x|θ

′

)) < 0 for all θ

′

< θ).

Condition FB (which stands for “single crossing from below”) states that, for any x, the

payoﬀ diﬀerential u(θ, 1−P (x|θ)) from investing when all agents follow a cut-oﬀ strategy with

cut-oﬀ x crosses 0 once from below. The property clearly holds in the baseline model where

(i) r (θ, A) = 1 (A > 1 − θ) and (ii) g(θ) > 0 > b(θ) for all θ. It also holds when u(θ, A), in

addition to being non-decreasing in A as assumed above, is non-decreasing in θ.

Theorem 2*. Suppose that p(x|θ) is log-supermodular and Condition FB holds. Then,

given any regular policy Γ = (S, π) satisfying the perfect-coordination property, there exists a

regular binary policy Γ

∗

= ({0, 1}, π

∗

) that also satisﬁes the perfect-coordination property and

such that, when agents play according to MARP under both Γ and Γ

∗

, for any θ, the probability

of default and the payoﬀs (for each agent and the policy maker) are the same under Γ

∗

and Γ.

Because Γ = (S, π) satisﬁes the perfect-coordination property, ∪

supp(π(θ)) can be parti-

tioned in two sets, S

and S

, such that, under Γ, all agents invest (alternatively, do not invest)

when receiving information s ∈ S

(alternatively, s ∈ S

), irrespectively of their private signals

x. The key step in the proof in the Appendix shows that the log-supermodularity of p (x|θ),

together with Condition FB, jointly imply that, under any policy, MARP is in cut-oﬀ strate-

gies. The reason is the same as the one discussed above for the baseline model. In turn, this

property implies that all agents continue to invest (alternatively, refrain from investing) when

the policy maker “pools the signals” and discloses only that s ∈ S

(alternatively, that s ∈ S

The arguments are similar to those leading to Theorem 2 above. The policy Γ

∗

= ({0, 1}, π

∗

)

We conjecture that, as long as the above suﬃcient statistic property holds, Theorems 2* and 3* below also

extend to settings in which the agents’ signals are not conditionally independent given θ. Whether the results

extend to some environments in which the suﬃcient statistic property is violated is an interesting question for

future work.

is then constructed by letting π

∗

(1|θ) = π(S

|θ) (and π

∗

(0|θ) = π(S

|θ)) for all θ. Contrary to

the baseline model, after the policy Γ

∗

discloses signal 1 (alternatively, signal 0), the regime

outcome need not be deterministic. Nonetheless, the probability of default is the same under

the two policies Γ and Γ

∗

and so are the payoﬀs.

4.2.3 Monotone Rules

First, we extend the deﬁnition of x

max

to accommodate for the fact that, in richer economies,

θ need not coincide with 0. That is, we let

max

≡ sup



x ∈ R :

u (θ, 1 − P (x|θ)) 1(θ > θ)p (x|θ) dF (θ) ≤ 0



. (8)

Next, we extend Condition M as follows.

Condition M*.

(1*) inf Θ(x

max

) ≤ θ;

(2*) For any θ

, θ

∈ [θ,

θ], with θ

< θ

, and x ≤ x

max

such that (a) u(θ

, 1 −P (x|θ

)) ≤ 0

and (b) x ∈ ϱ

(θ

, 1) − U

(θ

, 0)

(θ

, 1) − U

(θ

, 0)

p (x|θ

) u (θ

, 1 − P (x|θ

))

p (x|θ

) u (θ

, 1 − P (x|θ

))

. (9)

(3*) |u(θ, 1−P (x|θ))| is log-supermodular over



(θ, x) ∈ [θ,

θ] × R : u(θ, 1 − P (x|θ)) ≤ 0



Property (1*) is similar to Property (1) in Condition M in the baseline model but accom-

modates for the fact that, in richer economies, θ need not coincide with 0. Property (2*)

extends Property (2) in Condition M to the current environment with richer preferences in

which u(θ, A) and U

(θ, A) depend on A over and above the eﬀect that the latter variable

has on the regime outcome.

As in the baseline model, that payoﬀs (for each agent and the policy maker) are the same under Γ and

∗

follows from the fact that, for any θ, the probability that each agent invests is the same under the two

policies, along with the fact that signals are payoﬀ-irrelevant when ﬁxing the agents’ behavior.

The log-supermodularity of |u(θ, 1 − P (x|θ))| means that, for any x

′

, x

′′

∈ R, with x

′

< x

′′

, and any

′

, θ

′′

∈ Θ, with θ

′′

> θ

′

, such that u(θ

′′

, 1 − P (x

′

|θ

′′

)) ≤ 0,

u(θ

′′

, 1 − P (x

′′

|θ

′′

))u(θ

′

, 1 − P (x

′

|θ

′

)) ≥ u(θ

′′

, 1 − P (x

′

|θ

′′

))u(θ

′

, 1 − P (x

′′

|θ

′

)).

Property (3*) is a new condition that requires that, for any θ

′

< θ

′′

and x

′

< x

′′

such that

u(θ

′′

, 1 − P (x

′

|θ

′′

)) < 0,

u(θ

′′

, 1 − P (x

′

|θ

′′

))

u(θ

′

, 1 − P (x

′

|θ

′

))

≤

u(θ

′′

, 1 − P (x

′′

|θ

′′

))

u(θ

′

, 1 − P (x

′′

|θ

′

))

. (10)

Note that u(θ

′′

, 1 − P (x

′

|θ

′′

)) < 0 implies that u(θ

′

, 1 − P (x

′

|θ

′

)), u(θ

′

, 1 − P (x

′′

|θ

′

)), u(θ

′′

, 1 −

P (x

′′

|θ

′′

)) < 0. The condition thus requires that the relative reduction in the expected losses

stemming from the fundamentals improving from θ

′

to θ

′′

> θ

′

is larger when agents invest

if and only if x > x

′

than when they invest if and only if x > x

′′

> x

′

. The reduction in

the losses u(θ, 1 − P (x|θ)) combines the direct eﬀect of θ on u(θ, A) with the indirect eﬀect

of θ on A = 1 − P (x|θ) that obtains when the agents follow a cut-oﬀ strategy whereby they

invest if and only if their signals exceed x. The condition trivially holds in the baseline model

where u(θ, A) < 0 if and only if, given (θ, A), there is default (i.e., A ≤ 1 − θ), in which case

u(θ, A) = b(θ).

Theorem 3*. Suppose that p(x|θ) is log-supermodular and Conditions PC, FB, and M*

hold. Given any regular policy Γ, there exists a regular deterministic binary monotone policy

= ({0, 1}, π

) that satisﬁes the perfect-coordination property and such that, when the agents

play according to MARP under both Γ and Γ

, the policy maker’s ex-ante expected payoﬀ is

weakly higher under Γ

than under Γ.

To gain some intuition on the role played by the additional requirement in Condition M*

(property (3*)), ﬁrst observe that the conditions in the theorem imply that Theorems 1* and

2* hold. Given any policy Γ, there thus exists a binary policy Γ

′

= ({0, 1} , π

′

) satisfying the

perfect-coordination property and such that π

′

(1|θ) = 0 for all θ ≤ θ and π

′

(1|θ) = 1 for all

θ >

θ and such that the policy maker is weakly better oﬀ under Γ

′

than under Γ. The policy

′

can be constructed following the steps in the proofs of Theorems 1* and 2*. Now suppose

that Γ

′

is not a deterministic monotone rule (i.e., there is no

θ such that π

′

(1|θ) = 1(θ ≥

θ) for

F −almost all θ). As in Subsection 3.2, let U

′

(x, 1|x) be the expected payoﬀ of an agent with

signal x who, under the policy Γ

′

hears that s = 1, and who expects all other agents to invest

if and only if their signal exceeds x. Suppose that U

′

(x, 1|x) has a unique global minimum

x ≡ arg min

′

(x, 1|x), and that x ≤ x

max

(these properties are not assumed in the proof

but permit us to illustrate the role of property (3*) in Condition M* in the simplest possible

Figure 3: Construction of improving policy

Γ = ({0, 1} , ˜π).

terms). That Γ

′

satisﬁes the perfect-coordination property implies that U

′

(x, 1|x) > 0, for

otherwise it is rationalizable for some of the agents with signal x ≤ x not to invest, after

hearing that s = 1 (the arguments are similar to those in the baseline model). To make things

interesting suppose there exist two disjoint intervals of fundamentals Θ

−

, Θ

⊂ Θ(x), both

consistent with x, such that (i) sup Θ

−

≤ inf Θ

, (ii) u(θ, 1 − P (x|θ)) ≤ 0 for F -almost all

θ ∈ Θ

−

∪ Θ

, (iii) π

′

(1|θ) > 0 for F -almost all θ ∈ Θ

−

, and (iv) π

′

(1|θ) < 1 for F -almost all

θ ∈ Θ

(as we show in the Appendix, when these properties do not hold there exist trivial

improvements of the policy Γ

′

even when property (3*) in Condition M* does not hold).

Then, consider a binary policy

Γ = ({0, 1} , ˜π) constructed from Γ

′

by reducing the prob-

ability of the pass grade s = 1 over the interval Θ

−

and increasing it over the interval Θ

, as

in Figure 3. Let

∆S(x) ≡

+∞

−∞

u(θ, 1 − P (x|θ))p(x|θ)(˜π(1|θ) − π

′

(1|θ))dF (θ).

Suppose that the new policy

Γ is such that ∆S(x) = 0, which implies that U

(x, 1|x) > 0.

Property (3*), along with the fact that (a) ˜π(1|θ) − π

′

(1|θ) crosses zero from below, (b) p(x|θ)

is log-supermodular, and (c) Conditions FB holds, guarantees that ∆S(x) ≥ 0 for all x < x,

which in turn implies that U

(x, 1|x) > 0 for all x < x. The proof in the Appendix leverages

this property to show how to construct a sequence of perturbations of the policy Γ

′

leading

to a new binary policy Γ

= ({0, 1}, π

) that is deterministic and monotone and such that

(x, 1|x) > 0 for all x, which guarantees that Γ

also satisﬁes the perfect-coordination

property. That the new policy Γ

improves over the original one Γ then follows from the fact

that the policy maker’s payoﬀ satisﬁes property (2*) — the arguments for this last step are

similar to those leading to Theorem 3 in the baseline model.

4.3 Discussion: Role of multiplicity of receivers and exogenous pri-

vate information

It is worth contrasting the results about the sub-optimality of monotone rules (when Condition

M* is violated) to those for economies featuring either a single privately-informed receiver, or

multiple receivers with no exogenous private information.

Single receiver. With a single receiver, the optimal policy is a simple monotone pass/fail

policy with cutoﬀ equal to θ

∗

= 0. This is because, in this model, the policy maker’s and the

receiver’s payoﬀs are aligned (they both want to avoid default when possible). With a single

receiver, there is no risk of adversarial coordination and hence the optimal policy coincides

with the one that the designer would select if she trusted the receiver to play favorably to her.

Things are diﬀerent when preferences are misaligned. To see this, suppose the policy

maker’s payoﬀ is equal to W in case of no default, and L < W in case of default, with

W, L ∈ R constant, as in Examples 2 and 3 above. However, now suppose that the receiver’s

payoﬀ diﬀerential between investing and not investing is equal to −g in case of default and −b

in case of no default, with g > 0 > b. Such a payoﬀ diﬀerential may reﬂect the idea that the

receiver is a speculator whose payoﬀ is zero when he refrains from speculating (equivalently,

when he invests), is positive when he speculates and default occurs, and is negative when

he speculates and default does not occur. Using the results in Guo and Shmaya (2019),

one can then show that the optimal policy in this case has the interval structure: each type

x of the receiver is induced to play the action favorable to the policy maker (abstain from

speculating) over an interval of fundamentals [θ

(x), θ

(x)], with θ

(x) < 1 < θ

(x), for all x,

and with θ

(x) decreasing in x and θ

(x) increasing in x. Such a policy requires disclosing

more than two signals and hence cannot be implemented through a simple pass/fail test. In

contrast, with a continuum of heterogeneously informed receivers with the same payoﬀs as

in the variant above, the optimal policy is a pass-fail test that is typically non-monotone in

θ.

Furthermore, when the optimal policy is not monotone, it does not have the interval

structure, as each receiver with signal x is induced to invest over a non-connected set of

fundamentals. The reason for these diﬀerences is that, with a single receiver, to discourage

the latter from taking the adversarial action, the policy maker must persuade the receiver

that the fundamentals are likely to be above 1, in which case the attack is unsuccessful. With

multiple receivers, instead, the policy maker must persuade each receiver that enough other

receivers are not attacking, which, as shown above, is best accomplished by a non-monotone

policy that makes it diﬃcult for the receivers to commonly learn the fundamentals, when the

latter are between 0 and θ

Multiple receivers with no exogenous private information. When all receivers have the

same posterior beliefs, no matter whether payoﬀs are aligned or mis-aligned, under MARP,

each receiver plays the friendly action only if it is dominant to do so. The optimal policy is a

simple monotone pass/fail policy with cutoﬀ θ

∗

implicitly deﬁned by

∗

bdF (θ) +

∞

gdF (θ) = 0.

The reason why the optimal policy is monotone when the receivers possess no exogenous

private information is that the policy maker needs to convince each of them that θ is above 1

with suﬃciently high probability to make the friendly action dominant.

5 Conclusions

We consider the design of public information in coordination settings in which the designer

does not trust the receivers to play favorably to her. We show that, despite the fear of

adversarial coordination, the optimal policy induces all receivers to take the same action.

Importantly, while each agent can perfectly predict the action of any other agent, he is not

able to predict the beliefs that rationalize such actions. We identify conditions under which

the optimal policy has a pass/fail structure, as well as conditions under which the optimal

This is because, under MARP, all agents play the friendly action if and only if it is iteratively dominant

for them to do so, irrespective of the alignment in payoﬀs.

Mensch (2021) characterizes general conditions under which the optimal policy is monotone with a single,

uninformed, receiver. Goldstein and Leitner (2018) study an economy in which these conditions are not

satisﬁed and the optimal policy is non-monotone. The analysis in these works is very diﬀerent in that it does

not identify the role that coordination and the receivers’ private information play for the optimal policy.

policy is monotone, passing institutions with strong fundamentals and failing the others.

The results are worth extending in a few directions. The analysis assumes that the pol-

icy maker 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 inter-

esting to investigate how the structure of the optimal policy is aﬀected by the policy maker’s

uncertainty about the agents’ information sources.

Motivated by the applications the analysis is meant for (most notably, stress testing),

we have conﬁned attention to non-discriminatory 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.

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 policy maker 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), Calzolari and Pavan (2006b), and Dworczak (2020)).

See Dworczak and Pavan (2022) for a notion of robustness in information design that accounts for this

type of ambiguity.

For models of dynamic persuasion, see, among others, Ely (2017) and Basak and Zhou (2022).

Calzolari 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.

Appendix

Proof of Theorem 1*. Given any regular policy Γ = (S, π) and any n ∈ N, let T

(n)

be the set of strategies surviving n rounds of iterated deletion of interim strictly dominated

strategies (IDISDS), with T

(0)

denoting the entire set of strategy proﬁles a = (a

(·))

i∈[0,1]

where for any i ∈ [0, 1], a

(x, s) denotes the probability agent i invests, given (x, s). Let a

(n)

≡



(n),i

(·)



i∈[0,1]

∈ T

(n)

denote the most aggressive proﬁle surviving n rounds of IDISDS (that is,

the proﬁle in T

(n)

that is most adversarial to the policy maker, in the sense that it minimizes the

policy maker’s ex-ante payoﬀ).The proﬁles



(n)



n∈N

can be constructed inductively as follows.

The proﬁle a

(0)

≡



(0),i

(·)



i∈[0,1]

prescribes that all agents refrain from investing, irrespective

of (x, s). Next, let U

(x, s; a) denote the payoﬀ diﬀerential between investing and not investing

for agent i receiving information (x, s) when, under Γ, all other agents follow the strategy in

a. Then, a

(n),i

(x, s) = 0 if U



x, s; a

(n−1)



≤ 0 and a

(n),i

(x, s) = 1 if U



x, s; a

(n−1)



> 0.

MARP consistent with Γ is the proﬁle (a

(·))

i∈[0,1]

given by a

(·) = lim

n→∞

(n),i

(·), all i ∈ [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 a

the common

strategy that all agents follow under MARP consistent with Γ. For any θ and s ∈ supp(π(θ)),

aggregate investment under MARP consistent with Γ given (θ, s) is thus the same for any

x, x

′

∈ X (θ) and is given by A

(θ, s) ≡

(x, s) dP (x|θ).

Next, consider the policy Γ

= (S

, π

), S

≡ S×{0, 1}, that, for each θ, draws the public

signal s from the same distribution π(θ) ∈ ∆(S) as the original policy Γ, and then, for each s

it 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 Γ. That is, let 1



u(θ, A

(θ, s)) > 0



the indicator function, taking value 1 if θ is such that u(θ, A

(θ, s)) > 0, and 0 otherwise. For

any θ and any s ∈ supp(π(θ)), the new policy Γ

thus announces



s, 1



u(θ, A

(θ, s)) > 0



In the baseline model of Section 2, the sign of u(θ, A

(θ, s)) is uniquely determined by the

regime outcome r

(θ, s). In that environment, for any θ, and any s ∈ supp(π(θ)), the new

policy Γ

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 policy Γ

The proof is in three steps. Steps 1 and 2 show that any agent i who, given (x, s), ﬁnds

it dominant (alternatively, iteratively dominant) to invest under Γ also ﬁnds it dominant

(alternatively, iteratively dominant) to invest under Γ

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 a

also

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. First, we prove that, {(x, s) : U

(x, s; a) > 0 ∀a} ⊆ {(x, s) : U

(x, (s, 1); a) >

0 ∀a}, for all i ∈ [0, 1]. That is, any agent i who, under Γ, ﬁnds it dominant to invest,

given information (x, s), also ﬁnds it dominant to invest under Γ

when receiving information

(x, (s, 1)).

First, note that the supermodularity of the game implies that {(x, s) : U

(x, s; a) >

0 ∀a} = {(x, s) : U

(x, s; a

(0)

) > 0} and {(x, s) : U

(x, (s, 1); a) > 0 ∀a} = {(x, s) :

(x, (s, 1); a

(0)

) > 0}.

Now let Λ

(x, s) denote the distribution over Θ describing the beliefs of agent i ∈ [0, 1]

when receiving information (x, s) ∈ R×S under Γ, and Λ

(x, (s, 1)) the corresponding beliefs

under Γ

, when receiving information (x, (s, 1)) under Γ

. Bayesian updating implies that

(dθ|x, (s, 1)) =



u(θ, A

(θ, s)) > 0



(1|x, s)

(dθ|x, s), (11)

where Λ

(1|x, s) ≡

{θ∈Θ:u(θ,A

(θ,s))>0}

(dθ|x, s) is the total probability an agent with infor-

mation (x, s) assigns, under Γ, to fundamentals for which u(θ, A

(θ, s)) > 0.

Next, observe that, for any i ∈ [0, 1] and (x, s) ∈ R × S such that



x, s; a

(0)



u(θ, 0)Λ

(dθ|x, s) > 0, (12)

we also have that

(x, (s, 1); a

(0)

)Λ

(1|x, s) =

u(θ, 0)1



u(θ, A

(θ, s)) > 0



(dθ|x, s)

≥

u(θ, 0)Λ

(dθ|x

, s) = U

(x, 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 A along with the fact A

(θ, s) ≥ 0, which together imply that u(θ, 0) ≤ 0 for any θ

for which u(θ, A

(θ, s)) ≤ 0. The second equality follows from the deﬁnition of U



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

Γ, continues to ﬁnd it dominant to invest after receiving information (x, (s, 1)) under Γ

Step 2. Next, take any n > 1. Assume that, for any 1 ≤ k ≤ n − 1, any i ∈ [0, 1],

{(x, s) : U

(x, s; a) > 0 ∀a ∈ T

(k−1)

} ⊆ {(x, s) : U

(x, (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

(x, s; a) > 0 ∀a ∈ T

(n−1)

} ⊆ {(x, s) : U

(x, (s, 1); a) > 0, ∀a ∈ T

(n−1)

}. (14)

Intuitively, the result follows from the following two properties: (a) because the game is

supermodular, {(x, s) : U

(x, s; a) > 0 ∀a ∈ T

(n−1)

} = {(x, s) : U



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 Γ

); (b) a

(n−1)

is “less aggressive” than a

(n−1)

, in the sense that any

agent who, given (x, s), invests under a

(n−1)

also invests under Γ

when receiving information

(x, (s, 1)); and (c) the extra information that θ is 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

, meaning that any agent who, given (x, s), invests under a

(n−1)

, also invests under a

when

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 ∈ Θ

and s ∈ supp(π(θ)) such that u(θ, A

(θ, s)) > 0, a

(x, (s, 1)) ≡ lim

n→∞

(n)

(x, (s, 1)) = 1 for all

x. This follows directly from the fact that, as shown above, a

(x, s) = 1 ⇒ a

(x, (s, 1)) = 1.

The announcement that θ is 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 Γ

, the payoﬀ

diﬀerential from investing is strictly positive. Any agent i receiving information (s, 1) under

thus necessarily invests, no matter x. Under the new policy Γ

, all agents thus invest

when they learn that θ is such that u(θ, A

(θ, s)) > 0. That they all refrain from investing

when they learn that θ is 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 Γ

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 Γ

than under Γ 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 Γ (thus obtaining an expected payoﬀ of zero) invest under Γ

(obtaining an

expected payoﬀ u(θ, 1) > 0), and all agents who are investing under Γ 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

Γ (thus obtaining an expected payoﬀ of zero) continue not to invest under Γ

, whereas all

agents who are investing under Γ (obtaining a negative payoﬀ) now refrain from investing

thus obtaining a payoﬀ of zero.

Next, suppose that, under MARP consistent with Γ, for any θ and s ∈ supp(π(θ)),

the regime outcome is a deterministic function of (θ, s). Then, for any (θ, s), the sign of

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 Γ

induces the same

regime outcome as Γ.

To see that the policy maker is better oﬀ under Γ

than under Γ, for any set of signals S ⊆

S, any θ, let π

(S, 1|θ) (alternatively, π

(S, 0|θ)) denote the probability that the policy π

se-

lects signals (s, 1) (alternatively, (s, 0)) with s ∈ S. Then let, Π

(S, 1) ≡

(S, 1|θ) dF (θ)

(alternatively, Π

(S, 0) ≡

(S, 1|θ) dF (θ)) denote the ex-ante probability of announce-

ments (s, 1) (alternatively, (s, 0)) with s ∈ S, under the policy Γ

. Finally, for any S ⊆ S,

let Π

(S) ≡

π (S|θ) dF (θ) denote the ex-ante probability the policy Γ selects signals in S.

Condition PC implies that







u(θ, A

(θ, s)) > 0



(θ, 1) + 1



u(θ, A

(θ, s)) ≤ 0



(θ, 0)



(dθ|s)



(ds)

≥



(θ, A

(θ, s))Λ

(dθ|s)



(ds) .

Hence, the policy maker is better oﬀ under Γ

than under Γ.

The result in the theorem then follows by taking Γ

∗

= Γ

. Q.E.D.

Proof of Theorem 2*. The proof is in 2 steps. Step 1 shows that, when p(x|θ) is

log-supermodular, i.e., it satisﬁes MLRP, 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 Γ that satisﬁes the perfect-coordination property, one can construct

a binary policy Γ

∗

that also satisﬁes the perfect-coordination property and such that, for any

θ, the probability that each agent invests under Γ

∗

is the same as under Γ, which implies the

result in the theorem.

Step 1. Fix an arbitrary policy Γ = (S, π) and, for any pair (x, s) ∈ R × S, let Λ

(θ|x, s)

represent the endogenous posterior beliefs over Θ of each agent receiving exogenous informa-

tion x and endogenous information s. Next, let U

(x, s|k) ≡

u(θ, 1 − P (k|θ))Λ

(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

k and refrain from investing if it is below k). The following result establishes that, when the

distribution p(x|θ) from which the signals are drawn satisﬁes MLRP, and Condition FB holds,

no matter Γ, MARP is in cut-oﬀ strategies:

Lemma 1. Suppose that p(x|θ) is log-supermodular and that Condition FB holds. Given any

policy Γ = (S, π), for any s ∈ S, there exists ξ

Γ;s

∈ R such that MARP consistent with Γ

is given by the strategy proﬁle a

≡ (a

)

i∈[0,1]

such that, for any s ∈ S, x ∈ R, i ∈ [0, 1],

(x, s) = 1



x > ξ

Γ;s



with ξ

Γ;s

≡ sup{x : U

(x, s|x) ≤ 0} if {x : U

(x, s|x) ≤ 0} ̸= ∅, and

Γ;s

≡ −∞ otherwise. Moreover, the strategy proﬁle a

is a BNE of the continuation game

that starts with the announcement of the policy Γ.

Proof of Lemma 1. Fix the policy Γ = (S, π). For any s ∈ S, let ξ

Γ;s

(1)

≡ sup{x :

lim

k→∞

(x, s|k) ≤ 0}. Given the public signal s, it is dominant for any agent with private

signal x exceeding ξ

Γ;s

to invest. Next, recall that, for any n ∈ N, T

(n)

denotes the set

of strategy proﬁles that survive the ﬁrst n rounds of iterated deletion of interim strictly

dominated strategies (IDISDS), and a

(n)

≡



(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

Γ;s

(1)



for all (x, s) ∈ R × S, and all

i ∈ [0, 1], and minimizes the policy maker’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)Λ

(dθ|x, s) between investing and not investing crosses 0 only once and from below

at x = ξ

Γ;s

(1)

. The single-crossing property of

u(θ, 0)Λ

(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 → R crosses 0 only once from below at

θ = θ

(that is, h (θ) ≤ 0 for all θ ≤ θ

and h (θ) ≥ 0 for all θ > θ

). Let q : R

→ R

be a log-

supermodular function and suppose that, for any θ, there is an open interval ϱ

= (ϱ

, ¯ϱ

) ⊂ R

containing θ such that q (x, θ) > 0 for all x ∈ ϱ

and q (x, θ) = 0 for (almost) all x ∈ R \ ϱ

with the bounds ϱ

, ¯ϱ

non-decreasing in θ. Choose any (Lebesgue) measurable subset Ω ⊆ R

containing θ

and, for any x ∈ R, let Ψ(x; Ω) ≡

Ω

h(θ)q(x, θ)dθ. Suppose there exists x

⋆

∈ ϱ

such that Ψ(x

⋆

; Ω) = 0. Then, necessarily, Ψ(x; Ω) ≥ 0 for all x ∈ ϱ

with x > x

⋆

, and

Ψ(x; Ω) ≤ 0 for all x ∈ ϱ

with x < x

⋆

, with both inequalities strict if (a) {θ ∈ Ω : h (θ) ̸= 0}

has strict positive Lebesgue measure, (b) q is strictly log-supermodular over R

Proof of Property SCB. For any x ∈ R, let Ω

≡ {θ ∈ Ω : x ∈ ϱ

}. The monotonicity

of ϱ

in θ implies that Ω

is monotone in x in the strong-order sense. Pick any x

′

∈ ϱ

with

′

> x

⋆

. That x

⋆

and x

′

belong to ϱ

implies that θ

∈ Ω

⋆

∩ Ω

′

. Next, observe that

Ψ(x

′

; Ω) =

Ω

′

h(θ)q(x

′

, θ)dθ =

Ω

′

∩Ω

⋆

h(θ)q(x

′

, θ)dθ +

Ω

′

\Ω

⋆

h(θ)q(x

′

, θ)dθ

Ω

⋆

∩Ω

′

∩(−∞,θ

)

h(θ)q(x

⋆

, θ)

q(x

′

, θ)

q(x

⋆

, θ)

dθ +

Ω

⋆

∩Ω

′

∩(θ

,∞)

h(θ)q(x

⋆

, θ)

q(x

′

, θ)

q(x

⋆

, θ)

dθ

Ω

′

\Ω

⋆

h(θ)q(x

′

, θ)dθ

≥

q(x

′

, θ

)

q(x

⋆

, θ

)

Ω

⋆

∩Ω

′

∩(−∞,θ

)

h(θ)q(x

⋆

, θ)dθ +

Ω

⋆

∩Ω

′

∩(θ

,∞)

h(θ)q(x

⋆

, θ)dθ

Ω

′

\Ω

⋆

h(θ)q(x

′

, θ)dθ

≥

q(x

′

, θ

)

q(x

⋆

, θ

)

Ψ(x

⋆

; Ω)

| {z }

Ω

′

\Ω

⋆

h(θ)q(x

′

, θ)dθ ≥ 0.

The ﬁrst equality follows from the fact that q(x

′

, θ) = 0 for almost all θ ∈ Ω \ Ω

′

. The

second equality follows from the fact that Ω

′

can be partitioned into Ω

′

∩ Ω

⋆

and Ω

′

\ Ω

⋆

The third equality follows from noting that q(x

⋆

, θ) > 0 for all θ ∈ Ω

⋆

. The ﬁrst inequality

follows from the monotonicity of q(x

′

, θ)/q(x

⋆

, θ) over Ω

⋆

∩ Ω

′

as a consequence of q being

That q is strictly log-supermodular over R

also implies that q(x, θ) > 0 for all (x, θ) ∈ R

log-supermodular, along with the fact that θ

∈ Ω

⋆

∩ Ω

′

and the assumption that h crosses

0 once from below at θ = θ

. The second inequality follows from the fact that, for any θ ∈

(Ω

⋆

\ Ω

′

)∩(−∞, θ

), h(θ) ≤ 0, along with the fact that Ω

⋆

∩(θ

, +∞) = Ω

⋆

∩Ω

′

∩(θ

, ∞),

with the last property following from noting that the sets Ω

are ranked in the strong-order

sense. The last inequality follows from the observation that, for any θ ∈ Ω

′

\ Ω

⋆

, h (θ) ≥ 0,

which in turn is a consequence of (i) the monotonicity of the sets Ω

in x, (ii) the assumption

that h crosses 0 only once from below at θ = θ

, and (iii) the assumption that θ

∈ Ω

⋆

∩ Ω

′

Similar arguments imply that, for x < x

⋆

, Ψ(x; Ω) ≤ 0. The same arguments also imply

that, when (a) {θ ∈ Ω : h (θ) ̸= 0} has strict positive Lebesgue measure and (b) q is strictly

log-supermodular over R

, then Ψ(x; Ω) < 0 for all x < x

⋆

and Ψ(x; Ω) > 0 for all 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 mono-

tone in θ imply that, for any s ∈ S, there exists a unique sequence



Γ;s

(n)



n∈N

such that, for

any n ≥ 1, a

(n)

is such that a

(n),i

(x, s) = 1



x > ξ

Γ;s

(n)



for all i and all (x, s) ∈ R × S, with

each ξ

Γ;s

(1)

as deﬁned above, and with all other cut-oﬀs ξ

Γ;s

(n)

, n > 1, s ∈ S, deﬁned inductively by

Γ;s

(n)

≡ sup{x : U

(x, s|ξ

Γ;s

(n−1)

) ≤ 0}. Indeed, Condition FB together with Property SCB jointly

imply that U

(x, s|ξ

Γ;s

(n−1)

) =



θ, 1 − P



Γ;s

(n−1)

|θ



(dθ|x, s) crosses zero once from below

in x, and therefore U

(x, s|ξ

Γ;s

(n−1)

) > 0 if, and only if, x > ξ

Γ;s

(n)

Let T

≡ ∩

∞

n=1

denote the set of strategy proﬁles that survive IDISDS under Γ. The

most aggressive strategy proﬁle in T

is then given by a

(x, s) ≡ 1



x > ξ

Γ;s



for all i and all

(x, s) ∈ R × S, where, for any s ∈ S, ξ

Γ;s

≡ lim

n→∞

Γ;s

(n)

. The sequence (ξ

Γ;s

(n)

)

is monotone and

its limit is given by ξ

Γ;s

= sup{x : U

(x, s|x) ≤ 0} if {x : U

(x, s|x) ≤ 0} ̸= ∅, and ξ

Γ;s

≡ −∞

otherwise. This establishes the ﬁrst part of the lemma. That the proﬁle a

is a BNE for

the continuation game that starts with the announcement of the policy Γ follows from the

fact that, given any s∈ S, when all agents follow a cut-oﬀ strategy with cutoﬀ ξ

Γ;s

, the best

response for each agent i ∈ [0, 1] is to invest for x

> ξ

Γ;s

and to refrain from investing for

< ξ

Γ;s

. This completes the proof of the lemma. ■

Step 2. Now take any regular policy Γ = (S, π) satisfying the perfect-coordination prop-

erty. Given the result in Theorem 1, without loss of generality, assume that Γ = (S, π) is such

that S = {0, 1} ×

S, for some measurable set

S, and is such that (a) when the policy discloses

any signal s = (ˆs, 1), all agents invest and default does not happen, whereas (b) when the

policy discloses any signal s = (ˆs, 0), all agents refrain from investing and default happens.

Equipped with the result in Lemma 1, we show that, starting from Γ = (S, π), one can

construct a binary policy Γ

∗

= ({0, 1}, π

∗

) also satisfying the perfect-coordination property

and such that the probability of default under Γ

∗

is the same as under Γ. The policy Γ

∗

({0, 1}, π

∗

) is such that, for any θ, π

∗

(1|θ) =

π (d (ˆs, 1) |θ) . That is, for each θ, the binary

policy Γ

∗

recommends to invest with the same total probability as the original policy Γ

discloses signals leading all agents to invest.

We now show that, under Γ

∗

, when the policy announces that s = 1, the unique rationaliz-

able action for each agent is to invest. To see this, for any (x, 1) that are mutually consistent

given Γ

∗

, let U

∗

(x, 1|k) denote the expected payoﬀ diﬀerential for any agent with private

signal x, when the policy Γ

∗

announces s = 1, and all other agents follow a cut-oﬀ strategy

with cut-oﬀ k.

From the law of iterated expectations, we have that

∗

(x, 1|k) =

(x, (ˆs, 1)|k)ς

(dˆs|x, 1) (15)

where ς

(·|x, 1) is the probability measure over

S obtained by conditioning on the event (x, 1),

under Γ. For any signal s = (ˆs, 1) in the range of π, MARP consistent with Γ is such that

(x, (ˆs, 1)) = 1 all x ∈ R, and all i, meaning that investing is the unique rationalizable action

after Γ announces s = (ˆs, 1). Lemma 1 in turn implies that, for all s = (ˆs, 1) in the range

of π, ˆs ∈

S, all k ∈ R, U

(k, (ˆs, 1)|k) > 0. From (15), we then have that, for all all k ∈ R ,

∗

(k, 1|k) > 0. In turn, this implies that, given the new policy Γ

∗

, when s = 1 is disclosed,

under MARP consistent with Γ

∗

, all agents invest, that is, a

∗

(x, 1) = 1 all x, all i ∈ [0, 1].

It is also easy to see that, when the policy Γ

∗

discloses the signal s = 0, it becomes common

certainty among the agents that θ ≤ θ. Hence, under MARP consistent with Γ

∗

, after s = 0

is disclosed, all agents refrain from investing, irrespective of their private signals. The new

policy Γ

∗

so constructed thus (a) satisﬁes the perfect-coordination property, and (b) is such

that, for any θ, the probability of default under Γ

∗

is the same as under Γ. Q.E.D.

π (d (ˆs, 1) |θ) represents the total probability that the measure π(θ) assigns to signal (ˆs, 1).

Recall that (x, 1) are mutually consistent under Γ

∗

if p

∗

(x, 1) ≡

p(x|θ)π

∗

(1|θ)dF (θ) > 0.

Proof of Theorem 3*. The conditions in the theorem imply that Theorems 1* and

2* hold. Thus, assume that the policy Γ = (S, π) (a) is a regular (possibly stochastic)

“pass/fail”policy (i.e., S = {0, 1}, with π(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) = 0 for

all θ ≤ θ and π(1|θ) = 1 for all θ >

θ, and (c) satisﬁes the perfect-coordination property.

Theorems 1* and 2* imply that, if Γ does not satisfy these properties, there exists another

policy Γ

′

that satisﬁes these properties and yields the policy maker a payoﬀ weakly higher

than Γ. The proof then follows from applying the arguments below to Γ

′

instead of Γ.

Suppose that Γ is such that there exists no

θ such that π(1|θ) = 0 for F -almost all θ ≤

and π(1|θ) = 1 for F -almost all θ >

θ.

We establish the result by showing that there exists

a deterministic monotone policy Γ

= ({0, 1}, π

) satisfying the perfect-coordination property

that yields the policy maker a payoﬀ strictly higher than Γ.

Recall that, for the policy Γ to satisfy the perfect-coordination property, it must be that,

when the policy discloses the signal s = 1, U

(x, 1|x) > 0 for all x such that (x, 1) are mutually

consistent, where U

(x, 1|x) is the expected payoﬀ diﬀerential of an agent with signal x who

hears that s = 1 and who expects all other agents to follow a cut-oﬀ strategy with threshold

Let G denote the set of policies Γ

′

= (S, π

′

) that, in addition to properties (a) and (b)

above, are such that U

′

(x, 1|x) ≥ 0 for all x such that (x, 1) are mutually consistent.

For

any Γ ∈ G, let U

[Γ] denote the policy maker’s ex-ante expected payoﬀ when, under Γ, agents

invest after hearing that s = 1 and refrain from investing after hearing that s = 0. Denote by

arg max

Γ∈G

[

Γ] the set of policies that maximize the policy maker’s payoﬀ over G.

Step 1 below shows that any Γ ∈ arg max

Γ∈G

[

Γ] is such that π(1|θ) = 0 for F -almost all

θ ≤ θ

∗

and π(1|θ) = 1 for F -almost all θ > θ

∗

, with θ

∗

as deﬁned in (4). Step 2 then shows that

the policy maker’s payoﬀ under the optimal monotone policy Γ

∗

= ({0, 1}, π

∗

) with cut-oﬀ θ

∗

can be approximated arbitrarily well by a deterministic monotone policy Γ

= ({0, 1}, π

) ∈ G

that satisﬁes the perfect-coordination property, thus establishing the theorem.

If this not the case, then the deterministic monotone policy Γ

= ({0, 1}, π

) with cut-oﬀ

θ also satisﬁes

the perfect-coordination property and yields the policy maker the same payoﬀ as Γ, in which case the result

trivially holds.

As explained in the main text, some policies Γ

′

in G need 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.

That arg max

Γ∈G

[

Γ] ̸= ∅ follows from the compactness of G and the upper hemi-continuity of U

Step 1. Given any policy Γ, let

≡ {x : (x, 1) Γ-mutually consistent and U

(x, 1|x) = 0}.

Take any policy Γ

′

∈ G for which there exists no

θ such that π

′

(1|θ) = 0 for F -almost all θ ≤

and π

′

(1|θ) = 1 for F-almost all θ >

θ. Clearly, if X

′

= ∅, there exists another policy Γ

′′

∈ G

that yields the policy maker a payoﬀ strictly higher than Γ

′

Thus, assume that X

′

̸= ∅,

and let ¯x ≡ supX

′

. Claim A below shows that the set {θ ∈ Θ (¯x) : π

′

(1|θ) < 1} has strict

positive F -measure. Claim B shows that, given any Γ

′

∈ G for which the posterior beliefs of

the marginal agent with signal ¯x diﬀer from those obtained by Bayes rule conditioning on the

event that fundamentals are above some threshold

θ, there exists another policy Γ

′′

∈ G that

yields the policy maker a payoﬀ strictly higher than Γ

′

. Finally, Claim C shows that, under

the properties in Condition M*, the only policies Γ

′

∈ G that generate posterior beliefs for the

marginal agents with signal ¯x equal to those obtained from Bayes rule by conditioning on the

event that fundamentals are above some threshold

θ are such that π

′

(1|θ) = 0 for F -almost

all θ ≤ θ

∗

and π

′

(1|θ) = 1 for F -almost all θ > θ

∗

. Jointly, the three claims thus establish the

result that any policy Γ ∈ arg max

Γ∈G

[

Γ], is such that π(1|θ) = 0 for F -almost all θ ≤ θ

∗

and π(1|θ) = 1 for F -almost all θ > θ

∗

Given any x, let θ

(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, θ

(x) is well-deﬁned.

For any policy

Γ = ({0, 1} , π) ∈ G, let p

(x, 1) ≡

+∞

−∞

π(1|θ)p(x|θ)dF (θ) denote the joint probability density

of the exogenous signal x and the endogenous signal s = 1.

Claim A. For any Γ

′

= ({0, 1} , π

′

) ∈ G such 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) = 1 for F -almost all θ ∈

Θ (¯x). Property (1*) in Condition M* then implies that ¯x > x

max

, where x

max

is deﬁned

In fact, because there exists no such a

θ, there must exists a set (θ

′

, θ

′′

) ⊆ [θ,

θ] of F -positive measure over

which π

′

(1|θ) < 1. The policy Γ

′′

can then be obtained from Γ

′

by increasing π

′

(1|θ) over such a set. Provided

the increase is small, Γ

′′

∈ G. Because U

(θ, 1) > U

(θ, 0) over [θ,

θ], the policy maker’s payoﬀ under Γ

′′

strictly higher than under Γ

′

When the regime outcome is a function of A and θ only, as in the baseline model, θ

(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.

as in (8). In fact, if this was not the case, the monotonicity of Θ (·) would imply that

inf Θ (¯x) ≤ inf Θ (x

max

) < θ. That π

′

(1|θ) = 1 for F -almost all θ ∈ Θ (¯x) would then imply

that π

′

(1|θ) = 1 for a set of fundamentals θ < θ of strict positive F -measure, which is

inconsistent with the assumption that Γ

′

∈ G. Thus, necessarily, ¯x > x

max

Now suppose that inf Θ (¯x) ≥ θ. That π

′

(1|θ) = 1 for F -almost all θ ∈ Θ (¯x) means that,

from the perspective of an agent with signal ¯x, the information conveyed by the announcement

that s = 1 under Γ

′

is the same as under the monotone deterministic policy Γ



{0, 1}, π



with cut-oﬀ

θ = θ. As a result, U

′

(¯x, 1|¯x) = U

(¯x, 1|¯x). Because ¯x > x

max

, and because, by

deﬁnition of x

max

, U

(x, 1|x) > 0 for all x > x

max

, it must be that U

′

(¯x, 1|¯x) > 0, which

contradicts the assumption that ¯x ∈ X

′

. Hence, it must be that inf Θ (¯x) < θ. As explained

above, however, this is inconsistent with the assumption that Γ

′

∈ G. □

Next, for any Γ

′

= ({0, 1}, π

′

) ∈ G, let

≡ sup {θ ∈ Θ : ∃δ > 0 s.t. π

′

(1|θ

′

) < 1 for F -almost all θ

′

∈ [θ − δ, θ)} .

The result in Claim A above implies that θ

is such that θ

> inf Θ (¯x).

Claim B. Take any Γ

′

= ({0, 1} , π

′

) ∈ G such that X

′

̸= ∅. Suppose that

{θ ∈ (θ

, θ

) : π

′

(1|θ) > 0} has strict positive F -measure. (16)

Then, there exists another policy Γ

′′

∈ G that yields the policy maker a payoﬀ strictly higher

than Γ

′

Claim B essentially says that, if Γ

′

∈ G is not a deterministic monotone rule, and there

exists a ¯x such that U

′

(¯x, 1|¯x) = 0, then it is improvable.

Proof of Claim B. The proof below distinguishes two cases.

Case 1: θ < θ

(¯x) ≤ θ

. Consider the policy Γ

ϵ,δ

= ({0, 1}, π

ϵ,δ

) deﬁned by π

ϵ,δ

(1|θ) =

′

(1|θ) for all θ ≤ θ

(¯x + δ), with δ > 0 small so that θ

(¯x + δ) < θ

, and π

ϵ,δ

(1|θ) =

min{π

′

(1|θ) + ϵ, 1} for all θ > θ

(¯x + δ), with ϵ > 0 also small. To see that, when ϵ and

δ are small, Γ

ϵ,δ

∈ G, note that, by deﬁnition of θ

(·), for any x, and any θ > θ

(x),

u (θ, 1 − P (x|θ)) > 0. This property, together with the monotonicity of θ

(·), jointly imply

that, for any x ≤ ¯x + δ,

∞

−∞

u(θ, 1 − P (x|θ))



′

(1|θ)1 (θ ≤ θ

(¯x + δ)) + min{π

′

(1|θ) + ϵ, 1}1 (θ > θ

(¯x + δ))



p(x|θ)dF (θ)

≥

∞

−∞

u(θ, 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)) > 0 for θ > θ

(¯x + δ),

by deﬁnition of θ

(·). 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)) > 0 for all θ > θ

(¯x + δ). Because Γ

′

∈ G, the right-

hand side of (17) is non-negative.

Hence, for any x ≤ ¯x + δ such that (x, 1) are mutually

consistent under Γ

ϵ,δ

, because the left-hand side of (17) is equal to U

ϵ,δ

(x, 1|x)p

ϵ,δ

(x, 1) and

because, for such x, p

ϵ,δ

(x, 1) > 0, we have that U

ϵ,δ

(x, 1|x) ≥ 0. That U

ϵ,δ

(x, 1|x) ≥ 0 also

for all x > ¯x + δ such that (x, 1) are mutually consistent under Γ

ϵ,δ

follows from the fact that,

by deﬁnition of ¯x, for any x ≥ ¯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



ϵ,δ

(·)



whose elements J

ϵ,δ

(·) are given by J

ϵ,δ

(x) ≡

+∞

−∞

u(θ, 1 − P (x|θ))π

ϵ,δ

(1|θ)p(x|θ)dF (θ) is

continuous in ϵ in the sup-norm in a neighborhood of 0.

Because the new policy Γ

ϵ,δ

∈ G

is such that π

ϵ,δ

(1|θ) ≥ π

′

(1|θ) for all θ, with the inequality strict over a set of fundamentals

θ ∈ (θ,

θ] of F -positive measure, the policy maker’s payoﬀ under Γ

ϵ,δ

is strictly higher than

under Γ

′

, as claimed.

Case 2 : θ

< θ

(¯x). Consider the monotone deterministic policy Γ



{0, 1} , π



with

cut-oﬀ

θ = θ. Then, for any x ≥ ¯x,

+∞

−∞

u(θ, 1 − P (x|θ))π

(1|θ)p(x|θ)dF (θ) <

+∞

−∞

u(θ, 1 − P (x|θ))π

′

(1|θ)p(x|θ)dF (θ), (18)

where the inequality follows from the following facts: (i) π

(1|θ) = π

′

(1|θ) for F -almost all

θ ∈ (−∞, θ] ∪ [θ

, +∞) and (ii) π

(1|θ) = 1 ≥ π

′

(1|θ) for F -almost all θ ∈ (θ, θ

), with the

inequality strict over a set of fundamentals in (θ, θ

) of strictly positive measure under F ,

and (iii) u (θ, 1 − P (x|θ)) < 0 for θ ∈ (θ, θ

) (by the fact that θ

< θ

(¯x) ≤ θ

(x) along with

the deﬁnition and monotonicity of the function θ

(·)).

Either (x, 1) are not mutually consistent under Γ

′

, in which case the right-hand side of (17) is zero, or

they are mutually consistent, in which case the right-hand side of (17) is equal to U

′

(x, 1|x)p

′

(x, 1), which

is non-negative because p

′

(x, 1) > 0 and U

′

(x, 1|x) ≥ 0.

That is, ∀k > 0, ∃∆ > 0 so that ∀ 0 < ϵ < ∆, |J

ϵ,δ

(x) − J(x)| ≤ k, ∀x ≥ ¯x + δ.

Furthermore, (¯x, 1) are mutually consistent under Γ

′

, that is, p

′

(¯x, 1) > 0. Because

(1|θ) ≥ π

′

(1|θ) for all θ, (¯x, 1) are mutually consistent also under Γ

, i.e., p

(¯x, 1) > 0.

Observe that, when x = ¯x, the left-hand-side of (18) is equal to U

(¯x, 1|¯x)p

(¯x, 1) whereas

the right-hand-side is equal to U

′

(¯x, 1|¯x)p

′

(¯x, 1). By the deﬁnition of ¯x, U

′

(¯x, 1|¯x) = 0,

which then implies that U

(¯x, 1|¯x) < 0. By continuity of U

(x, 1|x) in x and the deﬁnition

of x

max

we thus have that ¯x < x

max

. This property in turn permits us to apply Properties

(1*) and (2*) of Condition M* below.

Next, let

≡ inf{θ ∈ Θ : ∃δ > 0 s.t. π

′

(1|θ) > 0, with π

′

(1|·) continuous over [θ, θ + δ)}.

By assumption, {θ ∈ (θ

, θ

) : π

′

(1|θ) > 0} has strict positive F -measure. If θ

≥ θ

, then

there exists another policy Γ

′′

for which θ

< θ

and such that (a) the policy maker’s payoﬀ

under Γ

′′

is the same as under Γ

′

and (b) U

′′

(x, 1|x) = U

′

(x, 1|x) for all x. The claim

(and ultimately the theorem) then follows from applying the arguments below to Γ

′′

instead

of Γ

′

. Thus, assume that θ

< θ

. Furthermore, note that u (θ

, 1 − P (¯x|θ

)) < 0.

Also

observe that inf Θ (¯x) < θ

. This follows from the fact that, as shown above, ¯x < x

max

, which

together with Property (1*) in Condition M* and the monotonicity of Θ (·) in x implies that

inf Θ (¯x) < θ. Because θ

≥ θ, we thus have that inf Θ (¯x) < θ

Recall that ¯x is the largest solution to U

′

(x, 1|x) = 0. This property, together with the

fact that Γ

′

∈ G implies that U

′

(x, 1|x) > 0 for all x > ¯x such that (x, 1) are mutually

consistent under Γ

′

. Next observe that, for all x ≥ ¯x, (x, 1) are mutually consistent under

′

. Because u(θ, 1 − P (¯x|θ)) > 0 for θ > θ

(¯x) and u(θ, 1 − P (¯x|θ)) < 0 for θ < θ

(¯x) (by

deﬁnition of θ

(¯x)) and because U

′

(¯x, 1|¯x) = 0, it must be that sup Θ (¯x) ≥ θ

(¯x). Because,

by deﬁnition of case (2), θ

(¯x) > θ

, this means that sup Θ (¯x) > θ

. The monotonicity of

Θ (·) implies that sup Θ (x) > θ

for all x ≥ ¯x. Because π

′

(1|θ) = 1 for F -almost all θ > θ

we thus have, for all x ≥ ¯x, (x, 1) are mutually consistent under Γ

′

(and hence U

′

(x, 1|x) is

well deﬁned for all such x).

The continuity of U

′

(·, 1|·) in x implies that, for any η ∈ (0, x

max

− ¯x), the function

′

(·, 1|·) is bounded away from zero over [¯x + η, x

max

]. That is, there exists K > 0 such

This follows from the deﬁnition of θ

(¯x) , along with Condition FB, and the fact that θ

< θ

(¯x).

that U

′

(x, 1|x) > K for all x ∈ [¯x + η, x

max

]. This property, along with (a) the continuity

of U

′

(·, 1|·) in x and (b) the fact that U

′

(¯x, 1|¯x) = 0 in turn imply that there exists η ∈

(0, x

max

− ¯x) such that U

′

(x, 1|x) ≥ U

′

(¯x + η, 1|¯x + η) > 0 for all x ∈ [¯x + η, x

max

Now ﬁx η ∈ (0, x

max

− ¯x) such that

′

(x, 1|x) ≥ U

′

(¯x + η, 1|¯x + η) > 0 ∀x ∈ [¯x + η, x

max

]. (19)

For any ϵ > 0 small, then let δ(ϵ) be implicitly deﬁned by

+ϵ

u(θ, 1 − P (¯x + η|θ))π

′

(1|θ)p(¯x + η|θ)dF (θ) =

−δ

u(θ, 1 − P (¯x + η|θ))(1 − π

′

(1|θ))p(¯x + η|θ)dF (θ).

(20)

Observe that, for ϵ > 0 small, δ(ϵ) is also small, and such that

+ ϵ < θ

− δ(ϵ). (21)

Also note that u(θ, 1 − P (¯x + η|θ)) < 0 for all θ ∈ [θ

, θ

]. This follows from the fact that

u(θ, 1 − P (¯x + η|θ)) > 0 only for θ ≥ θ

(¯x + η) > θ

(¯x) > θ

Consider the policy Γ

ϵ,η

= {{0, 1}, π

ϵ,η

} deﬁned by the following properties: (a) π

ϵ,η

(1|θ) =

′

(1|θ) for all θ /∈ {[θ

, θ

+ ϵ] ∪ [θ

− δ(ϵ), θ

]}; (b) π

ϵ,η

(1|θ) = 0 for all θ ∈ [θ

, θ

+ ϵ]; and

ϵ,η

(1|θ) = 1 for all θ ∈ [θ

− δ(ϵ), θ

]. Because, for any x ≥ ¯x, (¯x, 1) are mutually

consistent under Γ

′

, they are also mutually consistent under Γ

ϵ,η

. This follows from the fact

that π

ϵ,η

(1|θ) = 1 for F -almost all θ > θ

along with the fact that sup Θ (x) > θ

for all

x ≥ ¯x, as shown above. Hence, U

ϵ,η

(x, 1|x) is well-deﬁned for all x ≥ ¯x. Also observe that

ϵ,η

(¯x+η,1) need not coincide with p

′

(¯x + η, 1). However, Condition (20) implies that

ϵ,η

(¯x + η, 1|¯x + η)

sgn

= U

′

(¯x + η, 1|¯x + η) > 0.

We now show, for any η ∈ (0, x

max

− ¯x) satisfying Condition (19), ϵ > 0 satisfying Condi-

tion (21), and x such that (x, 1) are mutually consistent under Γ

ϵ,η

, U

ϵ,η

(x, 1|x) ≥ 0. Recall

that, by the deﬁnition of x

max

, for all x > x

max

, U

(x, 1|x) is well-deﬁned and strictly positive,

implying that

+∞

−∞

u(θ, 1 − P (x|θ))π

(1|θ)p(x|θ)dF (θ) > 0.

Also recall that, for any x, the payoﬀ diﬀerential u(θ, 1 − P (x|θ)) is negative for θ < θ

(x)

and positive for θ > θ

(x), and that, for any x > x

max

, θ

(x) > θ

max

) > θ

(¯x) > θ

Because the policy Γ

ϵ,η

is such that π

ϵ,η

(1|θ) = π

(1|θ) for all θ > θ

and π

ϵ,η

(1|θ) ≤ π

(1|θ)

for all θ < θ

, with the inequality strict over a set of strictly positive measure under F , we

have that

+∞

−∞

u(θ, 1 − P (x|θ))π

ϵ,η

(1|θ)p(x|θ)dF (θ) >

+∞

−∞

u(θ, 1 − P (x|θ))π

(1|θ)p(x|θ)dF (θ).

Because the right-hand-side is strictly positive, for any x > x

max

, U

ϵ,η

(x, 1|x) > 0.

Next, for any θ ∈ [θ,

θ], let x

∗

(θ) be the signal threshold such that, when all agents

invest for x > x

∗

(θ) and refrain from investing for x < x

∗

(θ), the expected payoﬀ diﬀerential



θ, 1 − P



∗

(θ)|



is positive if and only if the fundamentals

θ are above θ. Observe

that, for any θ ∈ [θ,

θ], the existence of such a threshold follows from Condition FB, and that

∗

(θ) = −∞ and x

∗

(

θ) = +∞. Clearly, for any η ∈ (0, x

max

− ¯x) satisfying Condition (19),

ϵ > 0 satisfying Condition (21), and x ≤ x

∗

(θ

+ ϵ),

+∞

u (θ, 1 − P (x|θ)) p (x|θ) π

ϵ,η

(1|θ) dF (θ) > 0.

This is because, for any x ≤ x

∗

(θ

+ ϵ), θ

(x) ≤ θ

+ ϵ. The result then follows from the fact

that, for any x ≤ x

∗

(θ

+ ϵ), π

ϵ,η

(1|θ) = 0 for all θ ≤ θ

(x). Hence, for any x ≤ x

∗

(θ

+ ϵ)

such that (x, 1) are mutually consistent under Γ

ϵ,η

, U

ϵ,η

(x, 1|x) ≥ 0.

Next, observe that, for any x ∈ (x

∗

(θ

+ ϵ), x

∗

(θ

− δ (ϵ))],

+∞

u (θ, 1 − P (x|θ)) p (x|θ) π

ϵ,η

(1|θ) dF (θ) =

+∞

u (θ, 1 − P (x|θ)) p (x|θ) π

′

(1|θ) dF (θ)

−

+ϵ

u (θ, 1 − P (x|θ)) p (x|θ) π

′

(1|θ) dF (θ)

−δ(ϵ)

u (θ, 1 − P (x|θ)) p (x|θ) (1 − π

′

(1|θ)) dF (θ) ≥ 0.

To understand the inequality, ﬁrst observe that the ﬁrst integral on the right-hand side of the

equality is non-negative (if it was strictly negative then (x, 1) would be mutually consistent

under Γ

′

and U

′

(x, 1|x) < 0, which is inconsistent with the fact that Γ

′

∈ G). Second, observe

that the integrand function in the second integral on the right-hand side of the equality is

non-positive (this follows from the fact that, when x > x

∗

(θ

+ ϵ), u (θ, 1 − P (x|θ)) ≤ 0

for all θ ≤ θ

+ ϵ). Finally, the integrand function in the third integral on the right-hand

side of the equality is non-negative (this follows from the fact that, when x < x

∗

(θ

− δ(ϵ)),

u (θ, 1 − P (x|θ)) ≥ 0 for all θ > θ

− δ(ϵ)). Hence, for any such x, if (x, 1) are mutually

consistent under Γ

ϵ,η

, it must be that U

ϵ,η

(x, 1|x) ≥ 0.

Next, consider x ∈ (x

∗

(θ

− δ (ϵ)) , x

∗

(θ

)). For any x, let

∆S(x) ≡

+∞

u(θ, 1 − P (x|θ))p(x|θ)(π

ϵ,η

(1|θ) − π

′

(1|θ))dF (θ),

and, for any (x, θ), let q (θ, x) ≡ |u (θ, 1 − P (x|θ))| p (x|θ) . Note that, for any

x ∈ (x

∗

(θ

− δ (ϵ)) , x

∗

(θ

)) ,

(x) ∈ (θ

− δ (ϵ) , θ

), and

∆S(x) =

−δ(ϵ)

−u (θ, 1 − P (x|θ)) p (x|θ)



′

(1|θ) − π

ϵ,η

(1|θ)



dF (θ)

(x)

−δ(ϵ)

−u (θ, 1 − P (x|θ)) p (x|θ)



′

(1|θ) − π

ϵ,η

(1|θ)



dF (θ)

(x)

−u (θ, 1 − P (x|θ)) p (x|θ)



′

(1|θ) − π

ϵ,η

(1|θ)



dF (θ)

≥

−δ(ϵ)

q (θ, x)

q (θ, ¯x + η)



′

(1|θ) − π

ϵ,η

(1|θ)



dF (θ)

(x)

−δ(ϵ)

q (θ, x)

q (θ, ¯x + η)



′

(1|θ) − π

ϵ,η

(1|θ)



dF (θ)

q(θ

− δ(ϵ), x)

q(θ

− δ(ϵ), ¯x + η)

(x)

q (θ, ¯x + η)



′

(1|θ) − π

ϵ,η

(1|θ)



dF (θ)

≥

q(θ

− δ(ϵ), x)

q(θ

− δ(ϵ), ¯x + η)

q (θ, ¯x + η)



′

(1|θ) − π

ϵ,η

(1|θ)



dF (θ)

q(θ

− δ(ϵ), x)

q(θ

− δ(ϵ), ¯x + η)

∆S(¯x + η) = 0.

The ﬁrst equality follows from the deﬁnition of the ∆S(x) function. The ﬁrst inequality

follows from the fact that (i) for any θ ≤ θ

(x), u (θ, 1 − P (x|θ)) < 0, whereas for any

θ > θ

(x), u (θ, 1 − P (x|θ)) > 0, and (ii) for θ ∈ [θ

(x), θ

], π

′

(1|θ) ≤ π

ϵ,η

(1|θ). Together,

these properties imply that

(x)

−u (θ, 1 − P (x|θ)) p (x|θ)



′

(1|θ) − π

ϵ,η

(1|θ)



dF (θ)

≥ 0 ≥

q(θ

− δ(ϵ), x)

q(θ

− δ(ϵ), ¯x + η)

(x)

q (θ, ¯x + η)



′

(1|θ) − π

ϵ,η

(1|θ)



dF (θ) .

The second inequality follows from the fact that, π

′

(1|θ) − π

ϵ,η

(1|θ) turns from positive to

negative at θ = θ

− δ(ϵ) ≤ θ

(x), along with the fact that, for any θ ∈ [θ

, θ

(x)], the

function q(θ, x)/q(θ, ¯x + η) is non-increasing in θ as implied by the log-supermodularity of

|u (θ, 1 − P (x|θ))| p (x|θ) over {(θ, x) ∈ [0, 1] × R : u(θ, 1 − P (x|θ)) ≤ 0}, which in turn fol-

lows from Property (3*) of Condition M* and the assumption that p (x|θ) is log-supermodular.

The second equality follows from the fact that θ

(¯x + η) > θ

(¯x) > θ

, which implies that

u(θ, 1 − P (¯x + η|θ)) ≤ 0 for all θ ≤ θ

. Finally, the last equality follows from the fact that, by

construction of the policy Γ

ϵ,η

, ∆S(¯x + η) = 0. Hence, for any x ∈ (x

∗

(θ

− δ (ϵ)) , x

∗

(θ

)),

∆S(x) ≥ 0, which implies that, for any x in this range such that (x, 1) are mutually consistent

under Γ

ϵ,η

, U

ϵ,η

(x, 1|x) ≥ 0.

Similar arguments imply that, for any x ∈ [x

∗

(θ

), x + η],

∆S(x) =

−u (θ, 1 − P (x|θ)) p (x|θ) (π

′

(1|θ) − π

ϵ,η

(1|θ)) dF (θ)

q(θ,x)

q(θ,¯x+η)

q (θ, ¯x + η) (π

′

(1|θ) − π

ϵ,η

(1|θ)) dF (θ) ≥

q(θ

−δ(ϵ),x)

q(θ

−δ(ϵ),¯x+η)

∆S(¯x + η) = 0,

implying that, for such x too, if (x, 1) are mutually consistent under Γ

ϵ,η

, then U

ϵ,η

(x, 1|x) ≥ 0

(this result also follows from Property (3*) of Condition M* along with the log-supermodularity

of p (x|θ)).

Thus far, we have established that, for any x ∈ (−∞, x + η) ∪ (x

max

, +∞) such that (x, 1)

are mutually consistent under Γ

ϵ,η

, U

ϵ,η

(x, 1|x) ≥ 0. Below we show that there exists a ¯ϵ > 0

such that, for any η ∈ (0, x

max

− ¯x) satisfying Condition (19), and ϵ > 0 satisfying Condition

(21), with ϵ ∈ [0, ¯ϵ], the same is true also for any x ∈ [¯x+η, x

max

] such that (x, 1) are mutually

consistent under Γ

ϵ,η

. For any x, let

ϵ,η

(x) ≡

+∞

u (θ, 1 − P (x|θ)) p (x|θ) π

ϵ,η

(1|θ) dF (θ).

Note that, for any η, the function family



ϵ,η

(·)



is continuous in ϵ in the sup-norm, in a

neighborhood of 0. That is, for any z > 0, there exists κ > 0 such that, for any 0 < ϵ < κ,

and all x, |S

ϵ,η

(x) − S

0,η

(x)| ≤ z, where Γ

0,η

= Γ

′

By Condition (19), U

′

(x, 1|x) is

bounded away from zero over [¯x + η, x

max

]. Hence, there exists ¯ϵ > 0 small such that, for any

η ∈ (0, x

max

− ¯x) satisfying Condition (19), and ϵ ∈ [0, ¯ϵ], Condition (21) holds and, for any

This follows from the fact that |u(θ, A)| and p(x|θ) are both bounded.

x ∈ [¯x + η, x

max

], U

ϵ,η

(x, 1|x) > 0

Together, the results above thus imply that, for any η ∈ (0, x

max

− ¯x) satisfying Condition

(19), and ϵ ∈ [0, ¯ϵ], the policy Γ

ϵ,η

∈ G.

We now show that, when property (2*) in Condition M* holds, for any η ∈ (0, x

max

− ¯x)

satisfying Condition (19), and ϵ ∈ [0, ¯ϵ], the new policy Γ

ϵ,η

yields the policy maker an expected

payoﬀ strictly higher than Γ

′

. To see this, observe that, the policy maker’s payoﬀ under any

such policy is equal to

[Γ

ϵ,η

] =

+ϵ

−∞

(θ, 0)dF (θ) +

−δ(ϵ)

(θ, 1)dF (θ)

(θ

+ϵ,θ

−δ(ϵ))∪(θ

,+∞)



′

(1|θ)U

(θ, 1) + (1 − π

′

(1|θ))U

(θ, 0)



dF (θ) .

Diﬀerentiating U

[Γ

ϵ,η

] with respect to ϵ, and using the implicit function theorem to obtain

the derivative of δ(ϵ), we have that

[Γ

ϵ,η

]

dϵ

= f(θ

− δ)(1 − π

′

(1|θ

− δ))



(θ

− δ, 1) − U

(θ

− δ, 0)



× δ

′

(ϵ)

−f(θ

+ ϵ)π

′

(1|θ

+ ϵ)



(θ

+ ϵ, 1) − U

(θ

+ ϵ, 0)



= f(θ

+ ϵ)π

′

(1|θ

+ ϵ)



(θ

− δ, 1) − U

(θ

− δ, 0)



p(¯x+η|θ

+ϵ)u(θ

+ϵ,1−P (¯x+η|θ

+ϵ))

p(¯x+η|θ

−δ)u(θ

−δ,1−P (¯x+η|θ

−δ))

−f(θ

+ ϵ)π

′

(1|θ

+ ϵ)



(θ

+ ϵ, 1) − U

(θ

+ ϵ, 0)



Property (2*) in Condition M*, together with the fact that ¯x ≤ x

max

, guarantee that, for any

η ∈ (0, x

max

− ¯x) satisfying Condition (19), and ϵ ∈ (0, ¯ϵ], dU

[Γ

ϵ,η

]/dϵ > 0. We conclude that

the policy Γ

¯ϵ,η

∈ G yields the policy maker a payoﬀ strictly higher than Γ

′

. This completes

the proof of Claim S1-B. □

Claim C. Take any Γ

′

= ({0, 1} , π

′

) ∈ G such that X

′

̸= ∅ and



θ ∈ (θ

, θ

) : π

′

(1|θ) > 0



has zero F -measure. (22)

Then, π

′

(1|θ) = 0 for F -almost all θ ≤ θ

∗

and π

′

(1|θ) = 1 for F -almost all θ > θ

∗

Claim C says that, if Γ

′

∈ G is a deterministic monotone rule, and there exists a ¯x such

that U

′

(¯x, 1|¯x) = 0, then Γ

′

diﬀers from the optimal monotone rule Γ

∗

over at most a set of

fundamentals of zero F -measure.

Proof of Claim C. Let Γ

= ({0, 1} , π

) be the deterministic monotone policy with

cut-oﬀ θ

. Clearly, any x such that (x, 1) are mutually consistent under Γ

′

is such that

Recall that, as established above, for any x ≥ ¯x + η, (x, 1) are mutually consistent under Γ

ϵ,η

(x, 1) are also mutually consistent under Γ

. Furthermore, for any such x, U

′

(x, 1|x) =

(x, 1|x) (both properties follow because the two policies diﬀer only over sets of zero

F -measure).

Suppose that θ

> θ

∗

. Below we establish that, in this case, any x such that (x, 1) are

mutually consistent under Γ

is such that U

(x, 1|x) > 0. Clearly, for any x such that

(a) θ

(x) ≤ θ

, and (b) (x, 1) are mutually consistent under Γ

, U

(x, 1|x) > 0. Thus,

consider any x such that θ

(x) > θ

, and (b) (x, 1) are mutually consistent under Γ

. Note

ﬁrst that, for any such x, (x, 1) are mutually consistent also under Γ

∗

= ({0, 1} , π

∗

). This

is because π

∗

(1|θ) ≥ π

(1|θ) for all θ. Furthermore, for any such x,

+∞

u(θ, 1 − P (x|θ))p(x|θ)dF (θ) ≥

+∞

∗

u(θ, 1 − P (x|θ))p(x|θ)dF (θ) . (23)

This follows from the fact that u(θ, 1 − P (x|θ)) < 0 for all θ ∈ [θ

∗

, θ

]. Hence, for any such

x, because U

∗

(x, 1|x) ≥ 0, U

(x, 1|x) ≥ 0. Now take x = ¯x and recall that, by deﬁnition,

′

(¯x, 1|¯x) = 0. Because U

′

(¯x, 1|¯x) = U

(¯x, 1|¯x), this means that U

(¯x, 1|¯x) = 0.

Property (1*) of Condition M* then implies that inf Θ (¯x) < θ. Hence, for x = ¯x, the inequality

in (23) is strict, which in turn implies that U

∗

(¯x, 1|¯x) < 0, contradicting the assumption

that Γ

∗

∈ G. Therefore, it must be that θ

≤ θ

∗

. However, by deﬁnition of θ

∗

, if θ

< θ

∗

there exists an x such that (a) U

(x, 1|x) < 0, and (b) (x, 1) are mutually consistent under

. Because, for all such x, U

′

(x, 1|x) is well-deﬁned (i.e., (x, 1) are mutually consistent

also under Γ

′

) and U

(x, 1|x) = U

′

(x, 1|x), we thus have that U

′

(x, 1|x) < 0, which

contradicts the assumption that Γ

′

∈ G. Hence θ

= θ

∗

. This completes the proof of Claim

C. □

Step 2. Step 1 implies that arg max

Γ∈G

[

Γ] ̸= ∅ and that any Γ

∗

= ({0, 1} , π) with

∗

∈ arg max

Γ∈G

[

Γ] is such that π(1|θ) = 0 for F -almost all θ ≤ θ

∗

and π(1|θ) = 1 for

F -almost all θ > θ

∗

. The result in the theorem then follows from observing that, given any

∗

∈ arg max

Γ∈G

[

Γ], there exists a nearby deterministic monotone policy Γ

∈ G with

cut-oﬀ

θ = θ

∗

+ ˜ε, for ˜ε > 0 small, such that Γ

satisﬁes the perfect-coordination property

(i.e., U

(x, 1|x) > 0 all x such that (x, 1) are mutually consistent under Γ

The continuity

of U

[Γ

] in

θ then implies that, for ˜ε > 0 small, U

[Γ

] > U

[Γ], thus establishing the result

The arguments are the same as those used in the proof of Claim C for the case θ

> θ

∗

in the theorem. Q.E.D.

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