# Coalition-Proof Disclosure

Germán Gieczewski, Maria Titova

PDF: https://maria-titova.com/papers/CPD.pdf

## Abstract

*Acknowledgements:* We thank Nageeb Ali, Aislinn Bohren, Steve Callander, Jeff Ely, Johannes Hörner, Navin Kartik, Elliot Lipnowski, Jacopo Perego, and the audience members at the University of Pennsylvania, Penn State, QMUL-City Theory Workshop, Berlin Verifiability in Mechanism and Information Design Workshop, Gerzensee and MPSA. All errors are our own.

We study generalized verifiable disclosure by an informed sender with type-independent preferences. We first characterize all perfect Bayesian equilibria as individually rational partitions of the sender's types into coalitions ordered by payoff. An equilibrium survives our coalition-proofness criterion if no group of types can profitably coordinate an announcement that the receiver will interpret as coming precisely from the types willing to join it. Coalition-proof equilibria are partitions that are greedy: at each step, a coalition of the remaining types attains the highest payoff consistent with equilibrium. Given the generality of our setting, coalition-proof equilibria may fail to exist. We establish existence under restrictions on preferences and evidence: quasiconcavity with message completeness, betweenness, cheap talk with message completeness, and payoff degradation. Finally, we solve a benchmark in which evidence can identify arbitrary fractions of types. Its selected value is a piecewise linear tent over the belief simplex, providing a no-commitment disclosure counterpart to the values of Bayesian persuasion and cheap talk.

## 1 Introduction

Games of verifiable disclosure model communication by an informed sender who chooses which message to send from a set that depends on his private information. Applications include sellers supporting a claim about product quality with a warranty, firms deciding which audited results to report to investors, lawyers choosing which records to present in court, and job applicants documenting qualifications for employers.

The canonical models (Grossman, 1981; Milgrom, 1981) deliver the classic unraveling result: skepticism toward silence forces full revelation. However, richer disclosure games in which only partial certification is possible or evidence is coarse have also been studied extensively in the literature (Ben-Porath, Dekel, and Lipman, 2019; Glazer and Rubinstein, 2004; Hagenbach, Koessler, and Perez-Richet, 2014; Hart, Kremer, and Perry, 2017; Sher, 2011). In these latter models, partial revelation in rich patterns is possible, but it typically comes with severe multiplicity of equilibria driven by how the receiver interprets messages nobody sends. Most of the literature responds by selecting receiver-optimal equilibria that often coincide with the receiver's commitment solution (Ben-Porath, Dekel, and Lipman, 2019; Glazer and Rubinstein, 2004; Hart, Kremer, and Perry, 2017).

This paper takes a different approach. We study disclosure games under one substantive assumption: the sender’s preferences are type-independent. The setup is otherwise general: the sender’s payoff depends on the receiver’s posterior belief through a correspondence, and an arbitrary message mapping assigns each type a set of available messages. Rather than selecting equilibria on the receiver’s behalf, we ask which equilibria survive credible collective deviations by the informed party. A perfect Bayesian equilibrium (PBE) fails our test of *coalition proofness* if some group of sender types can announce a joint deviation (possibly mixed, possibly spread over several messages, possibly by reusing messages already sent on path) such that, when the receiver correctly anticipates exactly which types join the announcement, every participant gains. In the spirit of neologism proofness for cheap-talk games, the selection requires that credible coalitional deviations be correctly interpreted (Farrell, 1993).

We begin by characterizing all PBE. We show that all PBE strategies, coalition-proof or not, are *partition strategies*: they partition the type space into *coalitions* ordered by equilibrium payoff from high to low. Each coalition bundles a set of types, evidence that only they can produce, and a common payoff. Equilibrium analysis thus reduces to a recursive partition problem implemented by an algorithm, which at each step removes one coalition from the game until no types are left. Coalition-proof PBE are exactly the outputs of the *greedy* version of this algorithm in which, generally speaking, the coalition that attains the largest available payoff is removed at each step.

In this general setting coalition-proof PBE may fail to exist for the same reason that neologism-proof equilibria may fail to exist in cheap-talk games. We thus give sufficient conditions for existence. If the message mapping is *complete*, in the sense that the available evidence is closed under unions (Bertomeu and Cianciaruso, 2018), quasiconcavity of the upper envelope suffices. *Betweenness* (Hart, Kremer, and Perry, 2017) of a single-valued payoff guarantees existence without any restrictions on the message mapping. Under either of these two conditions, if the game is generic, in the sense that distinct pooling groups earn distinct payoffs, the coalition-proof PBE partition is essentially unique. With cheap talk also available, completeness suffices without quasiconcavity. Finally, *payoff degradation* guarantees existence whenever any coalition can be rebuilt with the same members and any lower payoff down to a common floor. This property holds under *free disposal* of the sender’s payoff, with no restriction on messages, or under cheap talk and *revelation aversion*, a property indicating that the sender’s worst outcome is having his type revealed.

Finally, we solve the model with a maximally rich message space built on the stochastic message-mapping game of Titova and Zhang (2025), where any combination of fractions of types can pool while excluding all others. There, coalition-proof PBE take a simple recursive form: the top coalition assembles the belief that maximizes the sender’s payoff on the current support, and the recursion proceeds down the type space. The sender’s ex-ante payoff is a piecewise linear function of the prior that we call the *tent*. The tent is the disclosure analogue of the concave closure of Bayesian persuasion (Kamenica and Gentzkow, 2011) and the quasiconcave envelope of cheap talk (Lipnowski and Ravid, 2020): it lies below the former and generally coincides with neither.

### Two Motivating Examples

The first example shows that receiver-optimal equilibria can rest on implausibly asymmetric interpretations of deviations wanted equally by all sender types.

**Example 1** (Implausible revelation in receiver-optimal equilibria). *A centrist incumbent (the sender) learns the state of the world  $\theta \in \{-1, 1\}$  drawn from a uniform prior; she can reveal it to the voter (the receiver) or not, choosing a message  $m \in \{\theta, \emptyset\}$. The voter sees  $m$  and elects a left-wing challenger ($a = -1$), a right-wing challenger ($a = 1$), or the incumbent ($a = S$). The incumbent is office-motivated and obtains  $\mathbb{1}_{a=S}$. The voter receives 1 if the elected challenger matches the state ($a = \theta$), 0 if not, and 0.9 from the safe option of reelecting the incumbent.*

[^1]: A refinement such as the intuitive criterion or D1 does not help here: the refinements give conditions under which to rule some types as deviators, but in the example just given, the types are equally plausible deviators.

This game has two kinds of PBE. In the first, both types send  $m = \emptyset$  and the incumbent is reelected. In the second, the incumbent is reelected with probability zero: the empty message is either off path and interpreted as coming disproportionately from one state, or it is sent disproportionately by one type on path. In any receiver-optimal equilibrium, the voter learns  $\theta$  with probability one, and the voter's belief after  $\emptyset$  places the probability at most at 0.1 or at least at 0.9 on  $\theta = 1$: the voter extracts full revelation by threatening to interpret the off-path message asymmetrically even though the two types have identical incentives to send it. For the incumbent, revealing the state is weakly dominated by silence: anything she says can be used against her. She could announce, "It is in my interest to reveal nothing, and I benefit from this announcement regardless of the state; so hearing nothing, keep your prior." Both types strictly benefit from making this announcement, and under the correct interpretation both gain.[^1] Coalition proofness eliminates every equilibrium except pooling on silence.

One may conjecture that limiting the receiver's ability to interpret deviations adversarially simply shifts surplus to the sender such that selecting ex-ante sender-optimal equilibria will achieve the same outcome. The second example shows it will not: coalition proofness is an interim concept, and profitable interim deviations can destroy ex-ante optimal pooling.

**Example 2** (Implausible ex-ante sender-optimal equilibria). *Consider a Grossman (1981) game with a nuisance dimension. Let  $\Theta = \{(L, A), (L, B), (H, A), (H, B)\}$  with a uniform prior. Type  $\theta$  can send any message  $m \in 2^\Theta$  with  $\theta \in m$. The receiver's best response to belief  $\mu$  is  $a^*(\mu) = \sqrt{\Pr_\mu(\theta(1) = H)} - 8[\Pr_\mu(\theta(2) = A) - 0.5]^2$, and the sender's payoff is  $a$.*

Revealing  $\theta(1) = H$  is good for the sender; revealing anything about the nuisance dimension  $\theta(2)$  hurts him. The receiver-optimal PBE outcome is full revelation supported by adversarial off-path beliefs. The ex-ante sender-optimal PBE has all types pool on  $m = \Theta$  with payoff  $\sqrt{0.5}$, supported by the same kind of asymmetric threat: off path, the receiver attributes the message  $\{(H, A), (H, B)\}$  disproportionately to a single type. But the high types can announce exactly that message, evidence only they can produce, and both gain under the correct interpretation, meaning that the pooling equilibrium is not coalition-proof. In the coalition-proof prediction, the high types separate along the payoff-relevant dimension and pool along the nuisance dimension, mirroring the unique equilibrium of the game with the nuisance dimension removed. When the sender wants to reveal some dimensions of the state and conceal others, coalition proofness delivers precisely that outcome.

## Related Literature

Our selection concept belongs to the family of belief-based refinements built on credible deviations: announcement proofness (Matthews, Okuno-Fujiwara, and Postlewaite, 1991), neologism proofness (Farrell, 1993), the closely related equilibria of Grossman and Perry (1986), and undefeated equilibrium (Mailath, Okuno-Fujiwara, and Postlewaite, 1993). Aybas and Callander (2024) apply announcement proofness to select the sender-optimal equilibrium of cheap talk in complex environments. These refinements allow deviations only to a single off-path message; our blocking coalitions may mix over several messages and use messages already on the equilibrium path, which is what makes the greedy characterization exact and existence obtainable under weaker conditions.

Bertomeu and Cianciaruso (2018) adapt neologism proofness to disclosure games. A Grossman–Perry–Farrell equilibrium (GPFE) is a pure-strategy PBE that admits no single-message self-signaling deviation. We allow mixed strategies, including the mixing that arises in truth-leaning equilibria. Their priority algorithm is analogous to our Algorithm 3, but its termination is only necessary for a GPFE to exist. Under betweenness, a coalition-proof PBE exists without further restrictions on evidence (Theorem 2). In their Example 7, the priority algorithm does not terminate and no GPFE exists, though a coalition-proof PBE does. Their model also allows message-dependent payoffs through disclosure costs or lemons discounts, which our belief-based payoff cannot represent.

Koessler and Skreta (2023) study information design by a privately informed designer and propose *interim optimality* as a selection criterion. Their single-receiver setting with type-independent preferences is comparable to our rich disclosure benchmark (Section 7), as both permit arbitrary splittings of the prior. The timing of deviations differs, however: their designer evaluates a mechanism before its random message is realized, whereas in our rich game a sender observes his payoff-irrelevant label and knows whether he is included in a coalition. Their mechanisms let types commit to mixing over messages that yield different payoffs; sender mixing in our equilibria requires indifference.

A second strand, closer to our approach, characterizes equilibria of disclosure games directly. Rappoport (2025) characterizes receiver-optimal equilibria. Wu (2022) characterizes all PBE under the assumptions that each piece of evidence has cheap-talk copies and any two pieces of evidence can be jointly presented. Our Proposition 1 characterizes all PBE for an arbitrary message mapping.

Our paper also contributes to the literature on disclosure with structured evidence. Hart, Kremer, and Perry (2017) study evidence games, which are disclosure games with a particular evidence structure and sender payoff. They propose an equilibrium refinement of truth leaning and show that it selects the receiver’s commitment outcome. Their key payoff condition, betweenness, is one of our existence conditions. We show in Section 6.2.1 that in evidence games every truth-leaning equilibrium outcome is also a coalition-proof PBE outcome. Thus, coalition proofness provides an alternative sender-side foundation for focusing on receiver-optimal outcomes in these games. Glazer and Rubinstein (2004), Sher (2011, 2014), and Ben-Porath, Dekel, and Lipman (2019) develop the commitment side of persuasion with evidence, and Hagenbach, Koessler, and Perez-Richet (2014) study disclosure among many players. Callander, Lambert, and Matouschek (2021) select the sender-optimal equilibrium in a verifiable-disclosure model of expert advice, and Farina et al. (2026) test selective disclosure experimentally.

Finally, our benchmarks connect disclosure to the values of neighboring communication models. With commitment, the sender’s value is the concave closure of Kamenica and Gentzkow (2011); with pure cheap talk, it is the quasiconcave envelope of Lipnowski and Ravid (2020), which coalition proofness selects as the unique prediction in the cheap-talk corner of our model. The tent of the rich-message benchmark (Titova and Zhang, 2025) is the disclosure counterpart of these objects.

The rest of the paper proceeds as follows. Section 2 introduces disclosure games; Section 3 develops coalitions and partitions. Section 4 characterizes PBE as individually rational partitions with non-increasing payoffs. Section 5 introduces coalition proofness and characterizes coalition-proof PBE as greedy partitions. Section 6 presents the existence results and shows that truth-leaning equilibria in evidence games are coalition-proof (Section 6.2.1). Section 7 solves the rich-message benchmark and constructs the tent. Section 8 concludes. Proofs are provided in the appendix.

## 2 Model

[^2]: For a finite set  $S$,  $\Delta S$  denotes the set of probability distributions over  $S$.

There are two players, a sender and a receiver. First, the sender learns his type  $\theta \in \Theta$  drawn from a common prior  $\mu^0 \in \Delta\Theta$.[^2] Then, he sends a message  $m \in \mathcal{M}$  from a non-empty set  $M(\theta)$, where  $M : \Theta \rightarrow 2^{\mathcal{M}} \setminus \{\emptyset\}$  is the message mapping. Finally, the receiver updates her belief and acts, and payoffs are realized. We employ the belief-based approach and abstract away from the receiver's action. The sender's payoff primitive is a correspondence  $V : \Delta\Theta \rightrightarrows \mathbb{R}$; its envelopes are  $\bar{v}(\mu) := \max V(\mu)$  and  $\underline{v}(\mu) := \min V(\mu)$. Throughout, we refer to the sender's payoff simply as "the payoff." In all sections except for Section 7, we maintain the following assumption:

**Assumption 1.** (A1)  $\Theta$  and  $\mathcal{M}$  are finite.

(A2)  $\mu^0$  has full support.

(A3)  $\mathcal{M} = \bigcup_{\theta \in \Theta} M(\theta)$.

(A4)  $V$  is upper hemicontinuous with non-empty compact interval values.

[^3]: For example, suppose the receiver's action space  $A$  is a non-empty compact Hausdorff space, the sender's payoff  $u_S : A \rightarrow \mathbb{R}$  is continuous, and for each  $\theta \in \Theta$, the receiver's payoff  $u_R(\cdot, \theta) : A \rightarrow \mathbb{R}$  is continuous. Then, among other properties,  $V$  is non-empty, compact-valued, convex-valued, and upper hemicontinuous (see Lemma J.1 in the online appendix).

Here, (A3) imposes that every message is feasible for at least one type. (A4) follows from Berge's maximum theorem under mild conditions.[^3] Under (A4), the upper envelope  $\bar{v}$  is upper semicontinuous, the lower envelope  $\underline{v}$  is lower semicontinuous, and  $V(\mu) = [\underline{v}(\mu), \bar{v}(\mu)]$.

Our main object of study is a disclosure game defined as follows:

**Definition 1.** A disclosure game  $G$  is a tuple  $(\Theta, \mathcal{M}, M, \mu^0, V)$  satisfying conditions (A1)–(A4).

**Strategies.** A sender strategy in game  $G$  is a map  $\sigma : \Theta \rightarrow \Delta\mathcal{M}$  such that  $\text{supp } \sigma(\cdot | \theta) \subseteq M(\theta)$. The *evidence* of  $\sigma$  is the set of messages it uses,

$$
X(\sigma) := \bigcup_{\theta \in \Theta} \text{supp } \sigma(\cdot | \theta).
$$

Message  $m$  is *on path* under  $\sigma$  if

$$
p_{\sigma}(m) := \sum_{\theta \in \Theta} \mu^0(\theta) \sigma(m | \theta) > 0;
$$

since  $\mu^0$  has full support by (A2), the set of on-path messages is exactly  $X(\sigma)$. For  $m \in X(\sigma)$, the induced belief  $\mu_{\sigma}(\cdot | m) \in \Delta\Theta$  is

$$
\mu_{\sigma}(\theta | m) := \frac{\mu^0(\theta) \sigma(m | \theta)}{p_{\sigma}(m)}.
$$

**Restricted games.** Let  $R$  be a non-empty subset of  $\Theta$. A restricted game on  $R$  is

$$
G|_R := (R, \mathcal{M}_R, M_R, \mu_R^0, V_R),
$$

where  $\mathcal{M}_R := \bigcup_{\theta \in R} M(\theta) \subseteq \mathcal{M}$  is the set of all messages available to types in  $R$;  $M_R(\theta) := M(\theta)$, treated as a subset of  $\mathcal{M}_R$;  $\mu_R^0 \in \Delta R$  is the conditional prior defined as

$$
\mu_R^0(\theta) := \frac{\mu^0(\theta)}{\mu^0(R)}, \text{ where } \mu^0(R) := \sum_{\theta \in R} \mu^0(\theta);
$$

[^4]: For finite sets  $S \subseteq S'$, let  $\iota_{S,S'} : \Delta S \rightarrow \Delta S'$  denote the zero-extension map from  $S$  to  $S'$: for each  $\nu \in \Delta S$  and  $x \in S'$,  $\iota_{S,S'}(\nu)(x) = \nu(x)$  if  $x \in S$  and  $\iota_{S,S'}(\nu)(x) = 0$  otherwise. Thus,  $V_R = V \circ \iota_{R,\Theta}$.

and  $V_R$  is the sender's payoff correspondence restricted to beliefs supported on  $R$.[^4]

Any restricted game  $G|_R$  is itself a disclosure game,  $G|_{\Theta} = G$, and restriction is transitive: for non-empty  $R' \subseteq R \subseteq \Theta$, we have  $(G|_R)|_{R'} = G|_{R'}$  (see Lemma J.2 in the online appendix). All the notions and arguments below are described for an arbitrary disclosure game  $G$  and therefore apply to every restricted game. When we consider a restricted game  $G|_R$, we add the subscript  $R$.

**Useful notation.** Write  $n := |\Theta|$  for the number of types and  $\delta_{\theta} \in \Delta \Theta$  for the point mass on  $\theta$. Given a non-empty set of types  $R \subseteq \Theta$  and messages  $X \subseteq \mathcal{M}$, the set of types in  $R$  that can send at least one message in  $X$  is the preimage of  $X$  relative to  $R$,

$$
M_R^{-1}(X) := \{\theta \in R \mid M(\theta) \cap X \neq \emptyset\},
$$

which we abbreviate as  $M^{-1}(X)$  when there is no possibility of confusion. We write

$$
P(m) := M^{-1}(\{m\})
$$

for the set of types that can send message  $m$. Note that  $P(m)$  is non-empty for each  $m \in \mathcal{M}$  by (A3). Since  $\mu^0$  has full support,  $\mu_R^0$  is supported exactly on  $R$. We identify distributions on  $R$, including posterior beliefs induced in  $G|_R$, with their zero extensions to  $\Delta \Theta$  whenever they are used in the original game; similarly, we identify distributions on  $\mathcal{M}_R$  with their zero extensions to  $\Delta \mathcal{M}$.

## 3 Coalitions and Partitions

Fix a disclosure game  $G = (\Theta, \mathcal{M}, M, \mu^0, V)$  and a non-empty set of types  $C \subseteq \Theta$. A *coalition strategy on  $C$*  is a sender strategy  $\sigma$  of the restricted game  $G|_C$. The posterior beliefs induced by  $\sigma$  are the beliefs  $\mu_{\sigma}(\cdot \mid m)$  already defined for strategies, which are computed in the restricted game  $G|_C$  and zero-extended to  $\Delta \Theta$  when used in the original game. Since game restriction is transitive, these beliefs depend only on  $C$  and  $\sigma$, not on which larger (restricted) game contains  $C$.

**Definition 2.** A coalition of a disclosure game  $G$  is a triple  $(C, \sigma, w)$  such that

- (C1)  $C \subseteq \Theta$  is non-empty;
- (C2)  $\sigma$  is a coalition strategy on  $C$;
- (C3) the evidence of  $\sigma$  is exclusive to  $C$:

$$
M^{-1}(X(\sigma)) \subseteq C;
$$

- (C4) every message in  $X(\sigma)$  yields the same sender payoff:

$$
w \in V(\mu_{\sigma}(\cdot \mid m)) \text{ for every } m \in X(\sigma).
$$

A coalition consists of the participating types, their strategy, and their common payoff. The coalition payoff is constant across all messages in the coalition's evidence and hence across messages sent by each coalition type. Note that the reverse inclusion  $C \subseteq M^{-1}(X(\sigma))$  holds automatically since each  $\theta \in C$  uses messages from  $M(\theta) \cap X(\sigma)$; consequently, condition (C3) becomes  $M^{-1}(X(\sigma)) = C$  and states that the types participating in the coalition are exactly the types that can produce the coalition's evidence. The beliefs in (C4) are well-defined because every  $m \in X(\sigma)$  is on path.

**Remark 1.** Fix a disclosure game  $G$  and a message  $m \in \mathcal{M}$, and let  $\sigma^m$  be a coalition strategy on  $P(m)$  under which every type that has access to message  $m$  sends it with probability one. Then for every  $w \in V(\mu_{P(m)}^0)$, the triple  $(P(m), \sigma^m, w)$  is a coalition of  $G$. In particular, every disclosure game admits a coalition.

*Proof.*  $P(m)$  is non-empty by (A3); in addition,  $m \in M(\theta)$  for each  $\theta \in P(m)$, so  $\sigma^m$  is a coalition strategy in  $G|_{P(m)}$  with  $X(\sigma^m) = \{m\}$. Next,  $M^{-1}(X(\sigma^m)) = P(m)$  by the definition of  $P(m)$. Finally, the belief induced by  $\sigma^m$  is  $\mu_{P(m)}^0$  by Bayes' rule. Consequently,  $(P(m), \sigma^m, w)$  is a coalition for every  $w \in V(\mu_{P(m)}^0)$.  $\square$

**Definition 3.** A partition of a disclosure game  $G$  is a finite sequence

$$
\Pi = \{(C_t, \sigma_t, w_t)\}_{t=1}^T
$$

such that  $C_1, \dots, C_T$  are pairwise disjoint, non-empty, and  $\Theta = \bigcup_{t=1}^T C_t$, and for each  $t$, the triple  $(C_t, \sigma_t, w_t)$  is a coalition in the residual game  $G_t := G|_{R_t}$, where  $R_t := \bigcup_{s=t}^T C_s$.

We refer to  $t$  as the *step* and  $C_t$  as the *cell*, and we write  $X_t := X(\sigma_t)$  for the evidence used at step  $t$. For a partition  $\Pi$, we call  $R_t$  the *residual set* at step  $t$. More generally, for any object  $Y_R$  attached to the restricted game  $G|_R$, we abbreviate  $Y_{R_t}$  as  $Y_t$; thus  $G_t = G|_{R_t}$,  $\mathcal{M}_t = \mathcal{M}_{R_t}$, and so on.

All partitions arise from the recursion formalized by the following algorithm:

---

**Algorithm 1:** Partition Algorithm

```

Let  $t := 1$  and  $R_1 := \Theta$;
while  $R_t \neq \emptyset$ 
|   select a coalition  $(C_t, \sigma_t, w_t)$  of  $G_t$;
|   let  $R_{t+1} := R_t \setminus C_t$  and  $t := t + 1$;
end

```

---

A *run* of the algorithm is one of its possible executions. It has one selection per step. The algorithm *terminates* when the type space is exhausted; its *output* is the selected sequence  $\{(C_t, \sigma_t, w_t)\}_{t=1}^T$. At every step of every run, a selection exists by Remark 1, each step removes a non-empty cell, and each output satisfies the definition of a partition.

**Remark 2.** Every disclosure game admits a partition, and every partition has at most  $n$  steps.

*Proof.* At each non-empty residual set, Remark 1 applied to the restricted game gives an available coalition. Each selected coalition has a non-empty cell, so each step removes at least one type. Since  $\Theta$  is finite, the recursion exhausts the type space after at most  $n$  steps, and the selected sequence is a partition by construction.  $\square$

By transitivity of game restriction  $G_t|_{C_t} = G|_{C_t}$, so  $\sigma_t$  is simply a coalition strategy on  $C_t$  with beliefs  $\mu_{\sigma_t}$  that do not depend on the larger game in which the coalition is viewed. Similarly, the common-payoff condition (C4) of the step  $t$  coalition is the same whether viewed in  $G_t|_{C_t}$,  $G|_{C_t}$, or  $G$. The only condition that uses the residual set  $R_t$  is exclusivity (C3): no type in  $R_t \setminus C_t$  can produce evidence in  $X_t$. However, types in  $C_1 \cup \dots \cup C_{t-1}$  may be able to produce such evidence.

**Definition 4.** Given a partition  $\Pi = \{(C_t, \sigma_t, w_t)\}_{t=1}^T$  of  $G$, its associated sender strategy  $\sigma^\Pi : \Theta \rightarrow \Delta \mathcal{M}$  is obtained by letting every  $\theta \in C_t$  use  $\sigma_t(\cdot | \theta)$  zero-extended to  $\Delta \mathcal{M}$. A sender strategy is a partition strategy if it is associated with some partition.

Note that  $\sigma^\Pi$  is indeed a sender strategy of  $G$: if  $\theta \in C_t$, then

$$
\text{supp } \sigma^\Pi(\cdot | \theta) = \text{supp } \sigma_t(\cdot | \theta) \subseteq M(\theta).
$$

Note, too, that distinct steps use disjoint evidence. If, on the contrary,  $m \in X_s \cap X_t$  with  $s < t$, then some type in  $C_t \subseteq R_s$  can send a message in  $X_s$  contradicting the exclusivity of step  $s$  coalition in  $G_s$. Thus, whenever  $m \in X_t$, only types in  $C_t$  send  $m$  under  $\sigma^\Pi$; in particular,  $m$  is on path.

## 4 Perfect Bayesian Equilibria

To define and characterize PBE of disclosure games, we introduce the concepts of feasible beliefs, skeptical payoffs, and individually rational partitions.

**Definition 5.** For message  $m \in \mathcal{M}$, the set of feasible beliefs is

$$
\mathcal{F}(m) := \{\mu_\sigma(\cdot | m) \mid \sigma \text{ is a sender strategy of } G \text{ with } m \in X(\sigma)\}.
$$

Given message  $m$, feasible beliefs are those that arise, via Bayes' rule, under some sender strategy that uses that message as evidence. In other words, these are the beliefs that are consistent with disclosure: if message  $m$  cannot be sent by some types, then all feasible beliefs have zero mass on those types; if some type can only send message  $m$, then, among other things, all feasible beliefs after that message have positive masses on that type. Each  $\mathcal{F}(m)$  is a non-empty compact convex polytope parametrized explicitly by the mixing weights of the types that can send  $m$  (see Lemma J.3 in the online appendix).

**Definition 6.** For a type  $\theta \in \Theta$, the skeptical payoff is

$$
\underline{u}(\theta) := \max_{m \in M(\theta)} \min_{\mu \in \mathcal{F}(m)} \underline{v}(\mu).
$$

[^5]: The maximum and minimum are attained because  $\underline{v}$  is lower semicontinuous and  $\mathcal{F}(m)$  is non-empty and compact (see Lemma J.3 in the online appendix).

The skeptical payoff is the payoff type  $\theta$  obtains from his best message when, at every message, the receiver holds the least favorable feasible belief and breaks ties against the sender.[^5] Throughout, the skeptical payoff  $\underline{u}(\theta)$  is the one computed in  $G$  itself. We say that a partition is individually rational if every type receives at least the skeptical payoff.

**Definition 7.** A partition  $\Pi = \{(C_t, \sigma_t, w_t)\}_{t=1}^T$  of  $G$  is individually rational (IR) if

$$
w_t \geq \underline{u}(\theta) \quad \text{for every } t \text{ and every } \theta \in C_t. \quad (\text{IR})
$$

Next we define the PBE strategy and equilibrium.

**Definition 8.** A sender strategy  $\sigma : \Theta \rightarrow \Delta \mathcal{M}$  is a PBE strategy if there exist a belief system  $\mu : \mathcal{M} \rightarrow \Delta \Theta$  and a sender payoff function  $r : \mathcal{M} \rightarrow \mathbb{R}$  such that

1. beliefs are feasible:

$$
\mu(\cdot | m) \in \mathcal{F}(m) \quad \text{for every } m \in \mathcal{M};
$$

2. beliefs are Bayesian on path:

$$
\mu(\cdot | m) = \mu_{\sigma}(\cdot | m) \quad \text{for every } m \in X(\sigma);
$$

3. the realized payoff is compatible with the payoff correspondence:

$$
r(m) \in V(\mu(\cdot | m)) \quad \text{for every } m \in \mathcal{M};
$$

4. the sender is sequentially rational:

$$
\text{supp } \sigma(\cdot | \theta) \subseteq \arg \max_{m \in M(\theta)} r(m) \quad \text{for every } \theta \in \Theta.
$$

A triple  $(\sigma, \mu, r)$  satisfying these four conditions is a PBE.

The first condition requires that even the off-path beliefs are feasible, that is, consistent with disclosure/rationalizable by some sender strategy. The function  $r$  incorporates the receiver's tie-breaking and mixing: after each message  $m$, the receiver selects the sender's payoff from the set  $V(\mu(\cdot | m))$. Every disclosure game admits a PBE (see Lemma J.4 in the online appendix).

If  $(\sigma, \mu, r)$  is a PBE, then sequential rationality makes  $r$  constant on  $\text{supp } \sigma(\cdot | \theta)$, equal to

$$
u(\theta) := \max_{m \in M(\theta)} r(m),
$$

which we call type  $\theta$ 's *equilibrium payoff* in that PBE.

The PBE characterization below rests on two simple observations. First, the partition strategy preserves the beliefs generated by its cells: if  $m \in X_t$, then under  $\sigma^{\Pi}$  only types in  $C_t$  send  $m$, so the Bayes posterior after  $m$  is the posterior generated by  $\sigma_t$  in  $G|_{C_t}$, viewed as a belief in  $\Delta \Theta$. Second, skeptical payoffs are the relevant off-path benchmark. In any PBE, type  $\theta$  receives at least  $\underline{u}(\theta)$; conversely, after any unused message, the receiver can select a feasible belief and a compatible payoff that give no type more than its skeptical payoff.

**Proposition 1.** A sender strategy  $\sigma$  is a PBE strategy if and only if  $\sigma$  is associated with an IR partition  $\Pi = \{(C_t, \sigma_t, w_t)\}_{t=1}^T$  with  $w_1 \geq \dots \geq w_T$. Furthermore, every PBE strategy is associated with an IR partition with strictly decreasing payoffs  $w_1 > \dots > w_T$.

Intuitively, in any PBE, we can group sender types by payoff in a strictly decreasing order and construct a coalition for each of those payoffs. Conversely, any IR partition with non-increasing payoffs can be supported as a PBE: on-path messages receive the cell payoff, while off-path messages are assigned skeptical feasible beliefs and payoffs. A type cannot profitably deviate off path because its cell payoff is at least its skeptical payoff; it cannot profitably deviate to earlier evidence because that evidence is unavailable in the relevant residual game; and it does not wish to deviate to later evidence because later cells receive weakly lower payoffs.

Call an IR partition with non-increasing payoffs a *PBE partition*. Algorithm 2 returns all PBE partitions: it is simply the partition algorithm that imposes PBE restrictions (i.e., IR and non-increasing payoffs) at each step.

---

**Algorithm 2:** PBE Partition Algorithm

Let  $t := 1$,  $R_1 := \Theta$, and  $w_0 := \infty$;  
**while**  $R_t \neq \emptyset$   
    | select a coalition  $(C_t, \sigma_t, w_t)$  of  $G_t$  with  $w_t \in [\max_{\theta \in R_t} \underline{u}(\theta), w_{t-1}]$;  
    | **halt** with no output if there is none;  
    | let  $R_{t+1} := R_t \setminus C_t$  and  $t := t + 1$;  
**end**

---

Note that Algorithm 2 uses a stronger (but ultimately equivalent) version of IR: at step  $t$, it imposes  $w_t \geq \max_{\theta \in R_t} \underline{u}(\theta)$  rather than only  $w_t \geq \max_{\theta \in C_t} \underline{u}(\theta)$. For partitions with non-increasing payoffs, this stronger condition does not change the set of resulting partitions, as shown in Lemma 1 below. The stronger condition is easier to work with recursively: the admissible interval  $[\max_{\theta \in R_t} \underline{u}(\theta), w_{t-1}]$  depends only on the residual set  $R_t$  and the previous cell's payoff  $w_{t-1}$, not on which candidate coalition  $(C_t, \sigma_t, w_t)$  is being considered.

**Lemma 1.** *Let  $\Pi = \{(C_t, \sigma_t, w_t)\}_{t=1}^T$  be a partition of  $G$  with  $w_1 \geq \dots \geq w_T$. Then  $\Pi$  is IR if and only if*

$$
w_t \geq \max_{\theta \in R_t} \underline{u}(\theta) \quad \text{for every } t.
$$

*Proof.* The displayed condition implies IR because  $C_t \subseteq R_t$. Conversely, if  $\Pi$  is IR and  $\theta \in R_t$, then  $\theta \in C_s$  for some  $s \geq t$, so  $\underline{u}(\theta) \leq w_s \leq w_t$  by IR and non-increasing payoffs. Taking the maximum over  $\theta \in R_t$  gives the displayed condition.  $\square$

Unlike the unrestricted partition algorithm, Algorithm 2 may halt because no admissible coalition payoff is available at some step. If it terminates, its output is a PBE partition. Since every disclosure game admits a PBE, at least one run of Algorithm 2 terminates and returns an associated PBE partition.

## 5 Coalition-Proof PBE

Fix a disclosure game  $G = (\Theta, \mathcal{M}, M, \mu^0, V)$. For a partition  $\Pi = \{(C_t, \sigma_t, w_t)\}_{t=1}^T$  of  $G$  and a payoff  $\tilde{w} \in \mathbb{R}$, let

$$
\Theta(\tilde{w}) := \bigcup_{t: w_t < \tilde{w}} C_t
$$

denote the set of types whose payoff in the partition is strictly below  $\tilde{w}$.

**Definition 9.** *Let  $\Pi$  be a partition of  $G$  and  $\tilde{w} \in \mathbb{R}$.*

- (i)  *$(\tilde{C}, \tilde{\sigma}, \tilde{w})$  is a blocking coalition of  $\Pi$  if  $\Theta(\tilde{w}) \neq \emptyset$  and  $(\tilde{C}, \tilde{\sigma}, \tilde{w})$  is a coalition of the restricted game  $G|_{\Theta(\tilde{w})}$.*
- (ii)  *$\Pi$  is coalition-proof if it admits no blocking coalition.*

A coalition-proof PBE partition is a PBE partition that is coalition-proof. A sender strategy is a coalition-proof PBE strategy if it is associated with some coalition-proof PBE partition.

Because the payoff correspondence  $V$  may be multi-valued, the realized payoffs  $w_t$  are part of the equilibrium outcome rather than pinned down by the strategy; coalition proofness is accordingly a property of the partition.

A blocking coalition is an announced deviation: the types in  $\tilde{C}$  declare that they will switch to strategy  $\tilde{\sigma}$, and if the receiver believes the announcement and updates accordingly after observing a message from  $X(\tilde{\sigma})$, the deviators obtain payoff  $\tilde{w}$. The restriction to  $G|_{\Theta(\tilde{w})}$  captures credibility: only types whose partition payoff is strictly below  $\tilde{w}$  will want to join or imitate the announcement. Among those types, the announced evidence must be exclusive to the deviators and yield the common payoff  $\tilde{w}$.

Unlike other belief-based refinements of PBE, coalition proofness allows the announced deviation to involve multiple messages at once since it may involve a mixed strategy. Furthermore, the announced deviation may include messages that are already on the equilibrium path. This causes no ambiguity about who deviates: every on-path user of such a message participates in the announced deviation (see Lemma B.1 in the appendix).

Because the greedy selection defined below picks, at each step, the largest available coalition payoff, we introduce notation for the payoffs that coalitions of a restricted game can attain. For any non-empty  $R \subseteq \Theta$, write

$$
\mathcal{W}_R := \{w \in \mathbb{R} \mid (C, \sigma, w) \text{ is a coalition of } G|_R \text{ for some } C \subseteq R\},
$$

which denotes the set of coalition payoffs attainable in the restricted game  $G|_R$. This set is non-empty and compact (see Lemma B.2 in the appendix).

**Definition 10.** A partition  $\Pi = \{(C_t, \sigma_t, w_t)\}_{t=1}^T$  of  $G$  is *greedy* if, for every  $t$  (with  $w_0 := \infty$),

$$
w_t = \max\left(\mathcal{W}_t \cap \left[\max_{\theta \in R_t} \underline{u}(\theta), w_{t-1}\right]\right),
$$

where  $\mathcal{W}_t$  is the coalition payoff set of  $G_t$.

Greedy partitions are outputs of Algorithm 3 defined below. The greedy partition algorithm is essentially the PBE partition algorithm with a stronger selection rule: at each residual game  $G_t$, the payoff must satisfy the PBE bounds  $\max_{\theta \in R_t} \underline{u}(\theta) \leq w_t \leq w_{t-1}$, and among the payoffs in  $\mathcal{W}_t$  satisfying them it must be the largest. If no such payoff exists, the run halts.

---

**Algorithm 3:** Greedy Partition Algorithm

Let  $t := 1$,  $R_1 := \Theta$, and  $w_0 := \infty$;

**while**  $R_t \neq \emptyset$

select a coalition  $(C_t, \sigma_t, w_t)$  of  $G_t$  with  $w_t = \max(\mathcal{W}_t \cap [\max_{\theta \in R_t} \underline{u}(\theta), w_{t-1}])$;

**halt** with no output if there is none;

let  $R_{t+1} := R_t \setminus C_t$  and  $t := t + 1$;

**end**

---

**Proposition 2.** A partition of  $G$  is a coalition-proof PBE partition if and only if it is greedy. Consequently, a sender strategy is a coalition-proof PBE strategy if and only if it is associated with some greedy partition.

Proposition 2 says that the no-blocking condition is exactly the greedy selection rule. Suppose that at step  $t$  the residual game  $G_t$  contains a coalition that can do better than the payoff actually assigned at that step without exceeding the payoff of the previous cell. Such a payoff is attractive exactly to the types remaining in the residual set: earlier cells already receive at least as much, whereas every type in  $R_t$  receives weakly less than  $w_t$  and hence strictly less than the proposed payoff. This coalition is therefore a credible announced deviation, hence a blocking coalition. Greediness rules out precisely these deviations by assigning, at each residual game, the largest coalition payoff compatible with the IR lower bound and the previous cell's payoff cap.

As might be expected from the literature on neologism proofness, coalition-proof PBE do not always exist, meaning that Algorithm 3 may halt no matter what choices are made at each step. However, as we show next, they exist under relatively weak conditions, covering many settings studied previously in the disclosure literature.

## 6 Existence of Coalition-Proof PBE

Fix a disclosure game  $G = (\Theta, \mathcal{M}, M, \mu^0, V)$. By Proposition 2, a coalition-proof PBE exists if and only if some run of Algorithm 3 terminates. This section gives sufficient conditions for existence: first on the upper envelope  $\bar{v}$  and the message mapping  $M$  (Section 6.1), then for the single-valued payoff only (Section 6.2), then for the message mapping only (Section 6.3), and finally on the lower envelope only (Section 6.4). Section 6.5 concludes the section by showing that a coalition-proof PBE always exists if  $n = 2$  and providing a minimal example of non-existence with three types.

The first three existence conditions have a common mechanism: they produce runs of Algorithm 3 along which the residual maximum  $\max \mathcal{W}_t$  is non-increasing, so a coalition is available at each step. The fourth existence condition, which we call payoff degradation, ensures that we can replicate residual coalitions with ones that have the same participants but a lower payoff; a coalition is then available at each step even if the residual maximum strictly increases.

[^6]: Genericity is needed in sections 6.1 and 6.2, in which structural restrictions such as quasiconcavity and/or quasiconvexity are imposed on the upper envelope  $\bar{v}$  of the sender's payoff correspondence. Functions of either kind are not closed under addition and hence do not form a vector space, so measure-theoretic notions of genericity such as prevalence (Hunt, Sauer, and Yorke, 1992) do not apply. We therefore use injectivity, which is topologically generic: within each such class, under the  $\|\cdot\|_\infty$  metric, the envelopes with injective  $C \mapsto \bar{v}(\mu_C^0)$  form an open dense set, since  $\Theta$  is finite.

Some of our results establish the essential uniqueness of coalition-proof PBE under the following genericity condition, which requires distinct type sets to yield distinct pooling payoffs[^6]:

**Definition 11.** *Game  $G$  is generic if the map  $C \mapsto \bar{v}(\mu_C^0)$  is injective on non-empty subsets  $C \subseteq \Theta$.*

### 6.1 Quasiconcavity and Message Completeness

The first set of conditions that guarantee existence consists of quasiconcavity of the upper envelope  $\bar{v}$  and completeness of the message mapping.

**Definition 12.** *The upper envelope  $\bar{v}$  is quasiconcave (QC) if, for all  $\mu, \mu' \in \Delta\Theta$  and  $\lambda \in (0, 1)$,*

$$
\bar{v}(\lambda\mu + (1 - \lambda)\mu') \geq \min\{\bar{v}(\mu), \bar{v}(\mu')\},
$$

*and it is strictly quasiconcave (QC\*) if the inequality is strict whenever  $\mu \neq \mu'$.*

**Definition 13.** *The message mapping is complete (M-C) if for all  $m, m' \in \mathcal{M}$  there exists  $m'' \in \mathcal{M}$  with*

$$
P(m'') = P(m) \cup P(m'),
$$

*where  $P(m) = M^{-1}(\{m\})$  is the set of types that can send  $m$.*

Completeness of the message mapping, as in Bertomeu and Cianciaruso (2018), requires that the family  $\{P(m) \mid m \in \mathcal{M}\}$  of all type sets that can be expressed by the message mapping is closed under unions. That is, if some message means “My type is in  $A$ ” and another means “My type is in  $B$,” then some message means “My type is in  $A$  or  $B$.” Roughly speaking, message completeness ensures that if a group of senders can form a coalition using multiple messages, then they can also form a coalition using a single message.

Under these conditions, the following lemma lets us restrict our attention to “pooling” coalitions:

**Lemma 2.** *If  $\bar{v}$  is quasiconcave and the message mapping is complete, then for every coalition  $(C, \sigma, w)$  of  $G$, there is a message  $m^*$  with  $P(m^*) = C$, and for any such  $m^*$, the pooling triple  $(C, \sigma^{m^*}, \bar{v}(\mu_C^0))$  is a coalition of  $G$  with*

$$
\bar{v}(\mu_C^0) \geq w.
$$

*Consequently, the largest coalition payoff in the game is the largest pooling payoff,*

$$
\max \mathcal{W}_\Theta = \max_{m \in \mathcal{M}} \bar{v}(\mu_{P(m)}^0),
$$

*and every coalition attaining  $\max \mathcal{W}_\Theta$  has a type set  $C$  with  $\bar{v}(\mu_C^0) = \max \mathcal{W}_\Theta$.*

Note that quasiconcavity of  $\bar{v}$  and completeness of the message mapping are both inherited by every restricted game  $G|_R$  because QC is an inequality over all of  $\Delta\Theta$  and thus holds at beliefs supported on  $R$, while the restricted preimages remain closed under unions. Hence Lemma 2 applies to each restricted game as well.

Under these two conditions a coalition-proof PBE exists, and under genericity it is essentially unique.

**Theorem 1.** *If  $\bar{v}$  is quasiconcave and the message mapping is complete, then a coalition-proof PBE exists. If, in addition,*

- (i)  *$\bar{v}$  is strictly quasiconcave, then no run of Algorithm 3 halts, and every run produces a coalition-proof PBE partition. A partition of  $G$  is a coalition-proof PBE partition if and only if  $w_t = \max \mathcal{W}_t$  and  $w_t \leq w_{t-1}$  for every  $t$, where  $w_0 := \infty$.*
- (ii) *the game is generic, then the coalition-proof PBE partition is essentially unique: any two coalition-proof PBE partitions of  $G$  have the same length  $T$, the same cells  $C_1, \dots, C_T$, and the same payoffs  $w_1, \dots, w_T$.*

The mechanism is a merging argument (see Lemma D.1 in the appendix). A greedy run could halt at step  $t$  only if the largest coalition payoff of the residual game exceeded  $w_{t-1}$, the payoff just removed. Message completeness would then allow us to merge the removed coalition with the better residual one into a single pooling coalition available one step earlier, and quasiconcavity would place the pooled payoff at least at  $w_{t-1}$, hence exactly at  $w_{t-1}$. Under strict quasiconcavity the pooled payoff would be strictly higher, which is impossible; under quasiconcavity it would sit at  $w_{t-1}$  on a strictly larger type set, which selecting a payoff-maximal coalition with the most types, or genericity, rules out. Removing a payoff-maximal coalition therefore never raises the residual maximum, and the greedy partition algorithm never halts.

### 6.2 Betweenness

[^7]: Hart, Kremer, and Perry (2017) provide a microfoundation: betweenness holds whenever the receiver's expected utility is single-peaked in her action at every belief, and for  $|\Theta| = 2$  it reduces to monotonicity of  $v$  along the simplex. There, the sender's payoff at a belief is the peak of the receiver's single-peaked expected utility. Bertomeu and Cianciaruso (2018) impose a related condition on a real-valued payoff, their convexity condition (CV): the value of a merged belief lies between the worst and best values of its components.

Next we impose no condition on the message mapping and, in return, strengthen the payoff restriction of Section 6.1 from quasiconcavity to betweenness: the payoff at a mixture of two beliefs lies between the payoffs at these beliefs. Betweenness is a property of a real-valued payoff, so throughout this subsection  $V = \{v\}$; the two envelopes coincide,  $\bar{v} = \underline{v} = v$, and  $v$  is continuous by (A4).[^7]

**Definition 14.** *The sender's payoff satisfies betweenness ($B$) if  $V = \{v\}$  and  $v$  is both quasiconcave and quasiconvex: for all  $\mu, \mu' \in \Delta\Theta$  and  $\lambda \in (0, 1)$,*

$$
\min\{v(\mu), v(\mu')\} \leq v(\lambda\mu + (1 - \lambda)\mu') \leq \max\{v(\mu), v(\mu')\}.
$$

*The payoff satisfies strict betweenness ($B^*$) if, in addition, both inequalities are strict whenever  $v(\mu) \neq v(\mu')$.*

Like quasiconcavity, betweenness is inherited by every restricted game. Quasiconvexity replaces message completeness: even when the types that can send a set of messages cannot pool on a single one, some coalition of these types pays at least their pooling value.

**Lemma 3.** *If the sender's payoff satisfies betweenness, then*

- (i) *every coalition  $(C, \sigma, w)$  of  $G$  satisfies  $w = v(\mu_C^0)$;*
- (ii) *for every non-empty  $R \subseteq \Theta$, the largest coalition payoff is the largest pooling value over relative preimages,*

$$
\max \mathcal{W}_R = \max_{\emptyset \neq X \subseteq \mathcal{M}_R} v(\mu_{M_R^{-1}(X)}^0),
$$

*and this value is attained by a coalition of  $G|_R$.*

For (i), the coalition's prior  $\mu_C^0$  is a convex combination of its on-path posteriors. Each of these posteriors yields  $w$  by (C4), so betweenness gives  $v(\mu_C^0) = w$. For (ii), exclusivity makes every coalition type set a relative preimage, so (i) gives the upper bound. To attain it, choose a maximizing message set  $X^*$  and form the auxiliary game with messages  $X^*$  and types  $M_R^{-1}(X^*)$. This game has a PBE partition with decreasing payoffs. Its cell priors average to the auxiliary prior, so quasiconvexity and (i) imply that its first cell pays at least the maximum pooling value. The first cell is also a coalition of  $G|_R$: any type that can send one of its messages belongs to the auxiliary type set, where the cell's exclusivity already applies. Thus this cell attains the bound.

Betweenness guarantees existence, and either strict betweenness or genericity ensures that every greedy run terminates.

**Theorem 2.** *If the sender's payoff satisfies betweenness, then a coalition-proof PBE exists. If, in addition,*

- (i) *the sender's payoff satisfies strict betweenness or  $G$  is generic, then no run of Algorithm 3 halts, and every run produces a coalition-proof PBE partition. A partition of  $G$  is a coalition-proof PBE partition if and only if  $w_t = \max \mathcal{W}_t$  and  $w_t \leq w_{t-1}$  for every  $t$, where  $w_0 := \infty$.*

- (ii) *G* is generic, then the coalition-proof PBE partition is essentially unique: any two coalition-proof PBE partitions of *G* have the same length *T*, the same cells  $C_1, \dots, C_T$, and the same payoffs  $w_1, \dots, w_T$.

As in Section 6.1, a greedy run can halt only if the residual maximum rises above the previous payoff. Suppose removing a payoff-maximal coalition with type set  $C$  and payoff  $w$  leaves a coalition with type set  $C'$  and payoff  $w' > w$. The merging argument shows that  $C \cup C'$  is a relative preimage with pooling value  $w$  (see Lemma E.1 in the appendix). Under strict betweenness, this is impossible, since the union's value lies strictly between  $w$  and  $w'$. Under genericity, it contradicts injectivity, since  $C$  and  $C \cup C'$  are distinct type sets with the same pooling value. In either case, removing a payoff-maximal coalition cannot raise the residual maximum, so no greedy run halts.

Under plain betweenness, the equality case can occur, so we select among payoff-maximal coalitions by ranking their induced priors within each level set of  $v$. This ranking ensures that a rise in the residual maximum implies that a higher-ranked payoff-maximal coalition was available at the previous step, contradicting the selection. For the construction, see Lemma E.2 in the appendix.

#### 6.2.1 Evidence Games and Truth-Leaning

[^8]: In Hart, Kremer, and Perry (2017), the receiver chooses a reward  $x \in \mathbb{R}$, the sender's payoff is  $x$  itself, and the receiver's expected utility is single-peaked in  $x$  at every belief; the unique peak  $v(\mu)$  is the sender's payoff, which is single-valued, quasiconcave, and quasiconvex.

An evidence game (Hart, Kremer, and Perry (2017)) is a disclosure game with a particular message mapping and a payoff that satisfies betweenness.[^8] The message mapping is a preorder on types: each type can reveal itself or any type below it. Since B holds, a coalition-proof PBE exists by Theorem 2. We show that coalition proofness provides a sender-side foundation for the receiver's optimal-commitment outcome.

**Definition 15.** *An evidence game is a disclosure game  $(\Theta, \mathcal{M}, M, \mu^0, V)$  such that*

- *the sender's payoff satisfies betweenness;*
- *the message mapping has an evidence structure, that is,  $\mathcal{M} = \Theta$, and  $M : \Theta \rightarrow 2^\Theta \setminus \{\emptyset\}$  satisfies*

(L1)  $\theta \in M(\theta)$  for every  $\theta \in \Theta$  (reflexivity);

(L2) if  $\theta' \in M(\theta)$  and  $\theta'' \in M(\theta')$, then  $\theta'' \in M(\theta)$  (transitivity).

Hart, Kremer, and Perry (2017) define truth-leaning equilibria as limits of equilibria of perturbed games in which revealing the whole truth carries an infinitesimal advantage. By their Proposition 1, such equilibria exist, and each has the same outcome as an equilibrium satisfying the following two conditions, which for ease of exposition we adopt as our definition.

**Definition 16.** *A PBE  $(\sigma, \mu, r)$  of an evidence game is truth-leaning if*

(A0) for every  $\theta \in \Theta$: if  $r(\theta) = \max_{m \in M(\theta)} r(m)$, then  $\sigma(\theta | \theta) = 1$;

(P0) for every off-path  $m \in \mathcal{M}$:  $\mu(\cdot | m) = \delta_m$.

Condition (A0) breaks ties in favor of the whole truth: if message  $\theta$  gives type  $\theta$  its highest reward, then that type sends  $\theta$  with probability one. Condition (P0) attributes an unused message  $m$  to type  $m$. Note that our definition of PBE requires that off-path beliefs are feasible; this requirement is consistent with (P0) as the fully revealing strategy induces belief  $\delta_m$  at message  $m$, so  $\delta_m \in \mathcal{F}(m)$.

We next show that every truth-leaning equilibrium of an evidence game is coalition-proof.

**Proposition 3.** *Let  $G$  be an evidence game. Then the sender's strategy in every truth-leaning equilibrium is a coalition-proof PBE strategy; consequently, by Theorem 1 of Hart, Kremer, and Perry (2017), the receiver's optimal-commitment outcome is a coalition-proof PBE outcome of  $G$.*

**Corollary 1.** *In a generic evidence game, every truth-leaning equilibrium and every coalition-proof PBE induce the same cells and the same payoffs, and this common outcome is the receiver's optimal-commitment outcome.*

To prove Proposition 3, we use the truth-telling dichotomy (Hart, Kremer, and Perry (2017), Proposition 2), which states that in any truth-leaning equilibrium, either type  $\theta$  sends its own message with probability one, or type  $\theta$  never sends its own message and  $v(\delta_\theta) < u(\theta)$. Consider a truth-leaning equilibrium  $(\sigma, \mu, r)$  and suppose, toward a contradiction, that there is a blocking coalition  $(\tilde{C}, \tilde{\sigma}, \tilde{w})$. From betweenness, that coalition pays  $\tilde{w} = v(\mu_{\tilde{C}}^0)$  (Lemma 3).

Next, partition the participants in  $\tilde{C}$  into those who send their own messages with probability one in the original equilibrium (call that set  $\tilde{C}^{\text{tr}}$) and those who do not. For each truth-telling member  $\theta' \in \tilde{C}^{\text{tr}}$, all types that pool on message  $\theta'$  in the original equilibrium must also be in  $\tilde{C}$. Indeed, they can send any message available to the truth-telling member (by transitivity) and receive the same payoff as  $\theta'$  in equilibrium, which is  $r(\theta') = u(\theta') < \tilde{w}$. Thus the on-path posterior after message  $\theta'$  is supported on a subset of  $\tilde{C}$  and pays strictly less than  $\tilde{w}$. Next, decompose  $\mu_{\tilde{C}}^0$  as a convex combination of the following beliefs. First, take all on-path posteriors  $\mu_\sigma(\cdot | \theta')$  for all truth-telling members  $\theta' \in \tilde{C}^{\text{tr}}$. These are supported on  $\tilde{C}$  and pay strictly less than  $\tilde{w}$. For each non-truth-telling member  $\theta'' \in \tilde{C} \setminus \tilde{C}^{\text{tr}}$, take the degenerate belief  $\delta_{\theta''}$, weighted by its remaining probability mass. By the truth-telling dichotomy, each such belief pays  $v(\delta_{\theta''}) < u(\theta'') < \tilde{w}$. Quasiconvexity of  $v$  thus gives  $v(\mu_{\tilde{C}}^0) < \tilde{w}$, a contradiction. Quasiconvexity of  $v$  thus gives  $v(\mu_{\tilde{C}}^0) < \tilde{w}$, a contradiction. The optimal-commitment conclusion follows because a truth-leaning equilibrium exists and yields the receiver's optimal-commitment outcome by Proposition 1 and Theorem 1 of Hart, Kremer, and Perry (2017). Corollary 1 follows from Theorem 2(ii).

These results provide an alternative foundation for selecting receiver-optimal outcomes in evidence games. Under genericity, ruling out profitable coalitional deviations selects the same outcome as truth-leaning or receiver commitment. Betweenness is central to this conclusion because it restricts the gains from pooling. Pooling can benefit some types at the expense of others, but the pooled payoff cannot exceed the payoffs of all the groups that pool. A coalition also cannot increase its common payoff by changing how its members mix over messages. In Hart, Kremer, and Perry (2017), this restriction follows from the receiver's expected utility being single-peaked in her action at every belief. We show that betweenness, together with the evidence structure, also ensures that the receiver's optimal-commitment outcome survives credible deviations by groups of sender types.

### 6.3 Cheap Talk and Message Completeness

This section gives sufficient conditions for existence under message completeness together with cheap talk without assuming quasiconcavity of the upper envelope or other conditions on  $V$.

**Definition 17.** *The message mapping has cheap-talk copies (M-CT) if, for every  $m \in \mathcal{M}$,*

$$
|\{m' \in \mathcal{M} \mid P(m') = P(m)\}| \geq n.
$$

Messages with the same preimage carry identical evidence, so choosing among them is cheap talk. Any disclosure game can be augmented to satisfy M-CT by adjoining, for each message, enough copies with the same preimage. The augmented game is again a disclosure game, and M-CT is inherited by every restricted game because the relative preimages of the copies coincide.

Our main tool is the quasiconcave closure of  $\bar{v}$, the smallest quasiconcave function lying above it. The quasiconcave closure is the construction by which Lipnowski and Ravid (2020), who call it the *quasiconcave envelope*, characterize the value of sender-optimal cheap talk; we apply it to the disclosure upper envelope  $\bar{v}$.

**Definition 18.** *The quasiconcave closure of the upper envelope is the function  $\bar{v}^{qc} : \Delta\Theta \rightarrow \mathbb{R}$  given by*

$$
\bar{v}^{qc}(\mu) := \max \left\{ \min_{1 \leq i \leq n} \bar{v}(p_i) \mid \lambda \in \Delta(\{1, \dots, n\}), p_1, \dots, p_n \in \Delta\Theta, \sum_i \lambda_i p_i = \mu \right\}.
$$

The maximum is attained because  $\bar{v}$  is upper semicontinuous by (A4) and the constraint set is compact.

With cheap-talk copies, a coalition is no longer tied to a single posterior. A coalition on a type set  $C$  can spread its members across the  $n$  copies of a message and induce any decomposition of its prior  $\mu_C^0$  into posteriors. Because the members have a common payoff, such a splitting pays only up to the value of  $\bar{v}$  at its least favorable posterior, and the splitting that makes this least value largest attains the closure  $\bar{v}^{qc}(\mu_C^0)$. Therefore, under M-CT a coalition can reach  $\bar{v}^{qc}$  at any type set, and together with message completeness this guarantees existence.

**Theorem 3.** *If the message mapping is complete and has cheap-talk copies, then a coalition-proof PBE exists.*

The proof relies on a few key lemmas proved in the appendix. It applies Theorem 1 to a companion game whose upper envelope is already quasiconcave: the *closure game*  $G^{qc}$, obtained from  $G$  by replacing  $\bar{v}$  with  $\bar{v}^{qc}$  (see Lemma F.1 in the appendix). Two facts link the two games. First, no coalition of  $G$  can pay more than  $\bar{v}^{qc}$  at its prior because its common payoff is bounded by  $\bar{v}$  at each posterior it induces and the prior is a convex combination of those posteriors (see Lemma F.2 in the appendix); this holds in any disclosure game. Second, M-CT lets a coalition reach  $\bar{v}^{qc}$  at its prior (see Lemma F.3 in the appendix). Hence  $G$  and  $G^{qc}$  have the same attainable coalition payoffs in every restricted game. Then, as  $\bar{v}^{qc}$  is quasiconcave by construction and  $G^{qc}$  inherits M-C from  $G$, Theorem 1 applies to  $G^{qc}$  and returns a coalition-proof PBE partition. By M-CT, each cell of that partition is matched in  $G$  by a coalition with the same type set and payoff, so the partition carries over to  $G$  step by step and is a coalition-proof PBE partition there as well. The appendix gives the construction.

A special case is pure cheap talk, where every message is available to every type and therefore carries no evidence. Coalition proofness selects the sender-optimal cheap-talk payoff  $\bar{v}^{qc}(\mu^0)$  of Lipnowski and Ravid (2020): the sender obtains it in every coalition-proof PBE, though the supporting equilibrium need not be unique.

**Corollary 2.** *Suppose  $P(m) = \Theta$  for every  $m \in \mathcal{M}$  and  $|\mathcal{M}| \geq n$. Then a coalition-proof PBE exists and is essentially unique: every coalition-proof PBE partition consists of the single cell  $\Theta$  with payoff  $\bar{v}^{qc}(\mu^0)$.*

### 6.4 Payoff Degradation

This section gives sufficient conditions for existence that leave the upper envelope  $\bar{v}$  entirely unconstrained and impose no conditions on the message mapping. The common mechanism is payoff degradation, which allows a coalition to lower its payoff while retaining its members. Write $v_{\min} := \min_{\Delta\Theta} \underline{v}$  for the minimum of the lower envelope, which is attained because  $\underline{v}$  is lower semicontinuous on the compact set  $\Delta\Theta$  by (A4).

**Definition 19.** *The game  $G$  satisfies payoff degradation if, for every non-empty  $R \subseteq \Theta$, every coalition  $(C, \sigma, w)$  of  $G|_R$, and every  $w' \in [v_{\min}, w]$, the restricted game  $G|_R$  admits a coalition  $(C, \sigma', w')$.*

Two conditions guarantee payoff degradation: free disposal, and revelation aversion combined with cheap-talk copies.

**Proposition 4.** *In each of the following cases  $G$  satisfies payoff degradation:*

- (i) *the sender has free disposal, meaning that  $\underline{v}(\mu) = v_{\min}$  for every  $\mu \in \Delta\Theta$;*
- (ii) *the message mapping has cheap-talk copies and the sender is revelation-averse, meaning that  $\underline{v}(\delta_\theta) = v_{\min}$  for every  $\theta \in \Theta$.*

Under free disposal, the sender’s worst-case payoff is the same at every belief, so any lower payoff can be assigned to a coalition’s existing posteriors with no change in messages. Under revelation aversion, the sender’s worst-case payoff is the same at every point belief, that is, when the receiver learns the sender’s type and responds adversarially. A coalition’s payoff can be lowered by splitting its posteriors into beliefs closer to point masses that all admit the target payoff. Cheap-talk copies supply the messages needed for these splittings.

**Theorem 4.** *If  $G$  satisfies payoff degradation, then no run of Algorithm 3 halts; consequently,  $G$  admits a greedy partition and a coalition-proof PBE exists. In particular, these conclusions hold in each case of Proposition 4.*

Previous existence results argue that Algorithm 3 terminates because removing a payoff-maximal coalition cannot raise the residual maximum. With payoff degradation, even if the residual maximum rises, the algorithm can continue by reducing the next coalition’s payoff to the previous one’s. The reduced payoff is still IR for the remaining types because the greedy choice in the previous step is.

### 6.5 The Binary Case and Non-Existence

[^9]: Adding to Example 3 a message that only type 1 can send gives  $M$  an evidence structure. This adds the coalition payoff  $v(\delta_1) = -0.1$  and changes nothing else, so a coalition-proof PBE still does not exist.

We conclude the existence section with three observations. First, Theorem 2, and Theorem 4 under free disposal, provide sufficient conditions for existence that constrain only the payoff correspondence  $V$, while Theorem 3 provides sufficient conditions that constrain only the message mapping  $M$. Second, with two types a coalition-proof PBE always exists. Third, our results are relatively “tight”: concavity and continuity of  $v$, single-valuedness, and either the availability of cheap talk or the evidence structure on  $M$  (Hart, Kremer, and Perry, 2017) are not sufficient for existence of a coalition-proof PBE. Example 3 illustrates this point.[^9]

**Theorem 5.** *If  $|\Theta| = 2$, then every disclosure game admits a coalition-proof PBE.*

Choose a coalition attaining the largest payoff  $w_1$. If it contains both types, the run ends. Otherwise, it consists of one type  $\theta$; the remaining type  $\theta'$  can finish unless  $\underline{v}(\delta_{\theta'}) > w_1$. In that case, every message of  $\theta'$  is shared with  $\theta$  since an exclusive message would support a coalition paying more than  $w_1$. Let  $\theta'$  send one such message and gradually mix  $\theta$  into it, keeping its remaining mass on its exclusive evidence. The shared-message posterior moves from  $\delta_{\theta'}$, where all payoffs exceed  $w_1$, to the prior, where full pooling admits only payoffs that are at most  $w_1$. By the interval values and upper hemicontinuity in (A4), some belief along this path admits  $w_1$. The other posteriors stay at  $\delta_\theta$, where  $w_1$  remains feasible, giving a coalition of both types at  $w_1$.

The following three-type example is therefore minimal.

**Example 3.** *Let  $\Theta = \{1, 2, 3\}$  with a uniform prior,  $\mathcal{M} = \{a, b\}$,  $M(1) = \{a, b\}$,  $M(2) = \{a\}$,  $M(3) = \{b\}$, and let  $V = \{v\}$  be single-valued with*

$$
v(\mu) := \min\{\mu_1 + 0.9\mu_3, -0.1\mu_1 + 1.1\mu_2 + 0.9\mu_3\}.
$$

As a minimum of two affine functions,  $v$  is concave and continuous. Types 2 and 3 are each locked into one message, and type 1 can “join” either of them. The relevant values are  $v(\frac{1}{2}, \frac{1}{2}, 0) = \frac{1}{2}$,  $v(\frac{1}{2}, 0, \frac{1}{2}) = 0.4$, and  $v(\delta_3) = 0.9$. The only feasible coalition payoffs are 0.5 (with  $C_1 = \{1, 2\}$  pooling on message  $a$) and 0.4 (with  $C_1 = \{1, 3\}$  pooling on message  $b$). In particular, there is no way to build a grand coalition since payoffs from the two messages cannot be equalized. The unique PBE partition,  $(\{1, 3\}, b, 0.4)$,  $(\{2\}, a, 0)$, is blocked by  $\{1, 2\}$  pooling on  $a$. However, greedy selection at step 1 leaves a second coalition with  $C_2 = \{3\}$  and  $w_2 = 0.9 > 0.5$, so every greedy run halts at step 2.

**Proposition 5.** *In Example 3, no coalition-proof PBE exists. Non-existence persists when the game is augmented with cheap-talk copies of both messages. Consequently, concavity of the payoff, continuity, single-valuedness, and cheap-talk copies are jointly insufficient for existence.*

## 7 Rich Disclosure Games and the Tent

This section provides an explicit characterization of coalition-proof PBE when the message space is maximally “rich.” In particular, we require not only that there is a message that allows any set of types to pool together and separate from others (as in Grossman (1981) and Milgrom (1981), where any subset  $m \subseteq \Theta$  such that  $\theta \in m$  is a valid message) but also that *fractions* of each type can pool while excluding otherwise identical copies of themselves.

One example of a formal model that accommodates the richness that we study here is the stochastic-message-mapping game of Titova and Zhang (2025). In that model, the sender observes a payoff-irrelevant label drawn from an atomless distribution in addition to his type  $\theta$. The sender can then condition his message on this label, which means that, for every nonzero vector  $\nu \leq \mu^0$, there is a message available to mass  $\nu(\theta)$  of type  $\theta$  senders for each  $\theta \in \Theta$. We do not introduce the labels and messages explicitly and instead use  $\nu$  to keep track of these masses.

Suppose that at some stage  $\nu(\theta)$  is the mass of type  $\theta$  senders who remain, with  $0 \neq \nu \leq \mu^0$, and let  $C := \text{supp } \nu$. A coalition is a nonzero vector  $\eta \leq \nu$, where  $\eta(\theta)$  is the mass of type  $\theta$  senders in the coalition. It induces belief  $\hat{\eta} := \eta / \eta(\Theta)$, where  $\eta(\Theta) := \sum_{\theta \in \Theta} \eta(\theta)$, and payoff  $\bar{v}(\hat{\eta})$. Because arbitrary fractions of the remaining types can pool, they can induce exactly the beliefs in

$$
\Delta_C := \{\mu \in \Delta\Theta : \text{supp } \mu \subseteq C\}.
$$

For each non-empty  $C \subseteq \Theta$, let  $\mu_C^* \in \arg \max_{\mu \in \Delta_C} \bar{v}(\mu)$  be the belief that maximizes the sender’s payoff over the beliefs supported on  $C$. For a nonzero residual mass vector  $\nu$  with  $C = \text{supp } \nu$, define

$$
\lambda^*(\nu) := \min_{\theta: \mu_C^*(\theta) > 0} \frac{\nu(\theta)}{\mu_C^*(\theta)}
$$

as the largest mass that can pool at that belief.

The game we study here is not a disclosure game in the sense of Definition 1, but it admits a direct analogue of the greedy partition Algorithm 3 that recovers all coalition-proof PBE. At each stage, the algorithm selects a coalition that maximizes the sender's payoff across all beliefs supported on the remaining types and removes at least one type from the game completely.

First, find the largest sender payoff in the game. We claim that the first coalition in a coalition-proof PBE reaches this payoff and fully removes some  $\theta \in \Theta$  from the game. Indeed, the first coalition must obtain  $\bar{v}(\mu_\Theta^*)$  since otherwise senders in the proportions  $\mu_\Theta^*$  could pool and all gain. To do so, it must induce the belief  $\mu_\Theta^*$, meaning its mass vector must be  $\eta_1 = \lambda \mu_\Theta^*$  for some  $\lambda > 0$. To obtain a coalition of maximal size, we pick the highest feasible  $\lambda$  such that  $\lambda \mu_\Theta^*(\theta) \leq \mu^0(\theta)$  for every  $\theta$, that is,  $\lambda = \lambda^*(\mu^0)$  (a smaller coalition would leave enough of every type to pool at the same belief again). Since  $\lambda$  is maximal,  $\lambda \mu_\Theta^*(\theta_i) = \mu^0(\theta_i)$  for some  $\theta_i$; hence all  $\theta_i$  senders participate in this coalition and type  $\theta_i$  is fully removed in step 1.

Let  $\nu_1 := \mu^0 - \eta_1$  be the masses left after the first coalition is removed, and let  $R_2 := \text{supp } \nu_1$. By construction,  $R_2$  is a strict subset of  $\Theta$. We repeat the same construction with  $\mu_{R_2}^* \in \arg \max_{\mu \in \Delta_{R_2}} \bar{v}(\mu)$: since all  $\theta_i$  senders were removed in stage 1, the best the remaining types can do is induce the belief  $\mu_{R_2}^*$.

We repeat until all types are assigned to a coalition. Thus, if type  $\theta_j$  is fully removed in stage 2, then  $R_3 := \text{supp } \nu_2$  is a strict subset of  $R_2$  and the next coalition induces  $\mu_{R_3}^*$, and so on. The algorithm terminates in at most  $|\Theta|$  steps. The mass partition is unique if the argmax at each step is unique.

---

**Algorithm 4:** Greedy Partition Algorithm for Rich Disclosure Games

```

Let  $t := 1$  and  $\nu_0 := \mu^0$;
while  $\nu_{t-1} \neq 0$ 
|   let  $R_t := \text{supp } \nu_{t-1}$  and select  $\mu_{R_t}^* \in \arg \max_{\mu \in \Delta_{R_t}} \bar{v}(\mu)$;
|   let  $\eta_t := \lambda^*(\nu_{t-1}) \mu_{R_t}^*$;
|   let  $\nu_t := \nu_{t-1} - \eta_t$  and  $t := t + 1$;
end

```

---

**Proposition 6.** *Let  $\eta_1, \dots, \eta_T$  be the output of Algorithm 4. Then*

- (i) *at each step,  $\eta_t$  attains the highest payoff available to the remaining types and exhausts at least one type in  $R_t$;*
- (ii) *the algorithm terminates after at most  $|\Theta|$  steps, the coalitions  $\eta_1, \dots, \eta_T$  partition  $\mu^0$, and their payoffs are weakly decreasing;*
- (iii) *if  $\bar{v}$  has a unique maximizer on every non-empty face of  $\Delta\Theta$,  $(\eta_1, \dots, \eta_T)$  is the unique mass partition satisfying (i) at every step.*

*In the finite-action stochastic-message-mapping model of Titova and Zhang (2025), this mass partition is induced by a coalition-proof PBE.*

Part (i) describes the structure of the partition: each coalition receives the highest payoff attainable by the types left at that stage. Indeed, if positive masses of each  $\theta \in R_t$  are still left, some fractions of them can pool appropriately to induce the constrained-optimal belief  $\mu_{R_t}^*$. Part (ii) gives existence: choosing each coalition to have maximal size assigns every sender to a coalition in at most  $|\Theta|$  steps. Part (iii) gives uniqueness if the maximizer of  $\bar{v}$  on every face is unique.

The PBE claim follows from Titova and Zhang (2025, Theorem 3). Each member  $\theta$  of coalition  $t$  obtains  $\bar{v}(\mu_{R_t}^*) \geq \bar{v}(\delta_\theta) \geq \underline{v}(\delta_\theta)$  because  $\delta_\theta \in \Delta_{R_t}$, and after receiving the coalition's message, the receiver chooses a best response to  $\mu_{R_t}^*$  that yields  $\bar{v}(\mu_{R_t}^*)$. To see that this PBE is coalition-proof, suppose a coalition blocks, and let  $t$  be the first equilibrium coalition from which it draws senders. All of its members were still in the game at step  $t$, so the belief it induces is supported on  $R_t$  and its payoff is at most  $\bar{v}(\mu_{R_t}^*)$, which the senders it draws from coalition  $t$  already receive.

### The Tent Function

Bayesian persuasion and cheap talk admit geometric characterizations of the sender's ex-ante payoff: the concave closure of  $\bar{v}$  in Kamenica and Gentzkow (2011) and its quasiconcave closure in Lipnowski and Ravid (2020). We now provide an analogous characterization for rich disclosure.

Assume for the remainder of this section that  $\bar{v}$  has a unique maximizer  $\mu_C^*$  on every non-empty face  $\Delta_C$. Then Proposition 6 gives a unique mass partition for every prior  $\mu \in \Delta\Theta$; when  $\mu$  is on the boundary, the procedure simply starts with the types in  $\text{supp } \mu$. If the resulting coalitions are  $\eta_1, \dots, \eta_T$, their expected payoff is the *tent* of  $\bar{v}$:

$$
v^{\text{tent}}(\mu) := \sum_{t=1}^T \eta_t(\Theta) \bar{v}(\hat{\eta}_t).
$$

![Two panels, (a) and (b), showing payoff functions. Both panels have a horizontal axis from 0 to 1. A dashed blue line represents a 'tent' function with a peak at mu*(theta_1). A solid black curve represents a payoff function v-bar. In panel (a), the solid curve is concave and lies below the tent, with its maximum at mu*(theta_1). In panel (b), the solid curve is quasiconcave and lies above the tent, with its maximum at mu*(theta_1). Both curves start at the same value at 0 and end at the same value at 1. The x-axis is labeled with 0, mu*(theta_1), and 1. Below the x-axis, mu^0(theta_1) is indicated.](https://maria-titova.com/papers-content/CPD/images/bd4617f25d15430eb78c2d6d75a99dde_img.jpg)

Figure 1: The tent with two types. The payoff functions  $\bar{v}$  (solid black) have the same maximum and endpoint values and therefore the same tent (dashed blue). Both are quasiconcave, so the solid curves also give the cheap-talk value. The persuasion value coincides with the tent in panel (a) and with the solid curve in panel (b).

Figure 1 illustrates the tent with two types and an interior-optimal belief  $\mu^*$. In a coalition-proof PBE, at least one type must obtain  $\bar{v}(\mu^*)$, or else senders of both types, in the correct proportion, could pool and obtain it. If the prior is  $\mu^*$, both types obtain this payoff. Suppose instead that there are more  $\theta_1$  senders than needed to induce  $\mu^*$. All  $\theta_2$  senders then pool with some  $\theta_1$  senders at  $\mu^*$, and the leftover  $\theta_1$  senders reveal their type. The tent therefore decreases linearly from  $\bar{v}(\mu^*)$  to  $\bar{v}(\delta_{\theta_1})$. The reverse applies when  $\theta_2$  is more numerous. With two types, only the optimal belief, its payoff, and the two endpoint payoffs determine the tent.

The tent can lie above or below the cheap-talk value but never above the persuasion value. The reason is that there is no “loyalty” across sender types: senders who can obtain  $\bar{v}(\mu^*)$  do so even if this leaves other senders worse off and lowers the ex-ante payoff.

With two types, at most one type remains after the first coalition is removed. With more types, different types may be exhausted first. Fix an ordering  $(\theta_{i_1}, \dots, \theta_{i_n})$  of the  $n$  types and let

$$
R_j := \Theta \setminus \{\theta_{i_1}, \dots, \theta_{i_j}\}, \quad j = 0, \dots, n-1.
$$

Thus,  $R_j$  is the set of types left after the first  $j$  types in the ordering have been exhausted. The associated constrained-optimal beliefs  $\mu_{R_0}^*, \dots, \mu_{R_{n-1}}^*$  span a possibly degenerate simplex. If one coalition exhausts several types, the skipped sets receive weight zero. A prior may have representations associated with different type orderings; Proposition 7 shows that all of them give the same payoff.

**Proposition 7.** *For every  $\mu \in \Delta\Theta$, there exist an ordering of the types and weights  $\omega_0, \dots, \omega_{n-1}$  such that*

$$
\mu = \sum_{j=0}^{n-1} \omega_j \mu_{R_j}^*, \quad \omega_j \geq 0, \quad \sum_{j=0}^{n-1} \omega_j = 1.
$$

*Every representation of this form, for any ordering of the types, gives the same value,*

$$
v^{\text{tent}}(\mu) = \sum_{j=0}^{n-1} \omega_j \bar{v}(\mu_{R_j}^*).
$$

*Consequently,  $v^{\text{tent}}$  is the unique function such that*

- (i)  $v^{\text{tent}}(\mu_C^*) = \bar{v}(\mu_C^*)$  for every non-empty  $C \subseteq \Theta$;
- (ii)  $v^{\text{tent}}$  is affine on the convex hull of the constrained-optimal beliefs associated with every ordering of the types.

Figure 2 shows the six simplices associated with the possible type orderings when there are three types. Each gray facet joins the unconstrained optimum  $\mu_\Theta^*$, the optimum on one two-type face, and a compatible singleton belief. The tent is affine on each facet and agrees with  $\bar{v}$  at its vertices. More generally, the  $n!$  possibly degenerate simplices generated by the type orderings cover  $\Delta\Theta$. Thus, the tent depends only on the constrained-optimal beliefs, the values of  $\bar{v}$  at those beliefs, and the prior.

## 8 Conclusion

In this paper, we characterized PBE in general, and properties of coalition-proof PBE in particular, for general disclosure games with type-independent sender preferences. Our general characterization shows that the building blocks of coalitions and partitions are basic building blocks of all equilibria in this class of games. Simply picking coalitions with constrained-maximal payoffs at each step of the algorithm for the construction of PBE guarantees coalition proofness. Our existence results show that coalition-proof PBE exist under various conditions, some of which apply only to the message space, others only to the payoff correspondence, and still others to a combination of the two. Our results extend Lipnowski and Ravid (2020)’s analysis of cheap talk. They show that the truth-leaning equilibria in Hart, Kremer, and Perry (2017) are closely related to coalition-proof

![A 3D plot showing a red curved surface representing the payoff function \bar{v} and a gray polyhedral surface representing the tent v^{tent}. The horizontal axes are labeled Pr(\theta = \theta_1) and Pr(\theta = \theta_2). The vertical axis is labeled \bar{v}. The tent has six facets and its vertices are labeled \mu_1^*, \mu_{12}^*, \mu_2^*, \mu_3^*, \mu_{13}^*, and \mu_{23}^*. The unconstrained optimum \mu^* is also marked on the base plane.](https://maria-titova.com/papers-content/CPD/images/0f79a59f3766fc341ff688a23692c1d9_img.jpg)

Figure 2: The tent  $v^{\text{tent}}$  (gray) of the payoff function  $\bar{v}$  (red) with three types. Each of its six facets joins the payoffs at three beliefs: the unconstrained optimum  $\mu^*$, the optimum  $\mu_{ij}^*$  on a two-type face  $\Delta_{\{\theta_i, \theta_j\}}$, and one of that face's vertices,  $\mu_i^* = \delta_{\theta_i}$  or  $\mu_j^* = \delta_{\theta_j}$. The tent is affine on each facet and agrees with  $\bar{v}$  at its vertices.

PBE in the class of games they study, and they characterize sender payoffs under a canonical game of maximally flexible disclosure, which can be thought of as analogous to canonical games of cheap talk and Bayesian persuasion Kamenica and Gentzkow (2011).

Our purpose was to provide a natural equilibrium selection concept as generally as possible for disclosure games. There are at least three avenues of future research. First, coalition-proof PBE may be guaranteed to exist under other sets of conditions that have not occurred to us. Second, we informally suggested in our examples (Example 1 and Example 2) that a desirable property of selected equilibria in disclosure games is that they respond sensibly to perturbations of the game (for example, the equilibrium outcome should not change when incorporating the opportunity to send messages that no sender type wants to send). It may be worth proposing a formal axiomatic approach to equilibrium selection along these lines and verifying whether coalition proofness or other selection criteria have good properties along this axis. Third, naturally, many games of disclosure that are of practical interest feature state-dependent sender payoffs. Our solution concept does not immediately generalize in this case because even senders using the same message will obtain different payoffs. However, the premise of coalitional deviations still appears sensible in principle and could be useful if there is a translation of it well adapted to this class of games.

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## A Proofs for Section 4

### Proof of Proposition 1

Suppose first that  $\sigma$  is supported as a PBE by  $(\mu, r)$. Let  $u(\theta) = \max_{m \in M(\theta)} r(m)$  be the associated equilibrium-payoff function defined above, and list its distinct values as  $w_1 > \dots > w_T$. Define

$$
C_t := \{\theta \in \Theta \mid u(\theta) = w_t\}, \quad \sigma_t := \sigma|_{C_t}, \quad R_t := \bigcup_{s=t}^T C_s.
$$

We claim that  $\Pi = \{(C_t, \sigma_t, w_t)\}_{t=1}^T$  is a partition. First, notice that the cells are non-empty, disjoint, and cover  $\Theta$, and  $\sigma_t$  is a coalition strategy on  $C_t$. It remains to show that  $(C_t, \sigma_t, w_t)$  is a coalition of  $G_t$, i.e., it satisfies conditions (C3) and (C4), for each  $t$. For (C3), note that every  $m \in X_t$  satisfies  $r(m) = w_t$, because it is used on path by a type in  $C_t$. If some  $\theta \in R_t$  can send a message  $m \in X_t$, then  $u(\theta) \geq r(m) = w_t$. But every type in  $R_t = \bigcup_{s \geq t} C_s$  has equilibrium payoff at most  $w_t$. Hence  $u(\theta) = w_t$, so  $\theta \in C_t$. Therefore  $M_t^{-1}(X_t) \subseteq C_t$. For (C4), fix  $m \in X_t$. Any type that sends  $m \in X_t$  under  $\sigma$  must have payoff  $w_t$, and hence belongs to  $C_t$. Thus the on-path posterior  $\mu_\sigma(\cdot \mid m)$  is the posterior generated by  $\sigma_t$  in  $G|_{C_t}$. Since PBE beliefs are Bayesian on path and  $r(m) \in V(\mu(\cdot \mid m))$, we get  $w_t = r(m) \in V(\mu_{\sigma_t}(\cdot \mid m))$. Therefore each  $(C_t, \sigma_t, w_t)$  is a coalition in  $G_t$, so  $\Pi$  is a partition with strictly decreasing payoffs.

It remains only to check IR. For every message  $m$,  $\mu(\cdot \mid m) \in \mathcal{F}(m)$  and  $r(m) \in V(\mu(\cdot \mid m))$, so  $r(m) \geq \min_{\nu \in \mathcal{F}(m)} \underline{v}(\nu)$. Maximizing over  $m \in M(\theta)$  gives  $u(\theta) \geq \underline{u}(\theta)$. Hence  $w_t \geq \underline{u}(\theta)$  for every  $\theta \in C_t$.

Conversely, let  $\Pi = \{(C_t, \sigma_t, w_t)\}_{t=1}^T$  be an IR partition with  $w_1 \geq \dots \geq w_T$, and let  $\sigma = \sigma^\Pi$. For each off-path message  $m \notin X(\sigma)$, choose  $\nu_m \in \mathcal{F}(m)$  attaining  $\min_{\nu \in \mathcal{F}(m)} \underline{v}(\nu)$  and set  $a_m := \underline{v}(\nu_m)$. Then  $a_m \in V(\nu_m)$  and  $a_m \leq \underline{u}(\theta)$  for every  $\theta$  with  $m \in M(\theta)$.

Define beliefs and payoffs by

$$
\mu(\cdot | m) = \begin{cases} \mu_\sigma(\cdot | m), & m \in X_t \text{ for some } t, \\ \nu_m, & m \notin X(\sigma), \end{cases} \quad r(m) = \begin{cases} w_t, & m \in X_t \text{ for some } t, \\ a_m, & m \notin X(\sigma). \end{cases}
$$

The constructed belief system  $\mu$  and payoff function  $r$  therefore satisfy the first three requirements in the definition of  $\sigma$  being a PBE strategy; it remains only to show that the sender is sequentially rational. Fix  $\theta \in C_t$. Every message in  $X_t$, and hence every message used by  $\theta$  under  $\sigma$, yields payoff  $w_t$. Every message in any earlier evidence set  $X_s$,  $s < t$, is not available to  $\theta$, since that would violate evidence exclusivity (C3) of the step- $s$  coalition. Every message in any later evidence set  $X_s$,  $s > t$, yields payoff  $w_s \leq w_t$. Every off-path feasible message yields payoff at most  $\underline{u}(\theta) \leq w_t$. Thus, type  $\theta$  does not have profitable deviations and  $\sigma$  is a PBE strategy.  $\square$

## B Proofs for Section 5

**Lemma B.1.** *Let  $\Pi = \{(C_t, \sigma_t, w_t)\}_{t=1}^T$  be a partition of  $G$  with non-increasing payoffs  $w_1 \geq \dots \geq w_T$, and let  $(\tilde{C}, \tilde{\sigma}, \tilde{w})$  be a blocking coalition of  $\Pi$  with evidence  $\tilde{X} := X(\tilde{\sigma})$. Then*

$$
M^{-1}(\tilde{X} \cap X_t) \cap C_t \subseteq \tilde{C} \quad \text{for every } t.
$$

*Proof.* Let  $\tau := \min\{t \mid w_t < \tilde{w}\}$  be the lowest index such that types in  $C_1 \cup \dots \cup C_{\tau-1}$  get at least  $\tilde{w}$  and types in  $R_\tau$  get strictly less than  $\tilde{w}$. Such an index exists because  $\Theta(\tilde{w}) \neq \emptyset$.

By the definition of a blocking coalition,  $(\tilde{C}, \tilde{\sigma}, \tilde{w})$  is a coalition of  $G|_{\Theta(\tilde{w})}$, where  $\Theta(\tilde{w}) = \bigcup_{t:w_t < \tilde{w}} C_t = R_\tau$. By evidence exclusivity of that coalition, we have  $M_\tau^{-1}(\tilde{X}) = \tilde{C}$, and thus  $M^{-1}(\tilde{X}) \cap R_\tau = \tilde{C}$  where  $\tilde{C} \subseteq R_\tau$.

Now, observe that for all  $t < \tau$, we have  $\tilde{X} \cap X_t = \emptyset$. Indeed, suppose  $m \in \tilde{X} \cap X_t$. Since  $m \in \tilde{X}$, some type  $\theta \in \tilde{C}$  has  $m \in M(\theta)$, and  $\theta \in \tilde{C} \subseteq R_\tau \subseteq R_{t+1} = R_t \setminus C_t$. But  $m \in X_t$  and exclusivity of the step- $t$  coalition give  $M_t^{-1}(X_t) = C_t$, so no type of  $R_t \setminus C_t$  can send  $m$, a contradiction. Hence  $M^{-1}(\tilde{X} \cap X_t) \cap C_t = \emptyset \subseteq \tilde{C}$.

For all  $t \geq \tau$, we have

$$
M^{-1}(\tilde{X} \cap X_t) \cap C_t \subseteq M^{-1}(\tilde{X}) \cap R_\tau = \tilde{C}.
$$

$\square$

**Lemma B.2.** *For every non-empty  $R \subseteq \Theta$, the coalition payoff set  $\mathcal{W}_R$  is non-empty and compact.*

*Proof.* Since  $G|_R$  is itself a disclosure game and  $\mathcal{W}_R$  is its set of coalition payoffs, it suffices to prove the claim for  $R = \Theta$; write  $\mathcal{W} := \mathcal{W}_\Theta$. Non-emptiness is Remark 1.

For each non-empty  $C \subseteq \Theta$, let  $\text{Coal}(C) := \{(\sigma, w) \mid (C, \sigma, w) \text{ is a coalition of } G\}$. We show that every  $\text{Coal}(C)$  is compact. Granting this, its image  $\mathcal{W}(C)$  under the continuous projection  $(\sigma, w) \mapsto w$  is compact, and  $\mathcal{W} = \bigcup_C \mathcal{W}(C)$  is a finite union of compact sets, hence compact.

Fix  $C$. The coalition strategies on  $C$  form the compact set  $S := \prod_{\theta \in C} \Delta M(\theta)$, and every coalition payoff lies in the interval  $I := [\min_{\Delta \Theta} \underline{v}, \max_{\Delta \Theta} \bar{v}]$, which is finite because, by (A4),  $\underline{v}$  is lower semicontinuous and  $\bar{v}$  upper semicontinuous on the compact set  $\Delta \Theta$. Thus  $\text{Coal}(C)$  is a subset of the compact set  $S \times I$, and it suffices to show that it is closed.

Let  $(\sigma_k, w_k) \in \text{Coal}(C)$  with  $(\sigma_k, w_k) \rightarrow (\sigma, w)$. The evidence sets  $X(\sigma_k)$  take finitely many values, so after discarding all but infinitely many terms, which leaves the limit unchanged,  $X(\sigma_k) = X$  for all  $k$. We claim  $(C, \sigma, w)$  is a coalition of  $G$. First,  $\sigma$  is again a coalition strategy on  $C$, since the constraints defining  $S$  are preserved in the limit. Second,  $X(\sigma) \subseteq X$: if  $\sigma(m|\theta) > 0$, then  $\sigma_k(m|\theta) > 0$  for all large  $k$, so  $m \in X(\sigma_k) = X$. Each  $(C, \sigma_k, w_k)$  satisfies exclusivity (C3), so  $M^{-1}(X) = M^{-1}(X(\sigma_k)) \subseteq C$; with  $X(\sigma) \subseteq X$, this gives  $M^{-1}(X(\sigma)) \subseteq C$, which is (C3) for  $(C, \sigma, w)$.

Third, fix  $m \in X(\sigma)$. The denominator  $p_\sigma(m) = \sum_{\theta \in C} \mu_C^0(\theta) \sigma(m|\theta)$  is positive and  $p_{\sigma_k}(m) \rightarrow p_\sigma(m)$, so the induced beliefs converge,  $\mu_{\sigma_k}(\cdot|m) \rightarrow \mu_\sigma(\cdot|m)$. Since  $m \in X(\sigma) \subseteq X = X(\sigma_k)$, condition (C4) for  $(C, \sigma_k, w_k)$  gives  $\underline{v}(\mu_{\sigma_k}(\cdot|m)) \leq w_k \leq \bar{v}(\mu_{\sigma_k}(\cdot|m))$. Letting  $k \rightarrow \infty$, the semicontinuity of the envelopes from (A4) yields  $w \in V(\mu_\sigma(\cdot|m))$. As  $m \in X(\sigma)$  was arbitrary,  $(C, \sigma, w)$  is a coalition of  $G$, so  $(\sigma, w) \in \text{Coal}(C)$. Hence  $\text{Coal}(C)$  is a closed subset of  $S \times I$, and therefore compact. Messages in  $X \setminus X(\sigma)$  simply drop out of the limit coalition, since the common-payoff condition is required only on the evidence actually used.  $\square$

### Proof of Proposition 2

($\Leftarrow$) Let  $\Pi$  be greedy. By Definition 10,  $w_t$  belongs to  $\mathcal{W}_t \cap [\max_{\theta \in R_t} \underline{u}(\theta), w_{t-1}]$  for every  $t$. Thus  $w_t \leq w_{t-1}$  and  $w_t \geq \max_{\theta \in R_t} \underline{u}(\theta)$  for every  $t$, so  $\Pi$  has non-increasing payoffs and is IR by Lemma 1. Hence  $\Pi$  is a PBE partition.

Suppose, toward a contradiction, that  $(\tilde{C}, \tilde{\sigma}, \tilde{w})$  is a blocking coalition of  $\Pi$. Let

$$
\tau := \min\{t \mid w_t < \tilde{w}\}.
$$

This index exists because  $\Theta(\tilde{w}) \neq \emptyset$. Since payoffs are non-increasing, the cells with payoff strictly below  $\tilde{w}$  are exactly  $C_\tau, \dots, C_T$: for  $s < \tau$,  $w_s \geq \tilde{w}$  by minimality of  $\tau$, while for  $s \geq \tau$,  $w_s \leq w_\tau < \tilde{w}$. Hence  $\Theta(\tilde{w}) = R_\tau$. By the definition of blocking,  $(\tilde{C}, \tilde{\sigma}, \tilde{w})$  is a coalition of  $G|_{\Theta(\tilde{w})} = G_\tau$, so  $\tilde{w} \in \mathcal{W}_\tau$. Moreover,  $\tilde{w} > w_\tau \geq \max_{\theta \in R_\tau} \underline{u}(\theta)$, and  $\tilde{w} \leq w_{\tau-1}$  by minimality of  $\tau$  (with  $w_0 = \infty$  if  $\tau = 1$). Thus

$$
\tilde{w} \in \mathcal{W}_\tau \cap [\max_{\theta \in R_\tau} \underline{u}(\theta), w_{\tau-1}], \quad \tilde{w} > w_\tau,
$$

contradicting Definition 10, which makes  $w_\tau$  the maximum of this set.

($\Rightarrow$) Let  $\Pi$  be a coalition-proof PBE partition, and fix  $t$. Since  $(C_t, \sigma_t, w_t)$  is a coalition of  $G_t$, we have  $w_t \in \mathcal{W}_t$. Since  $\Pi$  is a PBE partition, its payoffs are non-increasing and it is IR; by Lemma 1,  $w_t \geq \max_{\theta \in R_t} \underline{u}(\theta)$, and non-increasing payoffs give  $w_t \leq w_{t-1}$. Hence  $w_t$  belongs to the greedy constraint set at step  $t$.

Suppose  $w_t$  is not the maximum of that set. Then there is a payoff  $w \in \mathcal{W}_t$  such that

$$
w \in \mathcal{W}_t \cap [\max_{\theta \in R_t} \underline{u}(\theta), w_{t-1}], \quad w > w_t.
$$

Since  $w \in \mathcal{W}_t$, there exists a coalition  $(C, \sigma, w)$  of  $G_t$. For every  $s < t$, non-increasing payoffs give  $w_s \geq w_{t-1} \geq w$; for every  $s \geq t$, they give  $w_s \leq w_t < w$. Therefore  $\Theta(w) = R_t \neq \emptyset$. Since  $(C, \sigma, w)$  is a coalition of  $G_t = G|_{\Theta(w)}$, it is a blocking coalition of  $\Pi$, contradicting coalition proofness. Thus  $w_t$  is the maximum of the greedy constraint set at every step, and  $\Pi$  is greedy.

The strategy-level statement follows directly from the definitions: a strategy is a coalition-proof PBE strategy if and only if it is associated with some coalition-proof PBE partition, and the partition equivalence just proved is exactly equivalence with being associated with some greedy partition.  $\square$

## C Coalition-Optimal Equilibrium Partitions

This section develops an auxiliary strengthening of greedy partitions which we use to prove existence theorems.

Call a sequence  $\{(C_s, \sigma_s, w_s)\}_{s=1}^{t-1}$  (possibly empty) a *prefix* with residual  $R_t$  if, setting  $R_1 := \Theta$  and  $R_{s+1} := R_s \setminus C_s$, each  $(C_s, \sigma_s, w_s)$  is a coalition of  $G_s$  and  $R_t \neq \emptyset$. Initial segments of partitions are prefixes; conversely, a prefix is an unfinished run of Algorithm 1. Call such a prefix *greedy* if  $w_s = \max(\mathcal{W}_s \cap [\max_{\theta \in R_s} \underline{u}(\theta), w_{s-1}])$  at every step  $s$  (with  $w_0 := \infty$). Every initial segment of a greedy partition is a greedy prefix, and an unfinished run of Algorithm 3 is exactly a greedy prefix.

For  $m \in \mathcal{M}$, let  $F(m) := \{\theta \in \Theta \mid M(\theta) = \{m\}\}$  be the set of types *forced* to send  $m$.

**Lemma C.1.** *Let  $R_t$  be the residual of a prefix. Then:*

- (i) *for every  $m \in \mathcal{M}_t$, every type forced to send  $m$  remains:  $F(m) \subseteq R_t$;*
- (ii) *for every  $m \in \mathcal{M}_t$, the conditional prior on the residual senders of  $m$  is feasible:  $\mu_{P(m) \cap R_t}^0 \in \mathcal{F}(m)$;*
- (iii)  *$\mathcal{W}_t$  is non-empty and compact, and*

$$
\max \mathcal{W}_t \geq \max_{\theta \in R_t} \underline{u}(\theta).
$$

*Proof.* (i) Suppose  $\theta_F \in F(m) \cap C_s$  for some  $s < t$. Since  $M(\theta_F) = \{m\}$  and  $\emptyset \neq \text{supp } \sigma_s(\cdot \mid \theta_F) \subseteq M(\theta_F)$, we have  $m \in X_s := X(\sigma_s)$. As  $m \in \mathcal{M}_t$, some  $\theta \in R_t$  has  $m \in M(\theta)$; then  $\theta \in R_s$  and  $m \in M(\theta) \cap X_s$  give  $\theta \in M_s^{-1}(X_s) \subseteq C_s$  by exclusivity at step  $s$, contradicting  $\theta \in R_t \subseteq R_{s+1} = R_s \setminus C_s$.

(ii) Write  $P_t := P(m) \cap R_t$, which is non-empty since  $m \in \mathcal{M}_t$  and contains  $F(m)$  by (i). Let  $\sigma$  be a strategy under which every type in  $P_t$  sends  $m$  and every other type sends a message in  $M(\theta) \setminus \{m\}$; this is possible because no type outside  $P_t$  is forced to send  $m$, by (i). The senders of  $m$  are then exactly  $P_t$, so  $\mu_\sigma(\cdot \mid m) = \mu_{P_t}^0$, and hence  $\mu_{P_t}^0 \in \mathcal{F}(m)$.

(iii)  $\mathcal{M}_t$  is non-empty, so  $G_t$  admits a coalition by Remark 1 and  $\mathcal{W}_t \neq \emptyset$; it is compact by Lemma B.2, so the maximum exists. Fix  $\theta \in R_t$  and  $m \in M(\theta) \subseteq \mathcal{M}_t$. Pooling on  $m$  in  $G_t$  is a coalition with type set  $P_t = P(m) \cap R_t$  and payoff  $\bar{v}(\mu_{P_t}^0)$, so by (ii),

$$
\max \mathcal{W}_t \geq \bar{v}(\mu_{P_t}^0) \geq \underline{v}(\mu_{P_t}^0) \geq \min_{\mu \in \mathcal{F}(m)} \underline{v}(\mu).
$$

Taking the maximum over  $m \in M(\theta)$  gives  $\max \mathcal{W}_t \geq \underline{u}(\theta)$, and then the maximum over  $\theta \in R_t$.  $\square$

**Definition C.1.** A partition  $\Pi = \{(C_t, \sigma_t, w_t)\}_{t=1}^T$  of  $G$  is a coalition-optimal equilibrium (COE) partition if, for every  $t$  (with  $w_0 := \infty$),

$$
w_t = \max \mathcal{W}_t \quad \text{and} \quad w_t \leq w_{t-1}.
$$

The maximum exists by Lemma C.1(iii). Unlike the greedy selection, a COE partition maximizes the payoff with no upper constraint and then requires the resulting sequence to be non-increasing.

**Lemma C.2.** (i) *Every COE partition of  $G$  is a greedy partition, hence a coalition-proof PBE partition.*

- (ii) *A greedy partition of  $G$  is a COE partition if and only if  $\max \mathcal{W}_t \leq w_{t-1}$  for every  $t$.*

*Proof.* (i) Let  $\Pi$  be a COE partition and fix  $t$. The initial segment through step  $t - 1$  is a prefix with residual  $R_t$, so by Lemma C.1(iii),  $w_t = \max \mathcal{W}_t \geq \max_{\theta \in R_t} \underline{u}(\theta)$, and  $w_t \leq w_{t-1}$  by definition. Hence  $w_t$  lies in the greedy constraint set  $\mathcal{W}_t \cap [\max_{\theta \in R_t} \underline{u}(\theta), w_{t-1}]$  and equals the maximum of  $\mathcal{W}_t$, which contains that set; so  $w_t$  is the maximum of the constraint set, and  $\Pi$  is greedy. By Proposition 2,  $\Pi$  is a coalition-proof PBE partition.

(ii) Let  $\Pi$  be greedy. If  $\Pi$  is COE, then  $\max \mathcal{W}_t = w_t \leq w_{t-1}$  for every  $t$. Conversely, suppose  $\max \mathcal{W}_t \leq w_{t-1}$  for every  $t$. By Lemma C.1(iii),  $\max \mathcal{W}_t \geq \max_{\theta \in R_t} \underline{u}(\theta)$, so  $\max \mathcal{W}_t$  lies in the greedy constraint set, whose maximum is therefore  $\max \mathcal{W}_t$  itself; by greediness  $w_t = \max \mathcal{W}_t$, and  $w_t \leq w_{t-1}$  holds since  $w_t$  lies in the constraint set. So  $\Pi$  is COE.  $\square$

## D Proofs for Section 6.1

### Proof of Lemma 2

Consider coalition  $(C, \sigma, w)$  of  $G$. By exclusivity of evidence,  $M^{-1}(X(\sigma)) = C$. Iterating M-C over the finite set  $X(\sigma)$  delivers message  $m^*$  with  $P(m^*) = \bigcup_{m \in X(\sigma)} P(m) = M^{-1}(X(\sigma)) = C$. By Remark 1, the pooling triple  $(C, \sigma^{m^*}, w')$  is a coalition for every  $w' \in V(\mu_C^0)$, in particular, for  $w' = \bar{v}(\mu_C^0)$.

Next, in  $G|_C$, the prior  $\mu_C^0$  is a linear combination of the posteriors  $\mu_\sigma(\cdot | m)$  induced by  $\sigma$  at the messages  $m \in X(\sigma)$, with weights  $p_\sigma(m) > 0$  summing to one. Iterating QC gives

$$
\bar{v}(\mu_C^0) \geq \min_{m \in X(\sigma)} \bar{v}(\mu_\sigma(\cdot | m)) \geq w,
$$

where the last inequality follows from the common-payoff condition (C4) for coalition  $(C, \sigma, w)$.

For the identity, every pooling payoff  $\bar{v}(\mu_{P(m)}^0)$  is itself a coalition payoff by Remark 1. Also, by the first part of this proof, every coalition  $(C, \sigma, w)$  satisfies  $w \leq \bar{v}(\mu_C^0) = \bar{v}(\mu_{P(m^*)}^0)$, where  $P(m^*) = C$. Hence the largest coalition payoff is  $\max \mathcal{W}_\Theta = \max_{m \in \mathcal{M}} \bar{v}(\mu_{P(m)}^0)$, and any coalition attaining it has  $\bar{v}(\mu_C^0) = \max \mathcal{W}_\Theta$.  $\square$

The following merging lemma is used in the existence and essential-uniqueness results below. It shows that, under QC\* or genericity, removing a coalition that attains  $\max \mathcal{W}_R$  cannot raise the largest coalition payoff on the residual types  $R \setminus C$.

**Lemma D.1.** *Assume QC and M-C. Let  $R \subseteq \Theta$  be non-empty, let  $(C, \sigma, w)$  be a coalition of  $G|_R$  with  $w = \max \mathcal{W}_R$, and suppose  $R' := R \setminus C$  is non-empty and*

$$
\max \mathcal{W}_{R'} > \max \mathcal{W}_R.
$$

*Then there is a coalition of  $G|_R$  whose type set  $\tilde{C}$  satisfies  $C \subsetneq \tilde{C}$  and whose payoff is  $w$; moreover  $\bar{v}(\mu_C^0) = \bar{v}(\mu_{\tilde{C}}^0) = w$. If QC\* holds, or if  $G$  is generic, this configuration is impossible; that is,*

$$
\max \mathcal{W}_{R \setminus C} \leq \max \mathcal{W}_R
$$

*whenever a coalition with type set  $C$  attains  $\max \mathcal{W}_R$  and  $R \setminus C \neq \emptyset$.*

*Proof.* Since QC and M-C are inherited by restricted games (Section 6.1), Lemma 2 and Remark 1 apply in  $G|_R$  and  $G|_{R'}$; the sets  $\mathcal{W}_R$  and  $\mathcal{W}_{R'}$  are non-empty and compact by Lemma B.2, so both maxima exist.

By Lemma 2 in  $G|_R$, there is a message  $m''$  with  $P(m'') \cap R = C$  and  $\bar{v}(\mu_C^0) \in \mathcal{W}_R$; since  $w = \max \mathcal{W}_R$, this gives  $\bar{v}(\mu_C^0) = w$. Let  $(C', \sigma', w')$  be a coalition of  $G|_{R'}$  with  $w' = \max \mathcal{W}_{R'}$; by Lemma 2 in  $G|_{R'}$, there is a message  $m$  with  $P(m) \cap R' = C'$  and  $\bar{v}(\mu_{C'}^0) = w'$.

Apply M-C in  $G|_R$  to  $m''$  and  $m$: there is a message  $m'$  with

$$
P(m') \cap R = (P(m'') \cap R) \cup (P(m) \cap R) = C \cup (C' \cup (P(m) \cap C)) = C \cup C' =: \tilde{C},
$$

using  $P(m) \cap R = (P(m) \cap R') \cup (P(m) \cap C) = C' \cup (P(m) \cap C)$. Since  $C'$  is non-empty and disjoint from  $C$, we get  $C \subsetneq \tilde{C}$. By Remark 1 in  $G|_R$,  $(\tilde{C}, \sigma^{m'}, \tilde{w})$  is a coalition of  $G|_R$  for every  $\tilde{w} \in V(\mu_{\tilde{C}}^0)$.

Since  $C$  and  $C'$  are disjoint and non-empty,

$$
\mu_{\tilde{C}}^0 = \lambda \mu_C^0 + (1 - \lambda) \mu_{C'}^0, \quad \lambda := \frac{\mu_C^0(C)}{\mu_{\tilde{C}}^0(\tilde{C})} \in (0, 1).
$$

By QC,

$$
\bar{v}(\mu_{\tilde{C}}^0) \geq \min\{\bar{v}(\mu_C^0), \bar{v}(\mu_{C'}^0)\} = \min\{w, w'\} = w,
$$

while  $\bar{v}(\mu_{\tilde{C}}^0) \in \mathcal{W}_R$  gives  $\bar{v}(\mu_{\tilde{C}}^0) \leq w$. Hence  $\bar{v}(\mu_{\tilde{C}}^0) = w$  and  $(\tilde{C}, \sigma^{m'}, w)$  is the required coalition.

Under QC\*, the conditional priors  $\mu_C^0$  and  $\mu_{C'}^0$  have disjoint supports, so  $\mu_C^0 \neq \mu_{C'}^0$  and the QC inequality is strict; then  $\bar{v}(\mu_{\tilde{C}}^0) > w$, which contradicts  $\bar{v}(\mu_{\tilde{C}}^0) \leq w$. Under genericity,  $\bar{v}(\mu_{\tilde{C}}^0) = w = \bar{v}(\mu_C^0)$  gives  $\tilde{C} = C$, which contradicts  $C \subsetneq \tilde{C}$. In either case  $\max \mathcal{W}_{R'} > \max \mathcal{W}_R$  is impossible.  $\square$

### Proof of Theorem 1

Existence. Assume M-C and QC. We construct a COE partition recursively. Let  $R_1 := \Theta$. Given a non-empty residual set  $R_t$, the maximum  $\max \mathcal{W}_t$  exists by Lemma C.1(iii). Let  $(C_t, \sigma_t, w_t)$  be a coalition of  $G_t$  with  $w_t = \max \mathcal{W}_t$  and  $C_t$  of largest cardinality among the type sets attaining  $\max \mathcal{W}_t$. Each step removes a non-empty cell, so the recursion stops and returns a partition  $\Pi$  with  $w_t = \max \mathcal{W}_t$  at every step.

We claim  $w_t \leq w_{t-1}$  for every  $t \geq 2$, so that  $\Pi$  is a COE partition. Suppose not, so  $\max \mathcal{W}_t = w_t > w_{t-1} = \max \mathcal{W}_{t-1}$  at some step. Apply Lemma D.1 with  $R = R_{t-1}$, the coalition  $(C_{t-1}, \sigma_{t-1}, w_{t-1})$, and  $R \setminus C_{t-1} = R_t$: since  $w_{t-1} = \max \mathcal{W}_{t-1}$  and  $\max \mathcal{W}_t > \max \mathcal{W}_{t-1}$, it gives a coalition of  $G_{t-1}$  with payoff  $w_{t-1}$  and type set  $\tilde{C} \supsetneq C_{t-1}$. This contradicts the choice of  $C_{t-1}$  as a type set of largest cardinality attaining  $\max \mathcal{W}_{t-1}$. Hence  $\Pi$  is a COE partition, so by Lemma C.2(i) it is a coalition-proof PBE partition and  $\sigma^\Pi$  a coalition-proof PBE strategy; in particular a coalition-proof PBE exists, proving existence.

(i) Assume M-C and QC\*. We show by induction on  $t$  that every greedy choice satisfies  $w_t = \max \mathcal{W}_t$. Suppose  $w_s = \max \mathcal{W}_s$  for every step  $s < t$  of a greedy prefix, and let  $R_t \neq \emptyset$. We first bound  $\max \mathcal{W}_t$  from above by  $w_{t-1}$. If  $t \geq 2$, apply the QC\* case of Lemma D.1 with  $R = R_{t-1}$  and the step- $(t-1)$  coalition. By the inductive hypothesis this coalition attains  $\max \mathcal{W}_{t-1} = w_{t-1}$, so the lemma gives  $\max \mathcal{W}_t \leq w_{t-1}$. If  $t = 1$, the bound  $\max \mathcal{W}_t \leq w_0 = \infty$  is trivial. For the lower bound, Lemma C.1(iii) gives  $\max \mathcal{W}_t \geq \max_{\theta \in R_t} \underline{u}(\theta)$. The two bounds place  $\max \mathcal{W}_t$  in the greedy constraint set  $\mathcal{W}_t \cap [\max_{\theta \in R_t} \underline{u}(\theta), w_{t-1}]$. This set is non-empty, and its maximum is  $\max \mathcal{W}_t$. So every greedy choice at step  $t$  takes  $w_t = \max \mathcal{W}_t$, which completes the induction.

Since the greedy constraint set is non-empty at every step, no run of Algorithm 3 halts. Each run removes a non-empty cell, so it terminates in a greedy partition. By Proposition 2, that greedy partition is a coalition-proof PBE partition.

For the characterization, every coalition-proof PBE partition is greedy by Proposition 2, so the induction gives  $w_t = \max \mathcal{W}_t$, while  $w_t \leq w_{t-1}$  is part of the greedy constraint. Conversely, a partition satisfying these two conditions is coalition-optimal; by Lemma C.2(i) it is greedy, and by Proposition 2 it is a coalition-proof PBE partition. (ii) Assume M-C, QC, and genericity. The induction in (i) goes through with the genericity case of Lemma D.1 in place of the QC\* case, so every greedy partition again has  $w_t = \max \mathcal{W}_t$  at every step. Let  $\Pi$  and  $\Pi'$  be greedy partitions; we show by induction on  $t$  that they have the same residual sets and cells. Suppose  $R_t = R'_t$, which holds for  $t = 1$. If both are empty, the two partitions both have length  $t - 1$. Otherwise both continue, with  $w_t = w'_t = \max \mathcal{W}_t$. By the last claim of Lemma 2 applied in  $G_t$, the cells satisfy  $\bar{v}(\mu_{C_t}^0) = \max \mathcal{W}_t = \bar{v}(\mu_{C'_t}^0)$, so  $C_t = C'_t$  because  $G$  is generic, and then  $R_{t+1} = R'_{t+1}$. Hence  $\Pi$  and  $\Pi'$  have the same length, cells, and payoffs. By Proposition 2 the coalition-proof PBE partitions are exactly the greedy partitions, so any two coalition-proof PBE partitions of  $G$  have the same length, cells, and payoffs.  $\square$

## E Proofs for Section 6.2

### Proof of Lemma 3

(i) In  $G|_C$, the prior is the convex combination

$$
\mu_C^0 = \sum_{m \in X(\sigma)} p_\sigma(m) \mu_\sigma(\cdot | m),
$$

with positive weights on the on-path posteriors. By (C4) and single-valuedness, each posterior in this combination has value  $w$. Applying B to that convex combination yields  $w \leq v(\mu_C^0) \leq w$.

(ii) For every non-empty  $R \subseteq \Theta$, write

$$
v^*(R) := \max_{\emptyset \neq X \subseteq \mathcal{M}_R} v(\mu_{M_R^{-1}(X)}^0).
$$

The maximum is over finitely many sets, and each relative preimage is non-empty because every message in  $\mathcal{M}_R$  is available to some type in  $R$.

Every coalition  $(C, \sigma, w)$  of  $G|_R$  has type set  $C = M_R^{-1}(X(\sigma))$  by exclusivity and its converse. Hence Lemma 3(i), applied in the restricted game, gives  $w = v(\mu_C^0) \leq v^*(R)$, so  $\max \mathcal{W}_R \leq v^*(R)$.

Let  $X^* \subseteq \mathcal{M}_R$  attain  $v^*(R)$  and set  $C^* := M_R^{-1}(X^*)$. Consider the auxiliary game

$$
G' := (C^*, X^*, M', \mu_{C^*}^0, V_{C^*}), \quad M'(\theta) := M(\theta) \cap X^*.
$$

Note that  $G'$  is a disclosure game: by construction every type in  $C^*$  can send some message in  $X^*$, and every message in  $X^*$  is available to some type of  $C^*$; the remaining assumptions are inherited from  $G$. However,  $G'$  is not simply a restricted game of  $G$, because its message space is set to  $X^*$  rather than  $\mathcal{M}_{C^*}$, so some messages available to types in  $C^*$  are not available in  $G'$. Nevertheless, the definitions and equilibrium results apply to it.

Every disclosure game has a PBE partition with decreasing payoffs (Lemma J.4 and Proposition 1); applied to  $G'$  this gives a partition  $\{(\tilde{C}_\tau, \tilde{\sigma}_\tau, \tilde{w}_\tau)\}_{\tau=1}^k$  with  $\tilde{w}_1 \geq \dots \geq \tilde{w}_k$. Each cell is a coalition in a residual game of  $G'$, so Lemma 3 gives  $\tilde{w}_\tau = v(\mu_{\tilde{C}_\tau}^0)$. Since the cells partition  $C^*$,

$$
\mu_{C^*}^0 = \sum_\tau \frac{\mu^0(\tilde{C}_\tau)}{\mu^0(C^*)} \mu_{\tilde{C}_\tau}^0
$$

is a convex combination with positive weights. From quasiconvexity of  $v$, we get

$$
v^*(R) = v(\mu_{C^*}^0) \leq \max_\tau \tilde{w}_\tau = \tilde{w}_1.
$$

Next notice that  $(\tilde{C}_1, \tilde{\sigma}_1, \tilde{w}_1)$  is a coalition of  $G|_R$. Conditions (C1), (C2), and (C4) carry over because  $\tilde{\sigma}_1$  uses only messages in  $M'(\theta) \subseteq M(\theta)$  and induces the same posteriors when viewed in  $G|_R$. For exclusivity in  $G|_R$, take  $\theta \in R$  with  $M(\theta) \cap X(\tilde{\sigma}_1) \neq \emptyset$. Since  $X(\tilde{\sigma}_1) \subseteq X^*$, this type lies in  $C^*$; then  $M'(\theta) \cap X(\tilde{\sigma}_1) \neq \emptyset$, so exclusivity of the first cell in  $G'$  places  $\theta$  in  $\tilde{C}_1$. Thus  $\tilde{w}_1 \in \mathcal{W}_R$. Combining the inequalities,

$$
\tilde{w}_1 \leq \max \mathcal{W}_R \leq v^*(R) \leq \tilde{w}_1,
$$

so  $(\tilde{C}_1, \tilde{\sigma}_1, \tilde{w}_1)$  is a coalition of  $G|_R$  paying  $v^*(R)$.  $\square$

**Lemma E.1.** *Assume B. Let  $R \subseteq \Theta$  be non-empty, let  $(C, \sigma, w)$  be a coalition of  $G|_R$  with  $w = \max \mathcal{W}_R$, and suppose  $R' := R \setminus C$  is non-empty and  $G|_{R'}$  admits a coalition  $(C', \sigma', w')$  with  $w' > w$. Then*

$$
M_R^{-1}(X(\sigma) \cup X(\sigma')) = C \cup C' \quad \text{and} \quad v(\mu_{C \cup C'}^0) = w.
$$

*If in addition  $B^*$  holds, no such configuration exists, so*

$$
\max \mathcal{W}_{R \setminus C} \leq \max \mathcal{W}_R
$$

*whenever a coalition with type set  $C$  attains  $\max \mathcal{W}_R$  and  $R \setminus C \neq \emptyset$.*

*Proof.* First identify the type set generated by the combined evidence. Preimages distribute over unions:

$$
M_R^{-1}(X(\sigma) \cup X(\sigma')) = M_R^{-1}(X(\sigma)) \cup M_R^{-1}(X(\sigma')).
$$

The first term is  $C$  by exclusivity and its converse. For the second term,

$$
M_{R'}^{-1}(X(\sigma')) \setminus C = M_{R'}^{-1}(X(\sigma')) = C',
$$

and hence  $M_R^{-1}(X(\sigma')) \subseteq C \cup C'$. The union is therefore exactly  $C \cup C'$.

By Lemma 3,  $v(\mu_C^0) = w$  and  $v(\mu_{C'}^0) = w' > w$. The sets  $C$  and  $C'$  are disjoint and non-empty, so

$$
\mu_{C \cup C'}^0 = \lambda \mu_C^0 + (1 - \lambda) \mu_{C'}^0, \quad \lambda := \frac{\mu^0(C)}{\mu^0(C \cup C')} \in (0, 1).
$$

The quasiconcave half of B gives

$$
v(\mu_{C \cup C'}^0) \geq \min\{w, w'\} = w.
$$

On the other hand,  $C \cup C'$  is a relative preimage in  $G|_R$, so Lemma 3(ii) gives

$$
v(\mu_{C \cup C'}^0) \leq v^*(R) = \max \mathcal{W}_R = w.
$$

Thus  $v(\mu_{C \cup C'}^0) = w$.

If  $B^*$  holds, the same convex combination must satisfy the strict quasiconcave inequality, because  $v(\mu_C^0) = w \neq w' = v(\mu_{C'}^0)$. This would give  $v(\mu_{C \cup C'}^0) > w$, contradicting the equality just proved. Therefore no coalition of  $G|_{R'}$  can pay more than  $\max \mathcal{W}_R$.  $\square$

Existence under plain betweenness in Theorem 2 rests on the following ordering of each level set of  $v$.

**Lemma E.2.** *Assume B and fix  $y \in \mathbb{R}$. Write  $L := \{\mu \in \Delta\Theta \mid v(\mu) = y\}$,  $L^{++} := \{v > y\}$, and  $L^{--} := \{v < y\}$. There is a complete and transitive relation  $\succeq$  on  $L$  such that:*

- (i) *If  $x \in L$,  $u \in L^{++}$,  $\alpha \in (0, 1)$, and  $z := \alpha x + (1 - \alpha)u \in L$, then  $z \succ x$.*

- (ii) If  $\bar{\mu} \in L$  and  $\bar{\mu} = \sum_{\tau} \alpha_{\tau} \mu_{\tau}$  is a convex combination with positive weights of beliefs  $\mu_{\tau} \in L \cup L^{--}$, then  $\mu_{\tau} \in L$  and  $\mu_{\tau} \succeq \bar{\mu}$  for some  $\tau$.

*Proof.* We use the following proper-separation fact. If two non-empty convex sets  $A_1, A_2 \subseteq \mathbb{R}^{\Theta}$  are disjoint, then their relative interiors are disjoint, so there is an affine functional  $h$  with  $h \leq 0$  on  $A_1$,  $h \geq 0$  on  $A_2$, and  $h$  not identically zero on  $A_1 \cup A_2$  (Rockafellar, 1970, Theorem 11.3).

The order is constructed recursively on convex subsets of the simplex. Fix a non-empty compact convex set  $F \subseteq \Delta\Theta$  and write

$$
S_F := L \cap F, \quad U_F := L^{++} \cap F, \quad B_F := L^{--} \cap F.
$$

Under B, these three sets are convex; so are  $L^+ := \{v \geq y\}$  and  $L^- := \{v \leq y\}$. We prove, by induction on the affine dimension  $d$  of  $F$, that there is a complete and transitive relation  $\succeq_F$  on  $S_F$  satisfying (i) and (ii), with  $(S_F, U_F, B_F)$  in place of  $(L, L^{++}, L^{--})$. The lemma is the case  $F = \Delta\Theta$. If  $S_F = \emptyset$, there is nothing to order, so assume  $S_F \neq \emptyset$.

*Base case  $d = 0$.* Then  $F$  is a singleton. Declare its points indifferent. Condition (i) is vacuous, because it would require the singleton to lie in both  $S_F$  and  $U_F$, and (ii) holds because every belief in the decomposition is the singleton.

*Inductive step.* Let  $d \geq 1$  and assume the claim for all non-empty compact convex sets of affine dimension below  $d$. In Cases 2–4, we construct the order in two layers. First, we assign each  $x \in S_F$  a rank  $\rho(x)$. We choose the rank so that, for every  $r \in \rho(S_F)$, all beliefs in  $S_F$  with rank  $r$  lie in a compact convex set  $F_r \subseteq F$  of affine dimension below  $d$. The inductive hypothesis gives an order  $\succeq_{F_r}$  on each such slice. We then order beliefs lexicographically:

$$
x \succeq_F x' \quad : \iff \quad \rho(x) > \rho(x') \quad \text{or} \quad (\rho(x) = \rho(x') \quad \text{and} \quad x \succeq_{F_{\rho(x)}} x').
$$

Thus, the rank compares beliefs in different slices, while the inductively constructed order compares beliefs within the same slice. The resulting relation is complete and transitive.

*Case 1:*  $U_F = B_F = \emptyset$. Declare indifference; (i) is vacuous and (ii) holds with any  $\mu_{\tau}$, all of which lie in  $S_F$.

*Case 2:*  $U_F \neq \emptyset$,  $B_F = \emptyset$. Separate the weak lower set  $L^- \cap F$  from the strict upper set  $U_F$. Thus there is an affine  $h$  with  $h \leq 0$  on  $L^- \cap F$,  $h \geq 0$  on  $U_F$, and  $h$  not identically zero on their union. Put  $\rho := h$  on  $S_F$  and  $F_r := \{x \in F \mid h(x) = r\}$. The functional  $h$  is non-constant on  $\text{aff}(F)$: if it were constant on  $F$, its value would be both non-positive on  $S_F$  and non-negative on  $U_F$, hence zero, contradicting proper separation. Each slice is therefore lower-dimensional.

For (i), write  $z = \alpha x + (1 - \alpha)u$  with  $x, z \in S_F$  and  $u \in U_F$. Then

$$
h(z) - h(x) = (1 - \alpha)(h(u) - h(x)) \geq 0.
$$

If the inequality is strict, then  $z \succ_F x$  by rank. If it is an equality, then  $x, z, u \in F_0$, and the inductive version of (i) inside  $F_0$  gives  $z \succ_F x$.

For (ii), all beliefs in the decomposition lie in  $S_F$, because  $B_F = \emptyset$. If some component has  $h(\mu_{\tau}) > h(\bar{\mu})$, it is ranked above  $\bar{\mu}$. Otherwise every component has  $h(\mu_{\tau}) \leq h(\bar{\mu})$, and the affine identity

$$
\sum_{\tau} \alpha_{\tau} h(\mu_{\tau}) = h(\bar{\mu})
$$

forces all components to have the same value  $r := h(\bar{\mu})$. They all lie in  $F_r$, so the inductive version of (ii) in  $F_r$  applies.

*Case 3:*  $U_F = \emptyset$,  $B_F \neq \emptyset$. Separate the strict lower set  $B_F$  from the weak upper set  $L^+ \cap F$. Let  $h$  be affine with  $h \leq 0$  on  $B_F$,  $h \geq 0$  on  $L^+ \cap F$, and not identically zero on their union. Define  $\rho$ and the slices as in Case 2. The same argument shows that  $h$  is non-constant on  $\text{aff}(F)$. Condition (i) is vacuous because  $U_F = \emptyset$.

For (ii), at least one component of the decomposition lies in  $S_F$. If all components lay in  $B_F$, the quasiconvex half of  $B$  would give

$$
v(\bar{\mu}) \leq \max_{\tau} v(\mu_{\tau}) < y,
$$

contradicting  $\bar{\mu} \in S_F$. Let  $r := h(\bar{\mu})$. If some component in  $S_F$  has  $h(\mu_{\tau}) > r$, it works. Otherwise every term in

$$
\sum_{\tau} \alpha_{\tau} (h(\mu_{\tau}) - r) = 0
$$

is non-positive: this is true for  $\mu_{\tau} \in S_F$  by assumption and for  $\mu_{\tau} \in B_F$  because  $h(\mu_{\tau}) \leq 0 \leq r$. Hence all terms vanish, every component lies in  $F_r$, and the inductive version of (ii) in  $F_r$  applies.

*Case 4:  $U_F \neq \emptyset$  and  $B_F \neq \emptyset$.* When both strict sides are non-empty, the two properties require different separators: one controls mixtures with beliefs in  $U_F$  for property (i), while the other controls decompositions involving beliefs in  $B_F$  for property (ii). Take affine separators  $a$  and  $b$, each proper in the sense above, such that

$$
a \leq 0 \text{ on } L^- \cap F, \quad a \geq 0 \text{ on } U_F, \quad b \leq 0 \text{ on } B_F, \quad b \geq 0 \text{ on } L^+ \cap F.
$$

On  $S_F$  we have  $a \leq 0 \leq b$. For  $\lambda \in [0, 1]$, put

$$
h_{\lambda} := \lambda a + (1 - \lambda)b.
$$

Then  $h_{\lambda} \geq 0$  on  $U_F$  and  $h_{\lambda} \leq 0$  on  $B_F$. Call  $x \in S_F$  *degenerate* if  $a(x) = b(x) = 0$; otherwise  $b(x) - a(x) > 0$. Define

$$
\rho(x) := \frac{b(x)}{b(x) - a(x)} \in [0, 1] \text{ for non-degenerate } x, \quad \rho(x) := 1 \text{ for degenerate } x.
$$

A direct calculation gives, for non-degenerate  $x \in S_F$,

$$
h_{\lambda}(x) = (\rho(x) - \lambda)(b(x) - a(x)).
$$

Thus the sign of  $h_{\lambda}(x)$  is the sign of  $\rho(x) - \lambda$, while every degenerate point has  $h_{\lambda}(x) = 0$  for all  $\lambda$. Let

$$
F_{\lambda} := \{x \in F \mid h_{\lambda}(x) = 0\}.
$$

Every point of  $S_F$  with rank  $\lambda$  lies in  $F_{\lambda}$, with degenerate points placed in  $F_1$.

We now check that every relevant slice is lower-dimensional. This is the only place where semicontinuity is used. Because  $v$  is upper semicontinuous,  $L^{--}$  is relatively open in  $\Delta\Theta$; since  $B_F$  is non-empty,  $\text{aff}(B_F) = \text{aff}(F)$. Fix  $\lambda$  with  $F_{\lambda} \cap S_F \neq \emptyset$. If  $h_{\lambda}$  were constant on  $F$, then evaluating it at a point of  $F_{\lambda}$  would make the constant zero. For  $\lambda \in (0, 1)$, the equality  $\lambda a + (1 - \lambda)b = 0$  on  $B_F$, together with  $a \leq 0$  and  $b \leq 0$  there, would force  $a = b = 0$  on  $B_F$. Since  $\text{aff}(B_F) = \text{aff}(F)$,  $b$  would vanish on all of  $\text{aff}(F)$, contradicting proper separation for  $b$. For  $\lambda = 1$, the same argument uses proper separation for  $a$; for  $\lambda = 0$, it uses proper separation for  $b$. Hence  $h_{\lambda}$  is non-constant on  $\text{aff}(F)$, and  $F_{\lambda}$  has affine dimension below  $d$.

For (i), put  $\lambda := \rho(x)$. Then  $h_{\lambda}(x) = 0$  and

$$
h_{\lambda}(z) = (1 - \alpha)h_{\lambda}(u) \geq 0.
$$

If  $h_\lambda(z) > 0$, then  $z$  has rank strictly above  $\lambda = \rho(x)$, so  $z \succ_F x$. If  $h_\lambda(z) = 0$, then  $x, z, u \in F_\lambda$. Any strict rank increase already proves the claim; otherwise the tie is resolved inside  $F_\lambda$, where the inductive version of (i) gives  $z \succ_F x$.

For (ii), first note as in Case 3 that some component lies in  $S_F$. Put  $\bar{\lambda} := \rho(\bar{\mu})$, so  $h_{\bar{\lambda}}(\bar{\mu}) = 0$. If some component in  $S_F$  has rank above  $\bar{\lambda}$, it works. Otherwise every component in  $S_F$  has non-positive  $h_{\bar{\lambda}}$  value, and every component in  $B_F$  has non-positive  $h_{\bar{\lambda}}$  value as well. Since

$$
\sum_{\tau} \alpha_{\tau} h_{\bar{\lambda}}(\mu_{\tau}) = h_{\bar{\lambda}}(\bar{\mu}) = 0,
$$

all these values are zero, so every component lies in  $F_{\bar{\lambda}}$. The inductive version of (ii) in  $F_{\bar{\lambda}}$  gives a component  $\mu_{\tau^*} \in S_{F_{\bar{\lambda}}}$  with  $\mu_{\tau^*} \succeq_{F_{\bar{\lambda}}} \bar{\mu}$. This component is weakly above  $\bar{\mu}$  in the order on  $F$, either by a higher rank or by the tie-breaker in  $F_{\bar{\lambda}}$. This completes the induction.  $\square$

### Proof of Theorem 2

*Existence.* We construct the desired partition recursively. Let  $R_1 := \Theta$. Given a non-empty residual set  $R_t$, set  $w_t := \max \mathcal{W}_t$; this maximum exists and is attained by Lemma 3(ii). Among the coalitions of  $G_t$  attaining  $w_t$, choose one with a maximal prior under the relation from Lemma E.2 for the level set  $\{v = w_t\}$. Such a coalition exists because only finitely many type sets, and hence only finitely many priors, can arise. Call the selected coalition  $(C_t, \sigma_t, w_t)$  and let  $R_{t+1} := R_t \setminus C_t$. Since each step removes a non-empty cell, the recursion returns a partition  $\Pi$  with  $w_t = \max \mathcal{W}_t$  at every step.

It remains to show that  $\{w_t\}$  is non-increasing. Suppose, toward a contradiction, that  $w_{s+1} > w_s$  for some step  $s$. Write

$$
y := w_s, \quad L := \{v = y\}, \quad L^{++} := \{v > y\}, \quad L^{--} := \{v < y\}.
$$

By Lemma 3,  $\mu_{C_s}^0 \in L$  and  $\mu_{C_{s+1}}^0 \in L^{++}$. Since the step- $s$  coalition attains  $\max \mathcal{W}_s$, Lemma E.1 applied in  $G_s$  gives

$$
D := C_s \cup C_{s+1} = M_{R_s}^{-1}(X_s \cup X_{s+1}) \quad \text{and} \quad v(\mu_D^0) = y,
$$

and  $\mu_D^0 = \alpha \mu_{C_s}^0 + (1 - \alpha) \mu_{C_{s+1}}^0$  with  $\alpha := \mu^0(C_s) / \mu^0(D) \in (0, 1)$.

Now form the following auxiliary game:

$$
G' := (D, X_s \cup X_{s+1}, \theta \mapsto M(\theta) \cap (X_s \cup X_{s+1}), \mu_D^0, V_D).
$$

Note that  $G'$  is a disclosure game by the same argument as in the proof of Lemma 3(ii): every type in  $D$  can send some message in  $X_s \cup X_{s+1}$  because  $D$  is the relative preimage, and every message in  $X_s \cup X_{s+1}$  is on path in one of the two coalitions and hence is available to some type in  $D$.

By Lemma J.4 and the strict-payoff form of Proposition 1,  $G'$  admits a PBE partition

$$
\{(\tilde{C}_\tau, \tilde{\sigma}_\tau, \tilde{w}_\tau)\}_{\tau=1}^k
$$

with

$$
\tilde{w}_1 > \dots > \tilde{w}_k.
$$

Lemma 3 gives  $\tilde{w}_\tau = v(\mu_{\tilde{C}_\tau}^0)$  for every  $\tau$, and the cells partition  $D$, so

$$
\mu_D^0 = \sum_{\tau} \beta_{\tau} \mu_{\tilde{C}_\tau}^0 \quad \text{with all } \beta_{\tau} > 0.
$$

As in the proof of Lemma 3(ii), the first cell  $(\tilde{C}_1, \tilde{\sigma}_1, \tilde{w}_1)$  is a coalition of the original residual game  $G_s$. Hence  $\tilde{w}_1 \leq \max \mathcal{W}_s = y$. On the other hand, the quasiconvex half of B applied to the last convex combination gives

$$
y = v(\mu_D^0) \leq \max_{\tau} \tilde{w}_{\tau} = \tilde{w}_1.
$$

Therefore  $\tilde{w}_1 = y$, and strict decrease gives  $\tilde{w}_{\tau} < y$  for all  $\tau \geq 2$. Equivalently,  $\mu_{\tilde{C}_1}^0 \in L$  and  $\mu_{\tilde{C}_{\tau}}^0 \in L^{--}$  for  $\tau \geq 2$.

Apply Lemma E.2 at level  $y$. Property (i), with  $x = \mu_{C_s}^0$,  $u = \mu_{C_{s+1}}^0$, and  $z = \mu_D^0$, gives

$$
\mu_D^0 \succ \mu_{C_s}^0.
$$

Property (ii), applied to the decomposition  $\mu_D^0 = \sum_{\tau} \beta_{\tau} \mu_{\tilde{C}_{\tau}}^0$, gives a cell on the level set whose prior is weakly above  $\mu_D^0$. Since the auxiliary partition has strictly decreasing payoffs, the only cell on level  $y$  is the first one. Thus

$$
\mu_{\tilde{C}_1}^0 \succeq \mu_D^0 \succ \mu_{C_s}^0.
$$

But  $(\tilde{C}_1, \tilde{\sigma}_1, y)$  is a coalition of  $G_s$  attaining  $\max \mathcal{W}_s$, so this contradicts the choice of  $(C_s, \sigma_s, w_s)$  as maximal in the level-set order among payoff-maximal coalitions at step  $s$.

Hence  $w_t \leq w_{t-1}$  at every step. The constructed partition is a COE partition (Definition C.1), so by Lemma C.2(i) it is greedy. By Proposition 2, it is a coalition-proof PBE partition. Therefore a coalition-proof PBE exists.

*Part (i).* Assume B\* holds or  $G$  is generic. We first prove the bound on residual maxima that drives the argument. Let a coalition of  $G|_R$  with type set  $C$  attain  $\max \mathcal{W}_R$, and let  $R' := R \setminus C$  be non-empty. We claim that

$$
\max \mathcal{W}_{R'} \leq \max \mathcal{W}_R.
$$

Under B\* this is exactly the strict-betweenness conclusion of Lemma E.1. Under genericity, suppose instead that  $G|_{R'}$  admits a coalition  $(C', \sigma', w')$  with  $w' > w := \max \mathcal{W}_R$. By Lemma E.1, the union  $C \cup C'$  is a relative preimage in  $G|_R$  and

$$
v(\mu_{C \cup C'}^0) = w.
$$

Also, Lemma 3 gives  $v(\mu_C^0) = w$. Genericity therefore forces  $C \cup C' = C$, which is impossible because  $C'$  is a non-empty subset of  $R \setminus C$. This proves the bound.

We show by induction that every greedy choice takes the unconstrained maximum in the current residual game. Suppose this holds before step  $t$, and let  $R_t \neq \emptyset$. If  $t \geq 2$, the step- $(t-1)$  coalition attains  $\max \mathcal{W}_{t-1}$  by the induction hypothesis, so the bound above gives

$$
\max \mathcal{W}_t \leq w_{t-1}.
$$

For  $t = 1$  this upper bound is vacuous because  $w_0 = \infty$. The lower bound is the standard one: by Lemma C.1(iii),

$$
\max \mathcal{W}_t \geq \max_{\theta \in R_t} \underline{u}(\theta).
$$

Thus  $\max \mathcal{W}_t$  belongs to the greedy constraint set

$$
\mathcal{W}_t \cap \left[ \max_{\theta \in R_t} \underline{u}(\theta), w_{t-1} \right],
$$

and it is the maximum of that set. The greedy choice at step  $t$  therefore has  $w_t = \max \mathcal{W}_t$, completing the induction.

The constraint set is non-empty at every step, so no run of Algorithm 3 halts. Each run removes a non-empty cell and therefore terminates in a greedy partition. By Proposition 2, every such partition is a coalition-proof PBE partition.

For the characterization, if a partition satisfies  $w_t = \max \mathcal{W}_t \leq w_{t-1}$  at every step, then it is a coalition-optimal partition in the sense of Definition C.1. By Lemma C.2(i) it is greedy, and by Proposition 2 it is a coalition-proof PBE partition. Conversely, any coalition-proof PBE partition is greedy. The induction above gives  $w_t = \max \mathcal{W}_t$  at every step, and the inequality  $w_t \leq w_{t-1}$  is part of the greedy constraint.

*Part (ii).* Let  $\Pi$  and  $\Pi'$  be coalition-proof PBE partitions of a generic game. They are both greedy by Proposition 2. We prove by induction on  $t$  that they have the same residual sets, payoffs, and cells. At  $t = 1$  the residual sets and previous payoffs agree. Suppose they agree at step  $t$. If the common residual set is empty, both partitions have length  $t - 1$. Otherwise both continue, and greediness gives the same payoff:

$$
w_t = w'_t = \max \left( \mathcal{W}_t \cap \left[ \max_{\theta \in R_t} \underline{u}(\theta), w_{t-1} \right] \right).
$$

By Lemma 3,

$$
v(\mu_{C_t}^0) = w_t = w'_t = v(\mu_{C'_t}^0),
$$

so genericity gives  $C_t = C'_t$. Hence  $R_{t+1} = R'_{t+1}$, and the induction continues. The two partitions have the same length, cells, and payoffs.  $\square$

### Proof of Proposition 3

Fix a truth-leaning equilibrium  $(\sigma, \mu, r)$  and let  $\Pi$  be its associated PBE partition from Proposition 1.

If message  $\theta'$  is on path, type  $\theta'$  sends it with probability one. To see this, choose a sender type  $\theta$  with  $\sigma(\theta' | \theta) > 0$. Sequential rationality gives  $u(\theta) = r(\theta')$. Since  $\theta$  can send message  $\theta'$, transitivity gives  $M(\theta') \subseteq M(\theta)$. Together with reflexivity, this yields

$$
r(\theta') \leq u(\theta') \leq u(\theta) = r(\theta').
$$

Hence  $r(\theta') = u(\theta')$, and (A0) gives  $\sigma(\theta' | \theta') = 1$. If type  $\theta$  does not send its own message with probability one, that message is therefore off path, and (P0) and (A0) give  $v(\delta_\theta) = r(\theta) < u(\theta)$.

Suppose  $(\tilde{C}, \tilde{\sigma}, \tilde{w})$  blocks  $\Pi$. By Definition 9 and exclusivity,

$$
\tilde{C} = M_{\Theta(\tilde{w})}^{-1}(X(\tilde{\sigma})), \quad u(\theta) < \tilde{w} \quad \text{for every } \theta \in \tilde{C},
$$

and Lemma 3 gives  $v(\mu_{\tilde{C}}^0) = \tilde{w}$. Let

$$
\tilde{C}^{\text{tr}} := \{\theta' \in \tilde{C} : \sigma(\theta' | \theta') = 1\}
$$

be the members who tell the whole truth in the original equilibrium.

Fix a truth-telling member  $\theta' \in \tilde{C}^{\text{tr}}$. Every actual type  $\theta$  that sends message  $\theta'$  in equilibrium must also belong to  $\tilde{C}$. Indeed,  $u(\theta) = r(\theta') = u(\theta') < \tilde{w}$, so  $\theta \in \Theta(\tilde{w})$. Type  $\theta'$  sends some message  $m \in M(\theta') \cap X(\tilde{\sigma})$  in the proposed deviation; transitivity makes this message available to type  $\theta$  as well, and exclusivity therefore includes  $\theta$  in  $\tilde{C}$. Thus the entire equilibrium pool at message  $\theta'$  consists of coalition members. Its posterior  $\mu_\sigma(\cdot | \theta')$  is supported on a subset of  $\tilde{C}$  and has value  $r(\theta') = u(\theta') < \tilde{w}$.

Decompose  $\mu_{\tilde{C}}^0$  into these equilibrium posteriors and the remaining mass at each non-truth-telling member. Let

$$
\begin{aligned}\alpha_{\theta'} &:= \frac{p_{\sigma}(\theta')}{\mu^0(\tilde{C})} & (\theta' \in \tilde{C}^{\text{tr}}), \\ \beta_{\theta} &:= \mu_{\tilde{C}}^0(\theta) \left( 1 - \sum_{\theta' \in \tilde{C}^{\text{tr}}} \sigma(\theta' | \theta) \right) & (\theta \in \tilde{C} \setminus \tilde{C}^{\text{tr}}).\end{aligned}
$$

These weights are nonnegative. Since each truth-telling member sends its own message with probability one, Bayes' rule gives the convex decomposition

$$
\mu_{\tilde{C}}^0 = \sum_{\theta' \in \tilde{C}^{\text{tr}}} \alpha_{\theta'} \mu_{\sigma}(\cdot | \theta') + \sum_{\theta \in \tilde{C} \setminus \tilde{C}^{\text{tr}}} \beta_{\theta} \delta_{\theta}.
$$

Every posterior in the first sum has value below  $\tilde{w}$, and every singleton in the second satisfies  $v(\delta_{\theta}) < u(\theta) < \tilde{w}$. Quasiconvexity gives  $v(\mu_{\tilde{C}}^0) < \tilde{w}$, a contradiction. Thus  $\sigma$  is a coalition-proof PBE strategy. The optimal-commitment conclusion follows from Proposition 1 and Theorem 1 of Hart, Kremer, and Perry (2017).  $\square$

### Proof of Corollary 1

An evidence game has  $V$  single-valued and  $v$  satisfying B (Definition 15); since  $G$  is generic, the essential-uniqueness clause of Theorem 2(ii) applies: all coalition-proof PBE partitions of  $G$  have the same cells and payoffs. Let  $(\sigma, \mu, r)$  be a truth-leaning equilibrium. By Proposition 3,  $\sigma$  is a coalition-proof PBE strategy, that is, associated with a coalition-proof PBE partition, and the outcome of the equilibrium, each type's payoff and pooling pattern, is that of its partition (Proposition 1). Essential uniqueness therefore makes these cells and payoffs common to every truth-leaning equilibrium and every coalition-proof PBE. Finally, Hart, Kremer, and Perry (2017) show that the truth-leaning outcome is the receiver's optimal-commitment outcome (their Theorem 1), so the common outcome is the commitment outcome.  $\square$

## F Proofs for Section 6.3

The proofs build an auxiliary game from the quasiconcave closure  $\bar{v}^{qc}$.

**Lemma F.1.** (i)  $\bar{v}^{qc}$  is upper semicontinuous and quasiconcave,  $\{\bar{v}^{qc} \geq y\} = \text{conv}\{\bar{v} \geq y\}$  for every  $y$, and  $\bar{v}^{qc} \geq \bar{v}$;

(ii) the qc-closure of  $G$,  $G^{qc} := (\Theta, \mathcal{M}, M, \mu^0, V^{qc})$  with  $V^{qc}(\mu) := [\min_{\Delta \Theta} \underline{v}, \bar{v}^{qc}(\mu)]$, is a disclosure game with upper envelope  $\bar{v}^{qc}$;

(iii) if  $G$  satisfies M-C, then so does  $G^{qc}$.

*Proof.* (i)  $\bar{v}^{qc}$  is the quasiconcave envelope of  $\bar{v}$  (Lipnowski and Ravid, 2020): by their Theorem 2 it is upper semicontinuous and quasiconcave, and by their securability characterization (Corollary 1)  $\{\bar{v}^{qc} \geq y\}$  is the closed convex hull of  $\{\bar{v} \geq y\}$. The two remaining points are immediate:  $\bar{v}^{qc} \geq \bar{v}$  because  $p_i = \mu$  is admissible in Definition 18, and that closed convex hull equals  $\text{conv}\{\bar{v} \geq y\}$  because  $\{\bar{v} \geq y\}$  is compact ($\bar{v}$  upper semicontinuous by (A4)). (ii) The interval  $V^{qc}(\mu) = [\min_{\Delta\Theta} \underline{v}, \bar{v}^{qc}(\mu)]$  is non-empty and compact, as  $\min_{\Delta\Theta} \underline{v} \leq \underline{v}(\mu) \leq \bar{v}(\mu) \leq \bar{v}^{qc}(\mu)$. It is upper hemicontinuous in  $\mu$: given an open  $U \supseteq V^{qc}(\mu)$, compactness gives  $\varepsilon > 0$  with  $[\min_{\Delta\Theta} \underline{v}, \bar{v}^{qc}(\mu) + \varepsilon] \subseteq U$, and upper semicontinuity of  $\bar{v}^{qc}$  keeps  $V^{qc}$  inside  $U$  near  $\mu$. Hence  $G^{qc}$  satisfies (A4); it inherits (A1)–(A3) from  $G$, its upper envelope is  $\bar{v}^{qc}$, and its lower envelope is the constant  $\min_{\Delta\Theta} \underline{v}$.

(iii) M-C depends only on  $(\Theta, \mathcal{M}, M)$, which  $G^{qc}$  shares with  $G$.  $\square$

**Lemma F.2.** *For every non-empty  $R \subseteq \Theta$  and every coalition  $(C, \sigma, w)$  of  $G|_R$,*

$$
w \leq \bar{v}^{qc}(\mu_C^0).
$$

*Proof.* For each  $m \in X(\sigma)$, condition (C4) gives  $w \in V(\mu_\sigma(\cdot | m))$, and hence  $\bar{v}(\mu_\sigma(\cdot | m)) \geq w$. The prior  $\mu_C^0 = \sum_{m \in X(\sigma)} p_\sigma(m) \mu_\sigma(\cdot | m)$  is a convex combination of these posteriors, so  $\mu_C^0 \in \text{conv}\{\bar{v} \geq w\} = \{\bar{v}^{qc} \geq w\}$  by Lemma F.1(i). Hence  $\bar{v}^{qc}(\mu_C^0) \geq w$.  $\square$

**Lemma F.3.** *Assume M-CT. For every non-empty  $R \subseteq \Theta$  and every message  $\bar{m} \in \mathcal{M}_R$, the restricted game  $G|_R$  has a coalition  $(C, \sigma, w)$  with type set  $C = P(\bar{m}) \cap R$  and payoff  $w = \bar{v}^{qc}(\mu_C^0)$.*

*Proof.* By Lemma F.1(i),  $\mu_C^0 = \sum_{i=1}^k \lambda_i p_i$  is a convex combination of beliefs  $p_i$  with  $\bar{v}(p_i) \geq w$, weights  $\lambda_i > 0$, and  $\text{supp } p_i \subseteq C$. We construct the coalition in two steps. First, we replace each  $p_i$  by a belief  $\tilde{p}_i$  on the segment from  $p_i$  to  $\mu_C^0$  with  $w \in V(\tilde{p}_i)$, keeping  $\mu_C^0$  a convex combination of the  $\tilde{p}_i$. Second, using the cheap-talk copies of  $\bar{m}$, we construct a coalition strategy on  $C$  that induces these beliefs.

*An intermediate belief on each segment.* Fix  $i$  and let  $\gamma(t) = (1-t)p_i + t\mu_C^0$  for  $t \in [0, 1]$. Then  $\bar{v}(\gamma(0)) = \bar{v}(p_i) \geq w$  and  $\bar{v}(\gamma(1)) = \bar{v}(\mu_C^0) \leq \bar{v}^{qc}(\mu_C^0) = w$. We claim that  $w \in V(\gamma(t))$  for some  $t$. If not, then every  $t$  has  $w \notin V(\gamma(t)) = [\underline{v}(\gamma(t)), \bar{v}(\gamma(t))]$, so  $\bar{v}(\gamma(t)) < w$  or  $\underline{v}(\gamma(t)) > w$. By upper semicontinuity of  $\bar{v}$  and lower semicontinuity of  $\underline{v}$  ((A4)), these two conditions define open subsets of  $[0, 1]$; they are disjoint and cover  $[0, 1]$. The first excludes 0 and the second excludes 1, so 0 belongs to the second and 1 to the first, and both are non-empty. This splits the connected interval  $[0, 1]$  into two disjoint non-empty open sets, which is impossible. Hence  $w \in V(\gamma(t_i))$  for some  $t_i$. Set  $\tilde{p}_i = \gamma(t_i)$; since  $p_i$  and  $\mu_C^0$  are supported on  $C$, so is  $\tilde{p}_i$.

If  $t_i = 1$  for some  $i$, then  $\tilde{p}_i = \mu_C^0$, so  $w \in V(\mu_C^0)$  and Remark 1, applied to  $\bar{m}$  in  $G|_R$, already gives a coalition  $(C, \sigma^{\bar{m}}, w)$; so assume  $t_i < 1$  for every  $i$. Put  $\beta_i := \lambda_i / (1 - t_i) > 0$. Then

$$
\sum_i \beta_i \tilde{p}_i = \sum_i \lambda_i p_i + \left( \sum_i \frac{\lambda_i t_i}{1 - t_i} \right) \mu_C^0 = \left( \sum_i \beta_i \right) \mu_C^0,
$$

so, after normalizing the  $\beta_i$  to sum to one,  $\mu_C^0 = \sum_i \beta_i \tilde{p}_i$  is a convex combination of the  $\tilde{p}_i$  with positive weights.

*Splitting across cheap-talk copies.* By M-CT there are  $k$  distinct copies  $m_1, \dots, m_k$  of  $\bar{m}$  sharing its preimage,  $P(m_i) = P(\bar{m})$, so  $P(m_i) \cap R = C$  and every type in  $C$  can send every copy. The plan is to spread the coalition across the copies so that copy  $m_i$  carries exactly the belief  $\tilde{p}_i$. To do this, let each  $\theta \in C$  send copy  $m_i$  with probability

$$
\sigma(m_i | \theta) := \frac{\beta_i \tilde{p}_i(\theta)}{\mu_C^0(\theta)} \quad (1 \leq i \leq k),
$$

which is well-defined since  $\mu_C^0(\theta) > 0$  by (A2). These probabilities sum to one over  $i$, because  $\sum_i \beta_i \tilde{p}_i = \mu_C^0$, so  $\sigma$  is a coalition strategy on  $C$. Copy  $m_i$  is then on path with probability  $p_\sigma(m_i) = \sum_{\theta \in C} \mu_C^0(\theta) \sigma(m_i | \theta) = \beta_i$, so the evidence is exactly  $\{m_1, \dots, m_k\}$, exclusive to  $C$  (that is,  $M^{-1}(X(\sigma)) \cap R = C$), and Bayes' rule returns the intended belief,  $\mu_\sigma(\cdot | m_i) = \tilde{p}_i$. Since  $w \in V(\tilde{p}_i)$  at every copy, every message in the coalition's evidence pays  $w$, so  $(C, \sigma, w)$  is a coalition of  $G|_R$.  $\square$

### Proof of Theorem 3

By Lemma F.1, the closure game  $G^{qc}$  satisfies M-C and QC, so the recursive construction in the existence part of the proof of Theorem 1 yields a COE partition of  $G^{qc}$. Write  $R_t$  for its residual sets,  $G_t^{qc} := G^{qc}|_{R_t}$, and  $\mathcal{W}_t^{qc}$  for the coalition payoff set of  $G_t^{qc}$, so  $w_t = \max \mathcal{W}_t^{qc}$  at every step. By Lemma 2 applied in  $G_t^{qc}$, we may take each cell to be a pooling cell  $(C_t, \sigma^{m_t}, w_t)$  with  $C_t = P(m_t) \cap R_t$  and  $w_t = \bar{v}^{qc}(\mu_{C_t}^0)$.

For each  $t$, Lemma F.3 applied to  $R_t$  and  $m_t$  yields a coalition  $(C_t, \tilde{\sigma}_t, w_t)$  of  $G_t$, with type set  $C_t = P(m_t) \cap R_t$  and payoff  $w_t = \bar{v}^{qc}(\mu_{C_t}^0)$. These cells and residual sets coincide with those of the COE partition of  $G^{qc}$, so  $\tilde{\Pi} := \{(C_t, \tilde{\sigma}_t, w_t)\}_{t=1}^T$  is a partition of  $G$, and its payoffs are non-increasing because those of the COE partition of  $G^{qc}$  are.

It remains to show  $w_t = \max \mathcal{W}_t$  in  $G$  at every step; with the non-increasing payoffs established above,  $\tilde{\Pi}$  is then a COE partition of  $G$. The payoff  $w_t$  is attained in  $G_t$  by  $(C_t, \tilde{\sigma}_t, w_t)$, so  $w_t \in \mathcal{W}_t$. For the upper bound, let  $(C, \sigma, w)$  be any coalition of  $G_t$. By Lemma F.2,  $w \leq \bar{v}^{qc}(\mu_C^0)$, so it suffices to show  $\bar{v}^{qc}(\mu_C^0) \leq w_t$. Iterating M-C over the finite evidence  $X(\sigma)$, as in the proof of Lemma 2, gives a message  $m^*$  with  $P(m^*) = \bigcup_{m \in X(\sigma)} P(m)$, so  $P(m^*) \cap R_t = M_t^{-1}(X(\sigma)) = C$. Pooling on  $m^*$  in  $G_t^{qc}$  is a coalition with payoff  $\bar{v}^{qc}(\mu_C^0)$, so  $\bar{v}^{qc}(\mu_C^0) \in \mathcal{W}_t^{qc}$  and hence  $\bar{v}^{qc}(\mu_C^0) \leq \max \mathcal{W}_t^{qc} = w_t$. Therefore  $w \leq w_t$, giving  $w_t = \max \mathcal{W}_t$. By Lemma C.2(i),  $\tilde{\Pi}$  is a coalition-proof PBE partition, so a coalition-proof PBE exists.  $\square$

### Proof of Corollary 2

Both message conditions hold:  $P(m'') = P(m) \cup P(m') = \Theta$  for any  $m, m', m''$  gives M-C, and since every message has preimage  $\Theta$, the assumption  $|\mathcal{M}| \geq n$  is exactly M-CT. Every non-empty  $X \subseteq \mathcal{M}$  has  $M^{-1}(X) = \Theta$, so by exclusivity (C3) and its converse the type set of any coalition of  $G$  is  $\Theta$; hence every partition has  $T = 1$  and  $C_1 = \Theta$. By Lemma F.2, every coalition payoff is at most  $\bar{v}^{qc}(\mu^0)$, and by Lemma F.3 (with  $R = \Theta$  and any  $\bar{m}$) the payoff  $\bar{v}^{qc}(\mu^0)$  is attained, so  $\max \mathcal{W}_\Theta = \bar{v}^{qc}(\mu^0)$. By Lemma C.1(iii),  $\max_{\theta \in \Theta} \underline{u}(\theta) \leq \bar{v}^{qc}(\mu^0)$. A single-cell partition  $\{(\Theta, \sigma, w_1)\}$  is greedy if and only if  $w_1 = \max(\mathcal{W}_\Theta \cap [\max_{\theta \in \Theta} \underline{u}(\theta), \infty)) = \max \mathcal{W}_\Theta = \bar{v}^{qc}(\mu^0)$. By Proposition 2, the coalition-proof PBE partitions are exactly these greedy partitions, so a coalition-proof PBE exists and is essentially unique.  $\square$

## G Proofs for Section 6.4

The proof of Proposition 4(ii) uses a more general condition. Call  $V$  *decomposable* if, for every  $q \in \Delta \Theta$  and every  $w' \in [v_{\min}, \bar{v}(q)]$,

$$
q \in \text{conv}\{\mu \in \Delta \Theta \mid \text{supp } \mu \subseteq \text{supp } q, w' \in V(\mu)\}.
$$

We show below that revelation aversion implies decomposability, and that decomposability together with cheap-talk copies implies PD.

### Proof of Proposition 4

Fix a non-empty  $R \subseteq \Theta$, a coalition  $(C, \sigma, w)$  of  $G|_R$, and  $w' \in [v_{\min}, w]$; in each case we produce a coalition  $(C, \sigma', w')$  of  $G|_R$, so that  $G$  satisfies PD. Write  $q_m := \mu_\sigma(\cdot | m)$  for the posterior at each  $m \in X(\sigma)$; by (C4),  $\bar{v}(q_m) \geq w \geq w'$, and  $\text{supp } q_m \subseteq P(m) \cap C$.

*Case (i).* Free disposal gives  $V(q_m) = [v_{\min}, \bar{v}(q_m)]$  for each  $m \in X(\sigma)$, and  $v_{\min} \leq w' \leq w \leq \bar{v}(q_m)$, so  $w' \in V(q_m)$. Hence  $(C, \sigma, w')$, with the same strategy  $\sigma$, is again a coalition of  $G|_R$: conditions (C1)–(C3) are unchanged and (C4) is the membership just shown. No assumption on the message mapping is used.

*Case (ii).* We first show that revelation aversion makes  $V$  decomposable. Fix  $q \in \Delta\Theta$  and  $w' \in [v_{\min}, \bar{v}(q)]$. If  $w' \in V(q)$, the claim is immediate. Otherwise, for each  $\theta_j \in \text{supp } q$, the intermediate-value argument in the proof of Lemma F.3, together with  $\underline{v}(\delta_{\theta_j}) = v_{\min} \leq w' \leq \bar{v}(q)$, gives

$$
q_j = (1 - t_j)q + t_j\delta_{\theta_j}, \quad t_j \in (0, 1], \quad w' \in V(q_j).
$$

Each  $q_j$  is supported in  $\text{supp } q$, and

$$
q = \sum_j \alpha_j q_j, \quad \alpha_j := \frac{q(\theta_j)/t_j}{\sum_k q(\theta_k)/t_k}.
$$

Thus  $V$  is decomposable.

Since  $w' \in [v_{\min}, \bar{v}(q_m)]$, decomposability writes each  $q_m$  as a convex combination of beliefs supported in  $\text{supp } q_m \subseteq P(m) \cap C$, each paying  $w'$. It remains to realize these beliefs as a coalition. Group the evidence by preimage: let the  $P_\rho$  be the distinct preimages among  $\{P(m) \mid m \in X(\sigma)\}$, and let  $\mu_\rho := \lambda_\rho^{-1} \sum_{m:P(m)=P_\rho} p_\sigma(m) q_m$  be the coalition's posterior conditional on sending a message with preimage  $P_\rho$, where  $\lambda_\rho := \sum_{m:P(m)=P_\rho} p_\sigma(m) > 0$  and  $\sum_\rho \lambda_\rho = 1$, so that  $\mu_C^0 = \sum_\rho \lambda_\rho \mu_\rho = \sum_{m \in X(\sigma)} p_\sigma(m) q_m$. Each  $\mu_\rho$  is a convex combination of beliefs supported on  $P_\rho \cap C$  and paying  $w'$, so by Carathéodory ($\Delta\Theta$  has affine dimension  $n-1$) at most  $n$  of them are needed. Since cheap-talk copies supply at least  $n$  messages with preimage  $P_\rho$, spreading the coalition across these copies as in the proof of Lemma F.3 yields a coalition strategy  $\sigma'$  on  $C$  whose on-path posteriors are exactly these beliefs; in particular every message in  $X(\sigma')$  pays  $w'$. Since  $X(\sigma')$  realizes the same preimages as  $X(\sigma)$,  $M_R^{-1}(X(\sigma')) = M_R^{-1}(X(\sigma)) \subseteq C$  by exclusivity (C3) for  $(C, \sigma, w)$. Hence  $(C, \sigma', w')$  is a coalition of  $G|_R$.  $\square$

### Proof of Theorem 4

Let  $\{(C_s, \sigma_s, w_s)\}_{s < t}$  be a greedy prefix with non-empty residual  $R_t$  (Section C); we show the greedy constraint set  $\mathcal{W}_t \cap [\max_{\theta \in R_t} \underline{u}(\theta), w_{t-1}]$  is non-empty (with  $w_0 := \infty$). Write  $\bar{w}_t := \max \mathcal{W}_t$, which exists and satisfies  $\bar{w}_t \geq \max_{\theta \in R_t} \underline{u}(\theta)$  by Lemma C.1(iii).

If  $\bar{w}_t \leq w_{t-1}$, then  $\bar{w}_t$  itself lies in the constraint set. Otherwise  $\bar{w}_t > w_{t-1}$, so  $t \geq 2$. Let  $(C, \sigma, \bar{w}_t)$  be a coalition of  $G_t$  attaining  $\bar{w}_t$. Every coalition payoff lies in the interval  $[v_{\min}, \max_{\Delta\Theta} \bar{v}]$  (see the proof of Lemma B.2), so  $w_{t-1} \in \mathcal{W}_{t-1}$  gives  $w_{t-1} \in [v_{\min}, \bar{w}_t]$. Because  $G$  satisfies PD, applied in  $G_t$  to this coalition with target  $w_{t-1}$, the residual game  $G_t$  admits a coalition  $(C, \sigma', w_{t-1})$, so  $w_{t-1} \in \mathcal{W}_t$. Moreover  $w_{t-1} \geq \max_{\theta \in R_{t-1}} \underline{u}(\theta) \geq \max_{\theta \in R_t} \underline{u}(\theta)$  by greediness at step  $t-1$  and  $R_t \subseteq R_{t-1}$, so  $w_{t-1}$  lies in the constraint set.

In either case the constraint set is non-empty; being the intersection of the compact  $\mathcal{W}_t$  with a closed set, it has a maximum, which a greedy choice at step  $t$  selects. Hence no run of Algorithm 3 halts. Each run removes a non-empty cell, so it terminates in a greedy partition, which by Proposition 2 is a coalition-proof PBE partition. The final claim follows from Proposition 4.  $\square$

## H Proofs for Section 6.5

### Proof of Theorem 5

Write  $\Theta = \{\theta, \theta'\}$  and  $w_1 := \max \mathcal{W}_\Theta$, which exists and satisfies  $w_1 \geq \underline{u}(\theta'')$  for each  $\theta'' \in \Theta$  by Lemma C.1(iii), so any coalition attaining  $w_1$  is an admissible greedy selection at step 1.

*Case 1: some coalition attaining  $w_1$  has type set  $\Theta$.* Selecting it completes the run at  $T = 1$; by Proposition 2 a coalition-proof PBE exists.

*Case 2: every coalition attaining  $w_1$  is a singleton.* Fix one,  $(\{\theta\}, \sigma_1, w_1)$; all its posteriors are  $\delta_\theta$, so  $w_1 \in V(\delta_\theta)$, and its evidence  $X_1$  satisfies  $X_1 \cap M(\theta') = \emptyset$  by exclusivity. Then  $\theta$  is not forced to any  $m \in M(\theta')$: if  $M(\theta) = \{m\}$  with  $m \in M(\theta')$, then  $X_1 \subseteq \{m\}$  meets  $M(\theta')$. Hence for every  $m \in M(\theta')$  there is a strategy in which only  $\theta'$  sends  $m$, so  $\delta_{\theta'} \in \mathcal{F}(m)$  and therefore

$$
\underline{v}(\theta') = \max_{m \in M(\theta')} \min_{\mu \in \mathcal{F}(m)} \underline{v}(\mu) \leq \underline{v}(\delta_{\theta'}).
$$

*Case 2a:  $\underline{v}(\delta_{\theta'}) \leq w_1$.* The residual game on  $\{\theta'\}$  has coalition payoff set  $V(\delta_{\theta'}) = [\underline{v}(\delta_{\theta'}), \bar{v}(\delta_{\theta'})]$: every strategy of  $G|_{\{\theta'\}}$  induces  $\delta_{\theta'}$  at each on-path message, and exclusivity is automatic in a one-type game. The step-2 constraint set  $V(\delta_{\theta'}) \cap [\underline{v}(\theta'), w_1]$  is non-empty, because  $\underline{v}(\delta_{\theta'}) \leq w_1$  and  $\underline{v}(\theta') \leq \underline{v}(\delta_{\theta'}) \leq \bar{v}(\delta_{\theta'})$, and two closed intervals intersect exactly when each lower endpoint is below the other's upper endpoint. The greedy run completes at  $T = 2$.

*Case 2b:  $\underline{v}(\delta_{\theta'}) > w_1$.* First, every  $m \in M(\theta')$  has  $P(m) = \Theta$: if  $P(m) = \{\theta'\}$, pooling  $\{\theta'\}$  on  $m$  is a coalition of  $G$  with payoff  $\underline{v}(\delta_{\theta'}) > w_1 = \max \mathcal{W}_\Theta$, a contradiction. Fix  $m \in M(\theta')$  and, for  $t \in [\mu^0(\theta'), 1]$, put  $q_t := t\delta_{\theta'} + (1-t)\delta_\theta$. The sets  $O_1 := \{t \mid \bar{v}(q_t) < w_1\}$  and  $O_2 := \{t \mid \underline{v}(q_t) > w_1\}$  are disjoint and relatively open in  $[\mu^0(\theta'), 1]$  by (A4). Suppose some  $t^*$  lies in neither, so  $w_1 \in V(q_{t^*})$. Let  $\theta'$  send  $m$  with probability one and let  $\theta$  send  $m$  with probability  $\alpha^* := \mu^0(\theta')(1-t^*)/(\mu^0(\theta)t^*) \in [0, 1]$, distributing its remaining mass proportionally to  $\sigma_1$  on  $X_1$. The on-path posteriors are  $q_{t^*}$  at  $m$, paying  $w_1 \in V(q_{t^*})$, and  $\delta_\theta$  at the messages of  $X_1$  (used if  $\alpha^* < 1$), paying  $w_1 \in V(\delta_\theta)$; the evidence has preimage  $\Theta$. So  $(\Theta, \cdot, w_1)$  is a coalition attaining  $w_1$, and Case 1 applies. Otherwise  $O_1 \cup O_2$  covers the interval; since  $1 \in O_2$  and the interval is connected,  $O_1 = \emptyset$, so  $O_2$  is the whole interval and in particular  $\underline{v}(\mu^0) > w_1$  at  $t = \mu^0(\theta')$. But pooling  $\Theta$  on  $m$  is a coalition of  $G$  with payoffs  $V(\mu^0) \ni \underline{v}(\mu^0)$, so  $\underline{v}(\mu^0) \leq \max \mathcal{W}_\Theta = w_1$, a contradiction.  $\square$

### Proof of Proposition 5

As a minimum of two affine functions,  $v$  is concave and continuous, so  $V = \{v\}$  satisfies (A4) and  $v$  is quasiconcave. On the edge  $\{\mu_3 = 0\}$, writing  $x := \mu_1$,  $v(x, 1-x, 0) = \min\{x, 1.1 - 1.2x\}$, which equals  $x$  exactly for  $x \leq \frac{1}{2}$; on the edge  $\{\mu_2 = 0\}$,  $v(x, 0, 1-x) = \min\{0.9 + 0.1x, 0.9 - x\} = 0.9 - x$.

*Coalition payoffs.* The preimages are  $P(a) = \{1, 2\}$,  $P(b) = \{1, 3\}$, and  $M^{-1}(\{a, b\}) = \emptyset$. A coalition of  $G$  with type set  $\{1, 2\}$  cannot use  $b$  (exclusivity:  $3 \in P(b)$), so both types send  $a$  and  $w = v(\frac{1}{2}, \frac{1}{2}, 0) = \frac{1}{2}$; likewise  $\{1, 3\}$  gives  $w = 0.4$. A coalition with type set  $\Theta$  must use both messages, with type 1 sending  $a$  with some probability  $\alpha \in [0, 1]$: the posterior at  $a$  lies on  $\{\mu_3 = 0\}$  with  $\mu_1$ -coordinate  $\alpha/(1+\alpha) \in [0, \frac{1}{2}]$, and the posterior at  $b$  on  $\{\mu_2 = 0\}$  with  $\mu_1$ -coordinate  $(1-\alpha)/(2-\alpha) \in [0, \frac{1}{2}]$. The common-payoff condition requires  $f(\alpha) := \alpha/(1+\alpha) + (1-\alpha)/(2-\alpha) = 0.9$; but  $f(0) = f(1) = \frac{1}{2}$  and  $f'(\alpha) = (1+\alpha)^{-2} - (2-\alpha)^{-2}$  vanishes only at  $\alpha = \frac{1}{2}$, where  $f(\frac{1}{2}) = \frac{2}{3} < 0.9$, so no such  $\alpha$  exists. Singletons fail exclusivity ($1 \in P(a) \cap P(b)$,  $2 \in P(a)$,  $3 \in P(b)$). Hence  $\mathcal{W}_\Theta = \{0.4, \frac{1}{2}\}$.

*Skeptical payoffs.* Type 2 is forced to  $a$, so by Lemma J.3  $\mathcal{F}(a) = \{(x, 1-x, 0) \mid x \in [0, \frac{1}{2}]\}$  and  $\min_{\mathcal{F}(a)} \underline{v} = 0$; likewise  $\mathcal{F}(b) = \{(x, 0, 1-x) \mid x \in [0, \frac{1}{2}]\}$  with  $\min_{\mathcal{F}(b)} \underline{v} = 0.4$. Thus  $\underline{v}(1) = \max\{0, 0.4\} = 0.4$,  $\underline{v}(2) = 0$, and  $\underline{v}(3) = 0.4$.

*Every greedy run halts.* At step 1 the constraint set is  $\mathcal{W}_\Theta \cap [0.4, \infty) = \{0.4, \frac{1}{2}\}$, so  $w_1 = \frac{1}{2}$  and  $C_1 = \{1, 2\}$, the unique coalition attaining it. The residual game on  $\{3\}$  has coalition payoff set  $V(\delta_3) = \{0.9\}$, and the step-2 constraint set is  $\{0.9\} \cap [0.4, \frac{1}{2}] = \emptyset$. Every run of Algorithm 3 halts, so by Proposition 2 no coalition-proof PBE exists.

*Cheap-talk copies.* Augment  $\mathcal{M}$  with three copies of each message; preimages are unchanged, so coalition type sets still lie in  $\{\{1,2\}, \{1,3\}, \Theta\}$. For any coalition  $(C, \sigma, w)$  of any restricted game,  $\mu_C^0$  is a positive convex combination of the on-path posteriors, each paying  $w$, so quasi-concavity gives  $w \leq v(\mu_C^0)$  as in Lemma 2. Posteriors of  $a$ -copies are supported in  $\{1,2\}$, where  $v \leq \max_{x \in [0,1]} \min\{x, 1.1 - 1.2x\} = \frac{1}{2}$; so every coalition using an  $a$ -copy, in particular every  $\Theta$ -coalition, has  $w \leq \frac{1}{2}$, while  $\{1,2\}$ - and  $\{1,3\}$ -coalitions have  $w \leq \frac{1}{2}$  and  $w \leq 0.4$. Hence the largest coalition payoff is still  $\frac{1}{2}$, attained by pooling  $\{1,2\}$  on an  $a$ -copy, and only by coalitions with type set  $\{1,2\}$: a  $\Theta$ -coalition attaining  $\frac{1}{2}$  would need every  $a$ -copy posterior equal to  $(\frac{1}{2}, \frac{1}{2}, 0)$, the unique point of the  $\{\mu_3 = 0\}$  edge with  $v = \frac{1}{2}$, and every  $b$ -copy posterior equal to  $(0.4, 0, 0.6)$, where  $0.9 - x = \frac{1}{2}$; matching the type-2 mass of  $\mu^0$  forces weight  $\frac{2}{3}$  on the former, leaving type-3 mass  $0.2 \neq \frac{1}{3}$. So every greedy run again selects  $C_1 = \{1,2\}$  at  $w_1 = \frac{1}{2}$; the residual coalition payoffs are  $V(\delta_3) = \{0.9\}$, and since adding copies only lowers the skeptical floor, the step-2 constraint set is again empty. Every run halts, and no coalition-proof PBE exists.  $\square$

## I Proofs for Section 7

**Lemma I.1.** *Let  $\nu: \Theta \rightarrow \mathbb{R}_{\geq 0}$  be nonzero and let  $\mu \in \Delta_{\text{supp } \nu}$. Then there is a nonzero  $\eta \leq \nu$  with  $\hat{\eta} = \mu$  and  $\text{supp}(\nu - \eta) \subsetneq \text{supp } \nu$.*

*Proof.* Set  $\lambda := \min_{\theta: \nu(\theta) > 0} \nu(\theta)/\mu(\theta)$  and  $\eta := \lambda\mu$. Since  $\mu \geq 0$  and  $\lambda > 0$, we have  $\eta \geq 0$. Then  $\eta \leq \nu$,  $\eta \neq 0$, and  $\hat{\eta} = \mu$. Because  $\eta \geq 0$, we have  $\text{supp}(\nu - \eta) \subseteq \text{supp } \nu$. If  $\theta'$  attains the minimum, then  $\eta(\theta') = \nu(\theta')$, so  $(\nu - \eta)(\theta') = 0$  and  $\theta'$  leaves the support after  $\eta$  is removed; hence the inclusion is strict.  $\square$

### Proof of Proposition 6

By Lemma I.1, the greedy increment induces  $\mu_{\text{supp } \nu}^*$, attains the maximum payoff over  $\Delta_{\text{supp } \nu}$, and removes at least one type from the support, proving (i). Iterating from  $\nu_0 = \mu^0$  therefore exhausts the prior in at most  $|\Theta|$  steps. The selected payoffs are weakly decreasing because the feasible faces shrink with the support, proving (ii). For (iii), if maximizers are unique, any partition satisfying (i) must induce  $\mu_{\text{supp } \nu_{t-1}}^*$  at step  $t$. Since the step also retires a type, feasibility forces its scale to be  $\lambda^*(\nu_{t-1})$, so the increment is the greedy increment. Induction from  $\nu_0 = \mu^0$  gives uniqueness.  $\square$

The remaining proofs maintain the uniqueness hypothesis of Proposition 6(iii), as in the tent subsection of Section 7:  $\bar{v}$  has a unique maximizer  $\mu_C^*$  over each face  $\Delta_C$.

**Lemma I.2.** *Let  $\emptyset \neq D \subseteq C \subseteq \Theta$. If  $\text{supp } \mu_C^* \subseteq D$, then  $\mu_D^* = \mu_C^*$.*

*Proof.* The belief  $\mu_C^*$  lies in  $\Delta_D$  and, since  $\Delta_D \subseteq \Delta_C$, maximizes  $\bar{v}$  over  $\Delta_D$; uniqueness of the maximizer over  $\Delta_D$  gives  $\mu_D^* = \mu_C^*$.  $\square$

### Proof of the representation and path-independence claims in Proposition 7

For a nonzero sub-measure  $\nu$, the greedy construction of Proposition 6 run from  $\nu$  produces increments  $\eta_t$  with  $\hat{\eta}_t = \mu_{\text{supp } \nu_{t-1}}^*$; call  $\sum_t \eta_t(\Theta) \bar{v}(\hat{\eta}_t)$  the *greedy value* of  $\nu$. Existence of the representation in Proposition 7 follows by running this construction from  $\mu$. The increments sum to  $\mu$, their masses sum to one, and their induced beliefs are the constrained peaks of the successive residual supports. If a step exhausts several types, add the omitted intermediate sets with zero weights; the resulting sets form a deletion order. We prove the following claim: for every family of non-empty sets  $E_0 \supseteq E_1 \supseteq \dots \supseteq E_K$  and weights  $\alpha_0, \dots, \alpha_K > 0$  with  $\nu = \sum_k \alpha_k \mu_{E_k}^*$, the value $\sum_k \alpha_k \bar{v}(\mu_{E_k}^*)$  equals the greedy value of  $\nu$. For the value claim, drop the zero-weight terms from any representation based on a deletion order. The remaining terms form a family of the kind just described.

First normalize the family. Let  $S := \text{supp } \nu$. For every  $k$,  $\alpha_k \mu_{E_k}^* \leq \nu$  gives  $\text{supp } \mu_{E_k}^* \subseteq E_k \cap S$, so  $\mu_{E_k \cap S}^* = \mu_{E_k}^*$  by Lemma I.2; replacing each  $E_k$  by  $E_k \cap S$  changes neither the summands nor the values, so assume  $E_k \subseteq S$  for every  $k$. Then  $S = \text{supp } \nu \subseteq \bigcup_k \text{supp } \mu_{E_k}^* \subseteq E_0$  forces  $E_0 = S$. Merging consecutive equal sets, whose peaks coincide, assume  $E_0 \supsetneq E_1 \supsetneq \dots \supsetneq E_K$.

We induct on  $K$. If  $K = 0$, then  $\nu = \alpha_0 \mu_S^*$, so  $\lambda^*(\nu) = \alpha_0$: the run retires all of  $\nu$  in one step, with greedy value  $\alpha_0 \bar{v}(\mu_S^*)$. If  $K \geq 1$, the first greedy increment is  $\lambda^*(\nu) \mu_S^*$  with  $\lambda^*(\nu) = \min_{\theta \in \text{supp } \mu_S^*} \nu(\theta) / \mu_S^*(\theta)$. On the one hand,  $\nu - \alpha_0 \mu_S^* = \sum_{k \geq 1} \alpha_k \mu_{E_k}^* \geq 0$, so  $\lambda^*(\nu) \geq \alpha_0$. On the other hand,  $\text{supp } \mu_S^* \not\subseteq E_1$: otherwise every summand of  $\nu$  would be supported in  $E_1$, giving  $S = \text{supp } \nu \subseteq E_1 \subsetneq S$. Any  $\theta^* \in \text{supp } \mu_S^* \setminus E_1$  has  $\mu_{E_k}^*(\theta^*) = 0$  for  $k \geq 1$, so  $\nu(\theta^*) = \alpha_0 \mu_S^*(\theta^*)$  and  $\lambda^*(\nu) \leq \alpha_0$. Hence the first increment is exactly  $\alpha_0 \mu_S^*$, contributing  $\alpha_0 \bar{v}(\mu_S^*)$, and the run continues from  $\nu_1 = \sum_{k \geq 1} \alpha_k \mu_{E_k}^*$, whose greedy value is  $\sum_{k \geq 1} \alpha_k \bar{v}(\mu_{E_k}^*)$  by the induction hypothesis.  $\square$

### Completion of the proof of Proposition 7

For (i), choose a deletion order containing  $C$  and assign weight 1 to  $C$; this representation has value  $\bar{v}(\mu_C^*)$, and the path-independence claim just proved gives the same value for every representation of  $\mu_C^*$. For (ii), beliefs represented using the same deletion order share the same list  $(\mu_{R_j}^*)$, so their tent values are the corresponding sums  $\sum_j (\cdot)_j \bar{v}(\mu_{R_j}^*)$, which are linear in the weight vector.

For uniqueness, let  $f$  satisfy (i) and (ii), fix  $\mu$, and choose a representation  $\mu = \sum_j \omega_j \mu_{R_j}^*$  based on a deletion order. Induct on the number of positive weights. If there is only one, then  $\mu = \mu_{R_j}^*$  for some  $j$, and (i) gives  $f(\mu) = \bar{v}(\mu_{R_j}^*) = v^{\text{tent}}(\mu)$. If there are at least two, choose an index  $J$  with  $\zeta := \omega_J \in (0, 1)$  and set  $\mu' := \sum_{j \neq J} \frac{\omega_j}{1-\zeta} \mu_{R_j}^*$. Then  $\mu'$  has a representation based on the same deletion order with one fewer positive weight and  $\mu = \zeta \mu_{R_J}^* + (1-\zeta) \mu'$. By (ii) and (i),  $f(\mu) = \zeta \bar{v}(\mu_{R_J}^*) + (1-\zeta) f(\mu')$, and the same calculation gives  $v^{\text{tent}}(\mu) = \zeta \bar{v}(\mu_{R_J}^*) + (1-\zeta) v^{\text{tent}}(\mu')$. The induction hypothesis gives  $f(\mu') = v^{\text{tent}}(\mu')$, so  $f(\mu) = v^{\text{tent}}(\mu)$.  $\square$

## J Technical Lemmas (for online publication)

The following lemma records the standard microfoundation under which a disclosure game's payoff correspondence  $V$  satisfies (A4): it is induced by a receiver who best-responds to her belief with continuous utilities over a compact action space.

**Lemma J.1.** *Let  $A$  be a non-empty compact Hausdorff space, let  $u_S: A \rightarrow \mathbb{R}$  be continuous, and, for each  $\theta \in \Theta$, let  $u_R(\cdot, \theta): A \rightarrow \mathbb{R}$  be continuous. For  $\mu \in \Delta\Theta$  and  $a \in A$  let  $U_R(a, \mu) := \sum_{\theta \in \Theta} \mu(\theta) u_R(a, \theta)$  and  $a^*(\mu) := \arg \max_{a \in A} U_R(a, \mu)$, and set  $V(\mu) := \text{conv}(u_S(a^*(\mu)))$, with envelopes  $\bar{v}(\mu) := \max_{a \in a^*(\mu)} u_S(a)$  and  $\underline{v}(\mu) := \min_{a \in a^*(\mu)} u_S(a)$, so that  $V(\mu) = [\underline{v}(\mu), \bar{v}(\mu)]$. Then  $V$  is non-empty, compact-valued, convex-valued, and upper hemicontinuous, so it satisfies (A4); moreover  $\bar{v}$  is upper semicontinuous and  $\underline{v}$  is lower semicontinuous.*

*Proof.* The map  $(a, \mu) \mapsto U_R(a, \mu)$  is continuous and  $A$  is compact, so by Berge's maximum theorem (Aliprantis and Border, 2006, Theorem 17.31) the best-response correspondence  $a^*$  is non-empty, compact-valued, and upper hemicontinuous, and the value functions  $\bar{v}$  and  $\underline{v}$  are upper and lower semicontinuous, respectively. Each  $V(\mu) = [\underline{v}(\mu), \bar{v}(\mu)]$  is therefore a non-empty compact interval, and the upper hemicontinuity of  $V$  follows from the semicontinuity of its two endpoints.  $\square$

**Lemma J.2.** *For every disclosure game  $G$  and non-empty  $R \subseteq \Theta$, the restricted game  $G|_R$  is itself a disclosure game, and  $G|_\Theta = G$. Moreover, restriction is transitive: for non-empty  $R' \subseteq R \subseteq \Theta$,*

$$
(G|_R)|_{R'} = G|_{R'}.
$$

*Proof.* The sets  $R$  and  $\mathcal{M}_R$  are non-empty and finite, each  $M_R(\theta) = M(\theta)$  is non-empty,  $\mathcal{M}_R = \bigcup_{\theta \in R} M(\theta)$  gives (A3), and  $\mu_R^0$  has full support on  $R$. The zero-extension  $\iota_{R,\Theta}$  is continuous, so  $V_R = V \circ \iota_{R,\Theta}$  is upper hemicontinuous and takes the same non-empty compact interval values as  $V$, giving (A4). Thus  $G|_R$  is a disclosure game. For  $R = \Theta$  we have  $\mu^0(\Theta) = 1$  and  $\iota_{\Theta,\Theta} = \text{id}$, so every component of  $G|_\Theta$  equals that of  $G$.

For transitivity, both sides have type space  $R'$, message space  $\bigcup_{\theta \in R'} M(\theta)$, and message mapping  $M$. The priors agree: for  $\theta \in R'$,

$$
(\mu_R^0)_{R'}(\theta) = \frac{\mu_R^0(\theta)}{\mu_R^0(R')} = \frac{\mu^0(\theta)/\mu^0(R)}{\mu^0(R')/\mu^0(R)} = \mu_{R'}^0(\theta).
$$

The payoff correspondences agree because  $\iota_{R,\Theta} \circ \iota_{R',R} = \iota_{R',\Theta}$.  $\square$

Fix a disclosure game  $G = (\Theta, \mathcal{M}, M, \mu^0, V)$  and a message  $m \in \mathcal{M}$. Recall that  $P(m) = M^{-1}(\{m\})$  is the set of types that can send  $m$, and  $F(m)$  is the set of types *forced* to send  $m$; note that  $F(m) \subseteq P(m)$.

**Lemma J.3.** *The set of feasible beliefs  $\mathcal{F}(m)$  is the set of all  $\mu \in \Delta\Theta$  such that*

- (i)  $\mu(\theta) = 0$  for every  $\theta \notin P(m)$;
- (ii)  $\mu(\theta) \mu^0(\theta') = \mu(\theta') \mu^0(\theta)$  for all  $\theta, \theta' \in F(m)$;
- (iii)  $\mu(\theta) \mu^0(\theta') \leq \mu^0(\theta) \mu(\theta')$  for all  $\theta \in P(m) \setminus F(m)$  and  $\theta' \in F(m)$.

*In particular,  $\mathcal{F}(m)$  is a non-empty compact convex polytope.*

*Proof.* Write  $P := P(m)$  and  $F := F(m)$; the set  $P$  is non-empty by (A3), and write  $\mu^0(F) := \sum_{\theta \in F} \mu^0(\theta)$. Every sender strategy  $\sigma$  satisfies  $\sigma(m|\theta) = 0$  for  $\theta \notin P$, since  $\text{supp } \sigma(\cdot|\theta) \subseteq M(\theta)$, and  $\sigma(m|\theta) = 1$  for  $\theta \in F$, since  $M(\theta) = \{m\}$.

*Parametrization.* Let  $\sigma$  be a strategy with  $m \in X(\sigma)$, and put  $\alpha_\theta := \sigma(m|\theta) \in [0, 1]$  for  $\theta \in P \setminus F$. The denominator of Bayes' rule is  $D := p_\sigma(m) = \mu^0(F) + \sum_{\theta \in P \setminus F} \mu^0(\theta) \alpha_\theta$, and the induced belief  $\mu := \mu_\sigma(\cdot|m)$  is

$$
\mu(\theta) = \begin{cases} \mu^0(\theta)/D, & \theta \in F, \\ \mu^0(\theta) \alpha_\theta / D, & \theta \in P \setminus F, \\ 0, & \theta \notin P. \end{cases}
$$

*Feasible beliefs satisfy (i)–(iii).* Let  $\mu \in \mathcal{F}(m)$, with parameter  $\alpha$  and denominator  $D$  as above. Condition (i) is immediate. For  $\theta, \theta' \in F$, both  $\mu(\theta) \mu^0(\theta')$  and  $\mu(\theta') \mu^0(\theta)$  equal  $\mu^0(\theta) \mu^0(\theta')/D$, so (ii) holds. For  $\theta \in P \setminus F$  and  $\theta' \in F$,

$$
\mu(\theta) \mu^0(\theta') = \frac{\mu^0(\theta) \alpha_\theta \mu^0(\theta')}{D} \leq \frac{\mu^0(\theta) \mu^0(\theta')}{D} = \mu^0(\theta) \mu(\theta'),
$$

since  $\alpha_\theta \leq 1$, so (iii) holds.

*Beliefs satisfying (i)–(iii) are feasible.* Let  $\mu \in \Delta\Theta$  satisfy (i)–(iii). Each  $\theta \in P \setminus F$  has  $M(\theta) \setminus \{m\} \neq \emptyset$, so it can send  $m$  with any chosen probability and place the remaining mass on  $M(\theta) \setminus \{m\}$; each  $\theta \in F$  sends  $m$; each  $\theta \notin P$  sends a message in  $M(\theta)$, none of which is  $m$.

If  $F = \emptyset$, then (i) gives  $\text{supp } \mu \subseteq P$. Set  $c := \max_{\theta \in \text{supp } \mu} \mu(\theta)/\mu^0(\theta) > 0$  and let each  $\theta \in P$  send  $m$  with probability  $\alpha_\theta := \mu(\theta)/(c \mu^0(\theta)) \in [0, 1]$. Then  $p_\sigma(m) = \sum_{\theta \in P} \mu^0(\theta) \alpha_\theta = 1/c > 0$  and  $\mu_\sigma(\theta|m) = \mu^0(\theta) \alpha_\theta c = \mu(\theta)$.

If  $F \neq \emptyset$, then  $\mu(\theta') > 0$  for every  $\theta' \in F$: otherwise (ii) makes  $\mu$  vanish on all of  $F$, and then (iii) makes  $\mu(\theta) \leq 0$  for every  $\theta \in P \setminus F$, so with (i) the belief  $\mu$  would have total mass zero. By (ii) the ratio  $\lambda := \mu(\theta')/\mu^0(\theta')$  is the same positive number for every  $\theta' \in F$. Put  $\alpha_\theta := \mu(\theta)/(\lambda \mu^0(\theta))$  for  $\theta \in P \setminus F$; then  $\alpha_\theta \geq 0$, and  $\alpha_\theta \leq 1$  by (iii). Since  $\mu$  has total mass one,

$$
1 = \sum_{\theta \in F} \lambda \mu^0(\theta) + \sum_{\theta \in P \setminus F} \lambda \mu^0(\theta) \alpha_\theta = \lambda \left( \mu^0(F) + \sum_{\theta \in P \setminus F} \mu^0(\theta) \alpha_\theta \right),
$$

so the denominator  $D$  equals  $1/\lambda$, and the strategy that sends  $m$  with probabilities  $\alpha$  induces  $\mu$  by the parametrization.

*Conclusion.* Taking the strategy under which every type in  $P$  sends  $m$  gives  $\mu_{P(m)}^0 \in \mathcal{F}(m)$, so  $\mathcal{F}(m)$  is non-empty. Conditions (i)–(iii) are finitely many linear equalities and weak inequalities, so they cut the simplex  $\Delta\Theta$  down to a bounded polyhedron. Hence  $\mathcal{F}(m)$  is a non-empty compact convex polytope.  $\square$

**Lemma J.4.** *Every disclosure game admits a PBE.*

*Proof.* We exhibit a triple  $(\sigma, \mu, r)$  satisfying the conditions of Definition 8 as a fixed point of a correspondence on the non-empty compact convex set

$$
K := \Sigma \times \prod_{m \in \mathcal{M}} \mathcal{F}(m) \times \prod_{m \in \mathcal{M}} I, \quad \Sigma := \prod_{\theta \in \Theta} \Delta M(\theta), \quad I := \left[ \min_{\Delta\Theta} \underline{v}, \max_{\Delta\Theta} \bar{v} \right].
$$

Here  $\Sigma$  is the set of sender strategies, a product of simplices; each  $\mathcal{F}(m)$  is a non-empty compact convex polytope (Lemma J.3); and  $I$  is a compact interval, finite because  $\underline{v}$  is lower semicontinuous and  $\bar{v}$  upper semicontinuous on the compact simplex  $\Delta\Theta$  by (A4), with  $V(\mu) = [\underline{v}(\mu), \bar{v}(\mu)] \subseteq I$  for every  $\mu \in \Delta\Theta$. Since  $\Theta$  and  $\mathcal{M}$  are finite,  $K$  is a subset of a finite-dimensional Euclidean space.

Define  $\Phi : K \rightrightarrows K$  by

$$
\Phi(\sigma, \mu, r) := S(r) \times \prod_{m \in \mathcal{M}} B_m(\sigma) \times \prod_{m \in \mathcal{M}} V(\mu(\cdot | m)),
$$

where

$$
S(r) := \prod_{\theta \in \Theta} \arg \max_{\tau \in \Delta M(\theta)} \sum_{m \in M(\theta)} \tau(m) r(m), \quad B_m(\sigma) := \begin{cases} \{\mu_\sigma(\cdot | m)\}, & m \in X(\sigma), \\ \mathcal{F}(m), & m \notin X(\sigma). \end{cases}
$$

Because a distribution maximizes a linear objective if and only if its support lies in the set of maximizing coordinates,

$$
\arg \max_{\tau \in \Delta M(\theta)} \sum_{m \in M(\theta)} \tau(m) r(m) = \{\tau \in \Delta M(\theta) \mid \text{supp } \tau \subseteq \arg \max_{m \in M(\theta)} r(m)\}. \quad (1)
$$

$\Phi$  has non-empty, compact, and convex values. Fix  $(\sigma, \mu, r) \in K$. For each  $\theta$, the component  $\arg \max_{\tau \in \Delta M(\theta)} \sum_m \tau(m) r(m)$  of  $S(r)$  is non-empty, because the continuous linear map  $\tau \mapsto \sum_m \tau(m) r(m)$  attains its maximum on the non-empty compact simplex  $\Delta M(\theta)$; by (1) it is a face of  $\Delta M(\theta)$, hence compact and convex. Each  $B_m(\sigma)$  is a subset of  $\mathcal{F}(m)$: on path it is the singleton  $\{\mu_\sigma(\cdot | m)\}$, with  $\mu_\sigma(\cdot | m) \in \mathcal{F}(m)$  by Definition 5; off path it is  $\mathcal{F}(m)$, which is non-empty, compact, and convex. Finally,  $V(\mu(\cdot | m)) = [\underline{v}(\mu(\cdot | m)), \bar{v}(\mu(\cdot | m))]$  is a non-empty compact interval contained in  $I$  by (A4). As products of these sets, the values of  $\Phi$  are non-empty, compact, and convex.

$\Phi$  has closed graph. Let  $(\sigma_k, \mu_k, r_k) \rightarrow (\sigma, \mu, r)$  in  $K$, and let  $(\sigma'_k, \mu'_k, r'_k) \in \Phi(\sigma_k, \mu_k, r_k)$  converge to  $(\sigma', \mu', r')$. We verify  $(\sigma', \mu', r') \in \Phi(\sigma, \mu, r)$  coordinate by coordinate.

- *Sender.* Since  $\sigma'_k \in S(r_k)$, for every  $\theta$  and every  $\tau \in \Delta M(\theta)$,

$$
\sum_{m \in M(\theta)} \sigma'_k(m | \theta) r_k(m) \geq \sum_{m \in M(\theta)} \tau(m) r_k(m).
$$

Each side is a finite sum of products of convergent sequences, so the inequality is preserved in the limit:  $\sum_m \sigma'(m | \theta) r(m) \geq \sum_m \tau(m) r(m)$  for every  $\tau$. Hence  $\sigma' \in S(r)$.

- *Beliefs.* Fix  $m$. If  $m \notin X(\sigma)$, then  $B_m(\sigma) = \mathcal{F}(m)$, and  $\mu'(\cdot | m) \in \mathcal{F}(m)$  because  $\mu'_k(\cdot | m) \in B_m(\sigma_k) \subseteq \mathcal{F}(m)$  for every  $k$  and  $\mathcal{F}(m)$  is closed. If  $m \in X(\sigma)$, then  $p_\sigma(m) > 0$. The on-path probability  $p_\sigma(m) = \sum_\theta \mu^0(\theta) \sigma(m | \theta)$  is continuous in  $\sigma$, so  $p_{\sigma_k}(m) \rightarrow p_\sigma(m) > 0$  and hence  $m \in X(\sigma_k)$  for all large  $k$. For such  $k$,  $B_m(\sigma_k) = \{\mu_{\sigma_k}(\cdot | m)\}$, so  $\mu'_k(\cdot | m) = \mu_{\sigma_k}(\cdot | m)$; and since the induced belief  $\mu_\sigma(\theta | m) = \mu^0(\theta) \sigma(m | \theta) / p_\sigma(m)$  is continuous in  $\sigma$  wherever  $p_\sigma(m) > 0$, we get  $\mu'_k(\cdot | m) \rightarrow \mu_\sigma(\cdot | m)$. By uniqueness of limits,  $\mu'(\cdot | m) = \mu_\sigma(\cdot | m) \in B_m(\sigma)$.
- *Payoffs.* For each  $m$,  $r'_k(m) \in V(\mu_k(\cdot | m))$  with  $\mu_k(\cdot | m) \rightarrow \mu(\cdot | m)$  and  $r'_k(m) \rightarrow r'(m)$. By (A4),  $V$  is upper hemicontinuous with closed values on the compact simplex  $\Delta\Theta$, so it has closed graph; therefore  $r'(m) \in V(\mu(\cdot | m))$.

By Kakutani's fixed-point theorem (Border, 1985, Corollary 15.3, p. 72),  $\Phi$  has a fixed point  $(\sigma, \mu, r)$. It satisfies the four conditions of Definition 8:  $\mu(\cdot | m) \in B_m(\sigma) \subseteq \mathcal{F}(m)$  gives feasibility; for  $m \in X(\sigma)$,  $B_m(\sigma) = \{\mu_\sigma(\cdot | m)\}$  gives the on-path Bayesian condition;  $r(m) \in V(\mu(\cdot | m))$  gives payoff compatibility; and  $\sigma \in S(r)$  together with (1) gives sequential rationality. Hence  $(\sigma, \mu, r)$  is a PBE.  $\square$