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Mathematical Social Sciences 55 (2008) 305 – 314 www.elsevier.com/locate/econbase

Lobbying with two audiences: Public vs private certification ☆ Frédéric Koessler Paris School of Economics (PSE), 48 Boulevard Jourdan, 75014 Paris, France Received 11 October 2006; received in revised form 6 October 2007; accepted 7 October 2007 Available online 23 October 2007

Abstract This paper compares public and private information certification in a simple class of communication games with one sender and two receivers. It is shown that, contrary to the cheap talk setting of [Farrell, J., Gibbons, R., 1989. Cheap talk with two audiences. American Economic Review 79, 1214–1223], allowing certifiable statements excludes mutual discipline (i.e., full information revelation in public but not in private) but allows for mutual subversion (i.e., full information revelation in private but not in public). In the latter case, the sender is always better off with public communication, while in other situations he may prefer either private or public communication. Compared to the previous models of strategic information revelation the paper also emphasizes the role of the “common belief ” consistency condition of the strong version of sequential equilibrium. © 2007 Elsevier B.V. All rights reserved. Keywords: Certifiable information; Cheap talk; Consistency of beliefs; Communication to multiple audiences JEL classification: C72; D82

1. Introduction In this paper we study a particular setting of lobbying activities with a single lobby and several decisionmakers. We refer to lobbying activities as meetings between the lobbyist (an informed interested party) and the decisionmakers in which the former try to influence the latter's choices by transmitting payoff-relevant information. For example, the lobbyist can be a leader of a country ☆ I thank the associate editor, the two anonymous referees, David Ettinger, Françoise Forges and Jérôme Mathis for interesting comments and suggestions. Financial support from an ACI grant by the French Ministry of Research is gratefully acknowledged. E-mail address: [email protected].

0165-4896/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.mathsocsci.2007.10.003

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possessing private information about the country's cost–benefit ratio for making concessions towards a rival country, and the decisionmakers the lobbyist's rival and his own public (Levy and Razin, 2004). Alternatively, the lobbyist may be a firm who must decide whether to reveal private information to a competitor and to the capital market (Gigler, 1994; Newman and Sanssing, 1993), a boss communicating to two candidates for a promotion (Farrell and Gibbons, 1989), or a company choosing between advertising publicly to a general audience and advertising privately to specific types of consumers. We consider the simple setting introduced by Farrell and Gibbons (1989), with two states of nature, two decisionmakers, two actions each, and no restriction on players' payoff functions except independence between the two decisionmakers. However, instead of assuming cheap talk (i.e., costless, non-binding, and non-certifying communication), we assume that lying is impossible due to severe consequences in case of later discovery, or we allow the lobbyist to make certifiable statements (for example, by producing unfalsifiable documents).1 This assumption is common in the literature on strategic information revelation in accounting and economics,2 and is particularly relevant for disclosures in the form of financial statements. It can also be especially appropriate in the above promotion and advertising examples. Technically, the only difference with cheap talk is that the set of messages available to the lobbyist is type dependent. We characterize the information revealed at equilibrium depending on whether meetings take place publicly or privately, and whether the information held by the lobbyist is certifiable or not. Mutual discipline refers to a situation where information is revealed to neither decisionmaker when communication is private, but a fully revealing equilibrium is played when communication takes place publicly. The opposite situation, called mutual subversion, refers to a situation where a fully revealing equilibrium is played with each decisionmaker when communication takes place privately, but information is not revealed when communication is public. In Farrell and Gibbons's (1989) binary model, mutual discipline is possible in the cheap talk (non-certifiable) communication case, but there cannot be mutual subversion (see Proposition 1). We show in Proposition 2 that the opposite holds in the case of communication with certifiable information: there cannot be mutual discipline but mutual subversion is possible. In this latter situation, the sender is always better off with public communication, while in other situations he may prefer either private or public communication (see Proposition 3). From a theoretical point of view, our study also emphasizes the role of belief consistency conditions that were irrelevant in previous work on strategic information revelation (e.g., Okuno-Fujiwara et al., 1990). More precisely, the conclusions above hold under the strong version of sequential equilibrium, while the weak version excludes mutual subversion by allowing the two decisionmakers to have different beliefs about the lobbyist's type off the equilibrium path. The remainder of the paper is organized as follows. Section 2 presents the basic class of games without communication. Section 3 defines the communication extension of the game and its weak and strong sequential equilibria. Section 4 (Section 5, respectively) characterizes the equilibria and compare them in the public and private communication cases under cheap talk (information certification, respectively). Section 5 also provides some preference orderings of the lobbyist over the communication protocols. Section 6 concludes. The proofs are relegated to the Appendix. 2. Silent game We consider one sender, S (the lobbyist), and two receivers, Q and R (the decisionmakers). The sender observes the state k ∈ K ≡ {k1, k2}, but the receivers do not. The prior probabilities of 1 2

Another similar literature assumes costly signaling to several audiences; see, e.g., Gertner et al. (1988). See, e.g., Okuno-Fujiwara et al. (1990), Verrecchia (2001) and, more recently, Giovannoni and Seidmann (2007).

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Table 1 Silent game between the lobbyist and the two decisionmakers Q

k1 k2

R

q1

q2

r1

r2

v1, x1 0, 0

0, 0 v2, x2

w1, y1 0, 0

0, 0 w2, y2

the states are Pr(k1) = p ∈ (0, 1) and Pr(k2) = 1 − p. As in Farrell and Gibbons (1989), Q has two actions, q1 and q2, and R has two actions, r1 and r2. The receivers' payoff, represented in the second coordinate of Table 1, only depends on the state and their own action. The sender's payoff is denoted by u(q, r; k) when the receivers' actions are q and r, and the sender's type is k. It is the sum of the payoff he gets from Q's action and the payoff he gets from R's action, as represented in the first coordinate of Table 1. For example, when Q chooses action q1 and R chooses action r2, the sender's payoff is u(q1, r2; k1) = v1 + 0 when his type is k1 and u(q1, r2; k2) = 0 + w2 when his type is k2. The parameters x1, x2, y1 and y2 are assumed strictly positive. Without communication, the sets of optimal (mixed) actions q(p) and r(p) of the decisionmakers Q and R are given by3 8 8 y2 x2 > > if pN¯y u ; if pN¯x u ; < fq1 g < fr1 g y1 þ y2 x1 þ x2 qðpÞ ¼ Dðfq ; q gÞ rðpÞ ¼ if p ¼ ¯ x; if p ¼ ¯y ; 1 2 > > : : Dðfr1 ; r2 gÞ fq2 g if pb ¯ x; fr2 g if pb ¯y : ð1Þ Excluding the non-generic case x¯ = y¯, we can assume without loss of generality that x¯ b y¯. 3. Communication game We consider the direct communication extension of the silent game in which before actions are taken, but after the state is revealed to the lobbyist, the latter can send to the decisionmakers a costless message in a finite set M(k) ≠ t when his type is k. The set of messages of the sender can be type-dependent in order to allow some messages to certify the lobbyist's type (see Section 5). Define M as ∪k∈K M(k). We say that communication is public when both decisionmakers observe the same message from the lobbyist. On the contrary, when the lobbyist can send a private, possibly different, message to each decisionmaker, then communication is said to be private.4 When communication is public, a “reasonable” consistency requirement would be that the receivers' beliefs about the state are the same because the receivers start exactly with the same prior beliefs, and the lobbyist's message is common knowledge. This is indeed true along any Nash equilibrium path (by Bayes' rule given the lobbyist's strategy), but also off the equilibrium path at a (strong) sequential or perfect Bayesian equilibrium (see, e.g., Fudenberg and Tirole, 3

With the usual abuse of notation in game theory, when a mixed action puts probability 1 on a single pure action we denote the mixed action by the corresponding pure action. 4 The fact that a decisionmaker knows whether the lobbyist sends a message to the other decisionmaker is irrelevant because there is no payoff interaction between the two decisionmakers. In more general settings, private communication may differ from secret communication.

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1991, condition B(iv) on page 332).5 We distinguish below this strong version of sequential equilibrium and the weak version putting no such restriction on beliefs off the equilibrium path (as defined, for example, in Mas-Colell et al., 1995, Section 9.C). Definition 1. A weak sequential equilibrium of the public communication game is a strategy profile (σ, τQ, τR) and a belief system (μQ, μR) where σ:K → Δ(M) is the sender's strategy satisfying supp[σ(k)] ⊆ M(k), τQ:M → Δ({q1, q2}) and τR:M → Δ({r1, r2}) are the receivers' strategies, and μQ:M → Δ(K) and μR:M → Δ(K ) are the receivers' belief functions, satisfying the following properties: (i) For every k ∈ K and m⁎ ∈ supp[σ(k)], m⁎ aarg max uðsQ ðmÞ; sR ðmÞ; kÞ; maM ðkÞ

(ii) For every m ∈ M , τQ(m) ∈ q(μQ(m)) and τR(m) ∈ r(μR(m)); (iii) For every m ∈ M , beliefs μQ(m) ∈ Δ(K) and μR(m) ∈ Δ(K) are obtained by Bayes's rule whenever possible, i.e., whenever Prσ (m) N 0;6 (iv) For every m ∈ M, if Prσ(m) = 0 then μQ(k|m) = μR(k|m) = 0 for every k ∉ M −1(m) ≡ {k ∈ K: m ∈ M(k)}. A strong sequential equilibrium satisfies also the common belief consistency condition: (v) For every m ∈ M, μQ(m) = μR(m) = μ(m). A sequential equilibrium of the private communication game is defined similarly except that the sender can send different and private messages to the receivers with strategies σQ:K → Δ(M) and σR:K → Δ(M) such that supp[σQ(k)] ⊆ M(k) and supp[σR(k)] ⊆ M(k). In that situation condition (v) does not apply, so weak and strong sequential equilibria coincide. In the rest of the paper, any statement referring to an “equilibrium” will apply to both the weak and strong version of sequential equilibrium. A fully revealing equilibrium is a sequential equilibrium in which the lobbyist reveals all his information, by sending with probability 1 a message m1 ∈ M (k1) when the state is k1 and m2 ∈ M (k2), m2 ≠ m1, when the state is k2. A nonrevealing equilibrium is a sequential equilibrium in which the lobbyist's strategy does not depend on his type. With only two states there is no other kind of pure strategy equilibrium. As in Farrell and Gibbons (1989), we focus on pure strategy equilibria, except when neither a fully revealing nor a non-revealing equilibrium exist (a situation that might arise in the public certification game). 4. Cheap talk In a cheap talk game any message can be sent whatever the sender's type: M(k1)=M(k2) = {m1, m2}. As is well known, in such games the set of (weak and strong) sequential equilibrium outcomes coincides with the set of Nash equilibrium outcomes, and there always exists a non-revealing equilibrium. More precisely, we have the following proposition (Farrell and Gibbons, 1989). 5

It is easy to see that a common belief for the receivers off the equilibrium path is a requirement of Kreps and Wilson's (1982) sequential equilibrium and Selten's (1975) perfect equilibrium because in the perturbed games receivers use the same trembling strategies of the sender to update their beliefs. 6 Prσ (m) is the probability of m induced by the lobbyist's strategy σ (and the prior p).

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Proposition 1. Consider the cheap talk game. There exists a fully revealing sequential equilibrium when the lobbyist communicates privately with the decisionmaker Q (R, respectively) if and only if v1 ≥ 0 and v2 ≥ 0 (w1 ≥ 0 and w2 ≥ 0, respectively). There exists a fully revealing equilibrium when the lobbyist communicates publicly with the two decisionmakers if and only if v1 + w1 ≥ 0 and v2 + w2 ≥ 0. Whatever the parameters, there always exists a non-revealing sequential equilibrium.7 Therefore, whenever there is a fully revealing equilibrium in each private cheap talk game, then there is also one in the public cheap talk game. In Farrell and Gibbons's (1989) terms, mutual subversion (full information revelation with both receivers in private but not in public) is not possible in this setting. However, the unique equilibrium outcome may be non-revealing with both receivers in private, but a fully revealing equilibrium may exist in public (take, e.g., v1 = w2 = 3 and v2 = w1 = − 1); this situation is called mutual discipline. A recent application of this effect includes, e.g., Levy and Razin (2004), in a binary model of conflict resolution between two countries in which (cheap talk) communication concerns the cost–benefit ratio from making concessions. It is also exploited in the literature in accounting and economics, for example concerning disclosures of news from a manager to both the capital and product markets (see, e.g., Evans and Sridhar, 2002). Farrell and Gibbons (1989) also perform some welfare analysis by comparing the sender's payoff at fully revealing and non-revealing equilibria in different situations. Clearly, since the decisionmakers' payoffs only depend on their own action and the sender's type, they always prefer full information revelation. However, while the sender always prefers the fully revealing equilibrium (when it exists) to the non-revealing equilibrium when the prior p lies outside the interval [x¯, y¯] (what Farrell and Gibbons call the “coherent” case), he may prefer either type of equilibrium in the other situations. In the following section we also provide some preference ordering for the sender in order to characterize situations in which he would prefer one communication protocol over another when types are certifiable. 5. Information certification In this section we assume that the set of messages available to the lobbyist depends on his type, so information is certifiable (or verifiable, provable) in the sense of, e.g., Green and Laffont (1986), Okuno-Fujiwara et al. (1990), Seidmann and Winter (1997), Forges and Koessler (2005, in press) and Giovannoni and Seidmann (2007). To simplify the exposition we assume that each state is certifiable, i.e., for every k ∈ K there exists mk ∈ M such that M −1(mk) = {k}. We also allow the lobbyist to remain silent (to send a non-certifying message), i.e., there exists a message m ¯ ∈ M(k1) ∪ M(k2) that can be sent whatever the sender's type.8 Clearly, under these assumptions all pure strategy equilibria can be achieved by letting the set of messages available to the lobbyist be M(k1) = {m1, m ¯} when the state is k1 and M(k2) = {m2, m ¯} when the state is k2. In this setting the conditions for a fully revealing equilibrium to exist are weaker than in the cheap talk case since a lobbyist's type is not necessarily able to imitate the other type's message (incentive compatibility constraints are weaker when information is certifiable). As the following proposition shows, the relationship between these conditions in the public and private communication situations is also different, and depends on the adopted equilibrium concept. In 7

In the public communication game, partially revealing equilibria, in mixed strategies, may also exist for generic parameters, but as in Farrell and Gibbons (1989) we concentrate on pure strategy equilibria when they exist. 8 Otherwise, the communication game is trivial and the unique possible outcome is fully revealing.

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addition, a non-revealing equilibrium does not always exist. It may even be the case that neither a fully revealing nor a non-revealing equilibrium exist when communication is public; in that situation the unique partially revealing equilibrium, which is in mixed strategies, is characterized. Proposition 2. • Consider the information certification game in which the lobbyist communicates privately. There exists a fully revealing sequential equilibrium with the decisionmaker Q (R, respectively) if and only if v1 ≥ 0 or v2 ≥ 0 (w1 ≥ 0 or w2 ≥ 0, respectively). There exists a non-revealing sequential equilibrium with the decisionmaker Q (R, respectively) if and only if v1 ≤ 0 when p b x¯ and v2 ≤ 0 when p N x¯ (w1 ≤ 0 when p by¯ and w2 ≤ 0 when p N y¯ , respectively).9 • Consider the information certification game in which the lobbyist communicates publicly. There exists a fully revealing weak sequential equilibrium if and only if conditions (i), (ii), (iii) or (iv) below holds. There exists a fully revealing strong sequential equilibrium if and only if conditions (i), (ii) or (iii) below holds. ðiÞv1 þ w1 z0 ðiiiÞw1 z0 and v2 z0

ðiiÞv2 þ w2 z0 ðivÞv1 z0 and w2 z0:

There exists a non-revealing sequential equilibrium if and only if v1 + w1 ≤ 0 when p b x¯, w1 ≤ 0 and v2 ≤ 0 when p ∈ ( x¯, y¯), and v2 + w2 ≤ 0 when p N y¯. There is no pure strategy sequential equilibrium if and only if p ∈ (x¯, y¯), v1 + w1 b 0, v2 + w2 b 0, and either (I) {w1 N 0, v2 b 0} or (II) {w1 b 0, v2 N 0}. In that case the unique equilibrium is in mixed 1p strategies. In situation (I) the sender's strategy is rðmjk ¯|k2 ) = 1 and the ¯ 1 Þ ¼ 1¯x ¯x p and σ(m v1 þw1 receivers' strategies after message m ¯ are sQ ðq1 jmÞ ¯ ) = 1. In situation (II) the ¯ ¼ v1 and τR(r2|m 1y p sender's strategy is σ(m ¯|k1) = 1 and rðm ¯ jk2 Þ ¼ ¯y ¯ 1p and the receivers' strategies after v2 þw2 message m ¯ are τQ(q1|m ¯) = 1 and sR ðr2 jmÞ ¯ ¼ w2 . Compared to the cheap talk situation, mutual discipline is impossible (whatever the equilibrium concept). Indeed, there is no fully revealing equilibrium in the private meetings if and only if v1, v2, w1 and w2 are strictly negative, which implies that conditions (i) to (iv) are not satisfied. The intuition of this result is the following. When there is no fully revealing equilibrium in private it means that neither decisionmaker has the ability to punish the lobbyist when he remains silent instead of fully certifying his information, whatever the decisionmakers' beliefs off the equilibrium path. This implies that full revelation of information cannot be sustained publicly since the decisionmakers' ability to punish the lobbyist is not improved when the lobbyist remains silent publicly. On the contrary, mutual subversion becomes possible with the strong version of sequential equilibrium (but not with the weak sequential equilibrium concept). For example, when v1 = w2 = x1 = x2 = y1 = 1, v2 = w1 = −2 and y2 = 2 there is a fully revealing strong sequential equilibrium in both private meetings but there is none in the public meeting. There is nonetheless a fully revealing weak sequential equilibrium. The intuition of this result is also easy. Under mutual subversion, if the lobbyist deviates by sending privately a non-certifying message to one of the decisionmakers, then he is punished with action q2 by decisionmaker Q and action r1 by decisionmaker R. When communication is public, this punishment is not credible anymore if we require the decisionmakers to have strongly consistent beliefs, because 9

We do not consider the non-generic priors p = ¯x and p = ¯y.

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action q2 is sequentially rationally with belief on k1 smaller than x¯ while action r1 is sequentially rationally with belief on k1 larger than y¯ N x¯. Our result has also some implication for strategic information revelation with multidimensional action spaces. Indeed, if there is only one decisionmaker with a two-dimensional action space {q1, q2} × {r1, r2}, and payoffs given by the sum of the payoffs in each dimension, the impossibility of mutual discipline means that full information revelation is possible only if information is fully revealed when each dimension is analyzed separately, and the possibility of mutual subversion means that full information revelation when each dimension is analyzed separately does not imply full information revelation in the multidimensional problem. When the unique equilibrium in public is in mixed strategies, then the unique equilibrium in private consists in full information revelation with one decisionmaker (R in situation (I) and Q in situation (II)) and no information revelation with the other decisionmaker (Q in situation (I) and R in situation (II)). We call this situation mixed subversion. Other situations that distinguish the equilibrium outcomes with the private and the public communication protocols are possible: onesided subversion (full information revelation with only one decisionmaker in private, and no information revelation in public) and one-sided discipline (full information revelation with only one decisionmaker in private, and full information revelation in public). The following proposition characterizes the sender's ex-ante preference ordering over the two communication protocols in all these situations. Proposition 3. Consider the information certification game. • Under mutual subversion the lobbyist strictly prefers to communicate in public. • Under one-sided discipline and mixed subversion the lobbyist strictly prefers to communicate in private. • Under one-sided subversion the lobbyist may prefer to communicate either in public or in private. When the lobbyist prefers the communication protocol with no information revelation (as in the mutual subversion situation), but cannot choose that protocol, an alternative for the lobbyist may be, if possible, to commit not to communicate (and to certify) his information.10 However, in our framework, it is very important that the lobbyist's commitment takes place at the ex ante stage, before he learns his type. 6. Conclusion In this paper we have studied strategic information certification in a simple model of communication with heterogeneous audiences. In general, contrary to the cheap talk case, public communication with certifiable information interfere with information revelation comparing to the case of private communication, at least under the assumption of strong belief consistency. In this situation of mutual subversion the lobbyist would always choose the public communication protocol, in which no information is revealed to the decisionmakers. How these conclusions hold in more general settings or in more specific applications is left for further research.

10

Players' ability to choose not to communicate has been investigated by Hurkens and Schlag (2003), but in a setting of complete information.

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Appendix A. Proofs Proof of Proposition 1. Directly from the incentive compatible constraints of the lobbyist.

□

Proof of Proposition 2. • The fully revealing sequential equilibrium in the private communication game with Q is supported by the fully revealing strategy σQ(m1|k1) = σQ(m2|k2) = 1 and the off the equilibrium path action τQ(m ¯) =q1 if v2 ≥ 0N v1, τQ(m ¯) =q2 if v1 ≥ 0 N v2, and any of the two if v1, v2 ≥ 0. If v1, v2 b 0 then, whatever the mixed action (in Δ({q1, q2}) of Q after message m ¯, the sender has a strict incentive to deviate from full information revelation. The action τQ(m ¯)=q1 can be made sequentially rational with, e.g., μQ(k1|m ¯) = 1, and action τQ(m ¯) =q2 with μQ(k1|m ¯)= 0. At a non-revealing equilibrium in the private communication game with Q the sender plays σQ(m ¯|k1)=σQ(m ¯|k2) = 1. If p b x¯ then Q plays τQ(m ¯) =q2 after message m ¯, so the sender of type k1 does not deviate by sending the (certifying) message m1 if and only if 0≥v1. The sender of type k2 never deviates since he gets the same payoff (v2) whatever the message he sends. Similarly, if p Nx¯ then τQ(m ¯) =q1 so the sender does not deviate if and only if 0 ≥v2. The same proof applies for the private communication game with R. • Full information revelation in public can be supported by the sender's strategy σ(m1|k1) = σ(m2|k2) = 1 and the following actions of the receivers off the equilibrium path, where τ ≡ (τQ, τR): τ (m ¯) = (q2, r2) in situation (i), τ (m ¯) = (q1, r1) in situation (ii), τ (m ¯) = (q1, r2) in situation (iii), and τ (m ¯ ) = (q2, r1) in situation (iv). The actions profiles in the three situations (i) to (iii) above can respectively be made sequentially rational with the common off the equilibrium path beliefs μ(k1|m ¯) ≤ x¯, μ(k1|m ¯) ≥ ¯y and μ(k1|m ¯ ) ∈ [x¯, ¯y]. Hence, we get a fully revealing strong sequential equilibrium in cases (i) to (iii). To show that there is no fully revealing strong sequential equilibrium in situation (iv) we must show that for every strongly sequentially rational mixed actions profile of the receivers after message m ¯ the sender deviates in case (iv) if neither (i), (ii) nor (iii) are satisfied. This situation implies v1 + w1 b 0, v2 + w2 b 0 and v1, w2 ≥ 0. Since a strongly sequentially rational mixed action profile puts zero probability on (q2, r1) (see the best responses in Eq. (1)), it is clear that in this situation at least one type of the sender wants to deviate by sending message m ¯. Nevertheless, the strategy profile in situation (iv) can be made weakly sequentially rational with heterogeneous beliefs μQ(k1|m ¯) ≤ x¯ for receiver Q and μR(k1|m ¯) ≥ y¯ N x¯ for receiver R. Next, we show that there is no fully revealing (weak and strong) sequential equilibrium when neither of the conditions (i) to (iv) above are satisfied. In that case, four situations are possible: (a) v2 + w2 b 0 and v1, w1 b 0, (b) v1 +w1 b 0, v2 +w2 b 0 and w1, w2 b 0, (c) v1 +w1 b 0, v2 +w2 b 0 and v1, v2 b 0, and (d) v1 +w1 b 0 and v2, w2 b 0. In situation (a), since v1, w1 b 0, type k1 does not deviate from full information revelation only if the receivers play τ (m ¯) = (q1, r1). But then, since v2 +w2 b 0, type k2 deviates and sends message m ¯ instead of m2. Situation (d) is symmetric. Next, consider situation (b). If v1 b 0 or v2 b 0 we are also in situations (a) or (d), so assume that v1, v2 ≥ 0. The sender does not deviate from full information revelation if and only if there is a mixed actions profile of the receivers,11 τQ(m ¯) = (α1, α2), τR(m ¯) = (β1, β2), where αi is Q's probability of playing action qi after m ¯ , and βi is R's probability of playing action ri after m ¯ , such that v1 þ w1 za1 b1 ðv1 þ w1 Þ þ a1 b2 v1 þ a2 b1 w1 ¼ a1 v1 þ b1 w1 ; v2 þ w2 za1 b2 w2 þ a2 b1 v2 þ a2 b2 ðv2 þ w2 Þ ¼ ð1  a1 Þv2 þ ð1  b1 Þw2 ; 11

It can be checked that the result also holds when the receivers can use correlated strategies.

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i.e., wv11 ð1  a1 Þ þ ð1  b1 ÞV0 and wv22 a1 þ b1 V0. Since v2 +w2 b 0 and w2 b 0 imply v2 /w2 N − 1, and v1 +w1 b 0 and v1 N 0 imply v1 /w1 N − 1, this yields a contradiction. Situation (c) is symmetric. At a non-revealing equilibrium in the public communication game, if p b x¯ then the receivers play τ (m ¯) = (q2, r2), so the sender does not deviate by sending the (certifying) message m1 if and only if 0 ≥ v1 + w1. If p ∈ (x¯, y¯) then the receivers play τ (m ¯) = (q1, r2), so the sender does not deviate if and only if v1 ≥ v1 + w1 and w2 ≥ v2 + w2. If p N y¯ then the receivers play τ (m ¯) = (q1, r1), so the sender does not deviate if and only if 0 ≥ v2 + w2. The conditions for no pure strategy sequential equilibrium to exist follow immediately from the existence conditions of fully revealing and non-revealing equilibria. Let τ (q, r|m ¯) be the probability that the receivers choose actions q and r after message m ¯. Since there is no pure strategy sequential equilibrium, m ¯ is played with strictly positive probability by at least one of the sender's type, so μ¯ ≡ μQ(k1|m ¯) = μR(k1|m ¯) is obtained by Bayes' rule. We detail the proof in situation (I). Situation (II) is similar. If μ¯ N ¯y then k2 sends m ¯ with probability 1 because v2 +w2 b 0. Hence, since the prior is p b ¯y, we cannot have the posterior μ¯ N ¯y whatever k1's strategy. If μ¯ ∈ (x¯, y¯) then k1 never sends m ¯ because v1 + w1 N v1, so μ¯ = 0, a contradiction. If μ¯ = y¯ then k1 never sends m ¯ or k2 sends m ¯ with probability 1, yielding to one of the previous contradictions. If μ¯ b x¯ then k1 sends m ¯ with probability 1 because v1 + w1 b 0. Again, this implies μ¯ N x¯ since p N x¯, a contradiction. So, the only possibility is μ¯ = ¯x , with τ (q1, r2|m ¯) + τ (q2, r2|m ¯) = 1. Since v2 b 0, this implies that k2 sends m ¯ with probability 1, v þw and k1 is indifferent between m1 and m ¯ if and only if sðq1 ; r2 jm ¯ Þ ¼ 1 v1 1 . Using Bayes' rule, it is easy to check that the sender's strategy rðmjk ¯ 1 Þ ¼ 1¯x ¯x 1p ¯ |k2) = 1 generates the p and σ(m required belief μ¯ = μ(k1|m ¯ ) = x¯. □ Proof of Proposition 3. Under mutual subversion we have v1 ≥ 0, w2 ≥ 0, v1 + w1 b 0, and v2 + w2 b 0. In private, information is fully revealed so the expected utility of the sender is p(v1 + w1) + (1 − p)(v2 + w2). In public, information is not revealed so the expected utility of the sender is p(v1 + w1) if p N ¯y , pv1 + (1  p)w2 if p ∈ (x¯, ¯y ), and (1 − p)(v2 + w2) if p b ¯x . Hence, the sender is always strictly better off in the second situation. Under mixed subversion, in situation (I), the expected utility of the sender and public  in private  1 v2 are respectively given by p(v1 + w1) + (1 − p)(w2) and pðv1 þ w1 Þ þ ð1  pÞ v1 w2vw , so he prefers 1 1 v2 the private communication protocol if w2 N v1 w2vw . Since v b 0 this is equivalent to w v b 0, which 1 1 2 1 is satisfied since w1 N 0 and v2 b 0. In situation (II), the expected utility of the sender in private and   public are respectively given by p(v1) + (1 − p)(v2 + w2) and p v1  ww1 v2 2 þ ð1  pÞðv2 þ w2 Þ, so he prefers the private communication protocol if ww1 v2 2 N0, which is satisfied since w1 b 0, v2 N 0 and w2 b 0. Under one-sided discipline we have either a fully revealing equilibrium in private with Q only or a fully revealing equilibrium in private with R only. We consider only the first situation (the second is similar). Then, we have w1 b 0, w2 b 0, {v1 ≥ 0 or v2 ≥ 0}, and {v1 + w1 ≥ 0 or v2 + w2 ≥ 0}. The expected utility of the sender in private is p(v1 + w1) + (1 − p)(v2) if p N ¯y and p(v1) + (1 − p)(v2 + w2) if p b ¯y , and p(v1 + w1) + (1 − p)(v2 + w2) in public. Since w1 b 0 and w2 b 0 he always prefers to communicate privately. Under one-sided subversion we have either a fully revealing equilibrium in private with Q only or a fully revealing equilibrium in private with R only. Again, we consider only the first situation. Then, we have w1 b 0, w2 b 0, {v1 ≥ 0 or v2 ≥ 0}, v1 + w1 b 0 and v2 + w2 b 0. The expected utility of the sender in private is as in the one-sided discipline case, and in public as in the mutual subversion case. Hence, the best communication protocol for the sender depends on the sign of v1 and v2. □

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