Jeanne Hagenbach
email: hagenbach@univ-paris1.fr

Frédéric Koessler

Antoni Calvó-Armengol

Olivier Compte

Wouter Dessein

Françoise Forges

David Martimort

Eduardo Perez

Francesco Squintani

Joel Sobel

Jean-Marc Tallon

Jean-Christophe Vergnaud

Strategic communication networks

Keywords: C72, D82, D83, D85 Cheap talk, coordination, partially verifiable types, public and private communication Cheap talk, coordination, partially verifiable types, public and private communication. JEL Classification: C72, D82, D83, D85

In this paper, we consider situations in which individuals want to choose an action close to others' actions as well as close to a payoff relevant state of nature with the ideal proximity to the common state varying across the agents. Before this coordination game with heterogeneous preferences is played, a cheap talk communication stage is offered to players who decide to whom they reveal the private information they hold about the state. The strategic information transmission taking place in the communication stage is characterized by a strategic communication network. We provide a direct link between players' preferences and the strategic communication network emerging at equilibrium, depending on the strength of the coordination motive and the prior information structure. Equilibrium strategic communication networks are characterized in a very tractable way and compared in term of efficiency. In general, a maximal strategic communication network may not exist and communication networks cannot be ordered in the sense of Pareto. However, expected social welfare always increases when the communication network expands. Strategic information transmission can be improved when group or public communication is allowed, and/or when information is certifiable.

Introduction

For social scientists, pressure to conform is a central instance of social influence. Since the work of [START_REF] Jones | The Economics of Conformism[END_REF], economists have acknowledged that, in many situations, the cost of non-conformist behavior can shape economic interactions. In the present paper, following [START_REF] Bernheim | A Theory of Conformity[END_REF] and [START_REF] Akerlof | Social Distance and Social Decisions[END_REF], the need for conformity is directly incorporated into individual preferences assuming that agents can be directly penalized for departing from behaviors accepted in their social group. In addition, the agents who have an interest in making a decision coordinated with that of others have heterogeneous preferences toward this decision. More precisely, we analyze situations in which individuals want to choose an action that is close to others' actions as well as close to a payoff relevant state of nature with the ideal proximity to this common state varying across the agents.

In the game we consider, players have access to independent sources of partial information about the true state of nature. They choose to whom they want to transmit their private information before playing the payoff-relevant coordination game with incomplete information. Within this stylized framework, our main object of study is the strategic information transmission that takes place during the communication stage preceding the decision stage. We characterize the strategic communication between players by a strategic communication network in the sense that a connection is formed from one individual to another if the former correctly transmits his private information to the latter. We provide a direct link between individuals' heterogeneous preferences and the emerging strategic communication network, depending on the type of communication that is allowed (private, public or group communication), the strength of the coordination motive and the prior information structure.

The situations in which agents have different "ideal actions" but an interest in coordinating their decisions with each other have the relevant features of many economic and social situations. Examples of actions taken within a social group and having bad social consequences if they turn out to be isolated include demand for education or effort towards environmental problems. 1 One can also think of financial analysts having, for one part, an interest in making predictions similar to that of others to be credible and, for another part, heterogeneous preferences towards such announcements.2 Inside a firm, decisions should be adapted to the market conditions and information about these conditions is often distributed among the members of the organization due to their specialization. On the one hand, the different divisions of the organization have to coordinate their decisions to maximize the firm's profit, but, on the other hand, each division may be biased in its decision because of career concerns, effort aversion or local adaptation costs. 3 In a market, firms have to take decisions, such as investment in order to launch a new product or amount of advertising expenses, that are the most appropriate to the underlying fundamentals. In addition, such firms may also have a "beauty contest" coordination motive arising from the strategic complementarities in the actions of all the firms in the market considered. In all these settings, a question arises about how players strategically share private information, and whether some physical communication links are worthless due to a lack of incentives between the sender(s) and the receiver(s) to correctly transmit it. When social welfare increases with information transmission, one could also search for communication protocols that stimulate strategic communication.

When individuals only differ in terms of knowledge, but not in term of preferences, it is theoretically well known how coordination and welfare is affected by the information structure, and in particular by the public or private nature of individuals' signals (see, e.g., [START_REF] Morris | Social Value of Public Information[END_REF][START_REF] Morris | Social Value of Public Information[END_REF][START_REF] Angeletos | Efficient Use of Information and Social Value of Information[END_REF][START_REF] Angeletos | Efficient Use of Information and Social Value of Information[END_REF]. The most efficient way to disseminate information about the fundamentals can therefore be investigated. With agents' goals aligned but physical or cost constraints on the number of communication links between agents, another object of study is to identify the most efficient communication structures. This problem has been analyzed in different settings in team theory and in coordination games with incomplete information by, among others, [START_REF] Marschak | Economic Theory of Teams[END_REF], Radner (1993), [START_REF] Jehiel | Information Aggregation and Communication in Organizations[END_REF], [START_REF] Chwe | Communication and Coordination in Social Networks[END_REF], Calvó-Armengol and Martí (2007a,b), and Morris and Shin (2007). A common feature of the papers cited above is that there is no conflict of interests between agents regarding the ideal state-contingent action profile. As a consequence, efficient networks are characterized under physical communication constraints. On the contrary, coordination situations we are interested in involve some conflicts of interest which is why we focus on networks arising in equilibrium under strategic communication constraints.

Since cheap talk communication is offered to players before they take their actions, our paper is methodologically related to the literature on strategic information transmission built on [START_REF] Crawford | Strategic Information Transmission[END_REF]. Our model includes multiple and interdependent decision-makers, all of them being endowed with private information, whereas most extensions of Crawford and Sobel's senderreceiver game with more than two players involve multiple senders (with no decision) but one uninformed receiver. 4One exception in the literature on cheap talk with multiple receivers (and only one informed sender) is the paper by [START_REF] Farrell | Cheap Talk with Two Audiences[END_REF] and some economic and accounting applications by [START_REF] Newman | Disclosure Policies with Multiple Users[END_REF], [START_REF] Gigler | Self-Enforcing Voluntary Disclosures[END_REF], [START_REF] Evans | Disclosure-Disciplining Mechanisms: Capital Markets, Product Markets, and Shareholder Litigation[END_REF] and [START_REF] Levy | It Takes Two: An Explanation for the Democratic Peace[END_REF]. 5 In [START_REF] Farrell | Cheap Talk with Two Audiences[END_REF] setting, the main question addressed is whether sending private or public messages to the receivers makes a difference. Farrell and Gibbons indeed illustrate a situation, called mutual discipline of public communication, in which information is revealed to neither decision-maker when communication is private but a fully revealing equilibrium is played when communication takes place publicly. Such an effect also arises in our setting, but, contrary to [START_REF] Farrell | Cheap Talk with Two Audiences[END_REF], the receivers we consider are not independent decisionmakers whose actions are separable in the sender's utility function. This enables us to identify another mutual discipline effect which is absent in [START_REF] Farrell | Cheap Talk with Two Audiences[END_REF] and that we call mutual discipline of coordination. This effect lies in the fact that, for a fix sender, when the set of his receivers gets larger, the incentive constraints to reveal his information to the original receivers become weaker. This implies that when the informational incentive constraints are satisfied for revealing information to a set of receivers, these constraints are not necessarily satisfied for information revelation to a strict subset of these receivers only. In particular, complete information revelation consisting in every player revealing his information to the whole set of players, can be the unique informative equilibrium.

Two recent papers are closely related to the present work in that they consider incentive conflicts over decisions and therefore endogenize communication between agents. [START_REF] Alonso | When Does Coordination Require Centralization?[END_REF] and [START_REF] Rantakaraki | Managing Change: Allocation of Decision Rights and Strategic Communication[END_REF] both analyze strategic communication in a two-divisions organization in which the decisions of the divisions must be responsive to local particularities as well as coordinated with each other. They compare different governance structures such as Decentralization, a case in which division managers communicate horizontally and make their respective decisions, and Centralization, a case in which decision managers communicate vertically with an independent headquarter who issues its decisions orders. Decision makers' payoffs are similar to the ones we consider but conflicts of interest regarding decisions are modeled in a different way. In [START_REF] Alonso | When Does Coordination Require Centralization?[END_REF] and [START_REF] Rantakaraki | Managing Change: Allocation of Decision Rights and Strategic Communication[END_REF], each division manager has an "ideal action" that depends on an idiosyncratic state and maximizes a weighted sum of his own division's profit and the one of the other division. These weights capture how biased each manager is towards his own division's profit. The focus is on determining the best organizational arrangement driven by these biases and by the relative importance of coordination need.

Our model, presented in Section 2, is a n-player coordination game with continuous, onedimensional action spaces. As in [START_REF] Morris | Social Value of Public Information[END_REF] and Calvó-Armengol and Martí (2007a,b), each player has a private signal about the fundamentals and incurs losses from a mismatch between his action and (i) his "ideal action" given by a parameter that depends on the underlying fundamentals, and (ii) others' actions. As in [START_REF] Crawford | Strategic Information Transmission[END_REF] or [START_REF] Dessein | Authority and Communication in Organizations[END_REF], each individual's ideal action is characterized by a systematic positive or negative bias. Biases vary across individuals and the profile of biases in the population is a measure of the conflict of interests faced by agents. Before players choose their action, they are offered a single stage to send costless messages to each other.

In Section 3, we first characterize the unique second-stage Bayes-Nash equilibrium decisions depending on the communication network induced by the first-stage communication strategy profile. This enables us to compare communication networks in terms of efficiency. While communication networks cannot be ordered in the sense of Pareto, even at the ex ante stage, expected social welfare always increases when the communication network expands. Next, we investigate the conditions for a communication network to be an equilibrium of the cheap talk extension of the game. In short, the incentive constraints for some player i to reveal his type to some subset of players R i are satisfied when player i's bias is close enough to the average bias of every subset of players in R i . Surprisingly, no maximal equilibrium network may exist meaning that there may be an equilibrium in which player i reveals his type only to players in R i , another equilibrium in which he reveals his type only to players in R ′ i , but no equilibrium in which he reveals his type to players in R i ∪ R ′ i . The tractable equilibrium characterization that we get also directly provides necessary and sufficient conditions for the complete social welfare maximizing network to be an equilibrium of the communication game.

In Section 4, the informational incentive constraints are weakened by considering other communication protocols. In Section 4.1, players are required to send the same message either to all the other players (public communication) or to a subset of these (group communication). With such communication forms, the informational incentive constraints for player i to reveal his type to a subset of players R i are satisfied whenever player i's bias is close enough to the average bias of players in R i , which is a weaker requirement than under private communication. Finally, in Section 4.2, we allow players to use messages that completely or partially certify their type. By providing sufficient conditions for full revelation of information in this case, we extend some results from the literature on strategic information revelation by [START_REF] Okuno-Fujiwara | Strategic Information Revelation[END_REF], [START_REF] Seidmann | Strategic Information Transmission with Verifiable Messages[END_REF] and [START_REF] Van Zandt | Monotone Equilibria in Bayesian Games of Strategic Complementarities[END_REF]. When types are completely certifiable, full revelation of information is obtained whatever the type profile and the communication protocol. On the contrary, when types are only partially certifiable, public communication is again more efficient than private communication, and full revelation of information is not guaranteed for every bias profile.

We conclude in Section 5. All proofs are relegated to the appendix.

Model 2.1 A Class of Coordination Games with Incomplete Information

Let N = {1, ..., n} be a finite set of agents. Each agent chooses an action a i ∈ A i = R. The action profile over all agents is denoted a = (a 1 , . . . , a n ). Each agent's payoff depends on the action profile and a state of nature θ. Before the game starts, nobody knows the state of nature, but each agent i ∈ N receives a private signal s i ∈ S i about θ. We assume that agents' types are independent and denote q i ∈ ∆(S i ) the prior probability distribution over agent i's set of types, for every i ∈ N . When the type profile is s = (s 1 , ..., s n ), the underlying state of nature is θ(s) ∈ R and agent i's payoff function is given by

u i (a 1 , ..., a n ; θ(s)) = -(1 -α)(a i -θ(s) -b i ) 2 - α n -1 j =i (a i -a j ) 2 .
(1)

The first component of agent i's utility function is a quadratic loss in the distance between his action a i and his ideal action θ(s) + b i . The second component is a miscoordination quadratic loss which increases in the average distance between i's action and other agents' actions. The constant α ∈ (0, 1) weights both sources of quadratic loss, i.e., it parameterizes agents' coordination motives arising from the strategic complementarity in their actions. The constant b i ∈ R parameterizes agent i's preference regarding his ideal action in the first component of his utility function. We allow the bias parameter b i to vary across individuals to reflect agents' conflict of interests with respect to their ideal actions. If all b i were equal, there would be no informational incentive problem and strategic information transmission would therefore be trivial.

Communication Game

Before the coordination game described below is played, but after each player has learnt his type, a communication stage is introduced in which players can send costless and private messages to each other. More precisely, every player i can send a different message m j i ∈ M i to every other player j = i, M i denoting the (nonempty) set of messages available to player i. Let m i = (m j i ) j =i ∈ (M i ) n-1 be the vector of messages sent by player i, and m i = (m i j ) j =i ∈ j =i M j ≡ M -i be the vector of messages received by player i.

In this communication game, player i's first stage communication strategy is a profile

σ i = (σ j i ) j =i with σ j i : S i → M i .
Let σ j i (m j i | s i ) be the probability (0 or 1) that player i sends message m j i to player j according to his strategy σ i when his type is s i . Player i's second-stage decision strategy is a mapping

τ i : S i × (M i ) n-1 × M -i → A i ,
where τ i (s i , m i , m i ) is the action chosen by player i when his type is s i ∈ S i , he sent the vector of private messages m i = (m j i ) j =i ∈ (M i ) n-1 and received the vector of private messages

m i = (m i j ) j =i ∈ M -i . Let τ (s, (m i ) i∈N ) = (τ i (s i , m i , m i ))
i∈N be the corresponding action profile.

At the end of the communication stage, a belief system is a profile µ = (µ j i ) i =j , where µ j i : M j → ∆(S j ) for every i ∈ N and j = i. Given player j's message m i j to player i, µ j i (s j | m i j ) is player i's belief about player j's type s j ∈ S j .

A perfect Bayesian equilibrium (PBE)6 of the communication game is a strategy profile (σ, τ ) = ((σ i ) i∈N , (τ i ) i∈N ) and a belief system µ satisfying the following properties:

(i) Sequential rationality in the communication stage. For all i, j ∈ N , i = j, and s i ∈ S i ,

σ j i (s i ) ∈ arg max m j i ∈M i s -i ∈S -i q -i (s -i ) u i τ (s, (σ -i (s -i ), σ -j i (s i ), m j i )); θ(s) ,
where q -i (s -i ) = j =i q j (s j ).

(ii) Sequential rationality in the action stage. For all i ∈ N , m i ∈ (M i ) n-1 and m i ∈ M -i ,

τ i (s i , m i , m i ) ∈ arg max a i ∈A i s -i ∈S -i µ i (s -i | m i ) u i (τ j (s j , (σ -i j (s j ), m i j ), (σ j -i (s -i ), m j i ))) j =i , a i ; θ(s) ,
where µ i (s -i | m i ) = j =i µ j i (s j | m i j ). (iii) Belief consistency. For all i, j ∈ N , i = j, and m i j ∈ supp[σ i j ],

µ j i (s j | m i j ) = σ i j (m i j | s j )q j (s j ) t j ∈S j σ i j (m i j | t j )q j (t j )
.

Results

In this section, in order to characterize information transmission networks that emerge from the cheap talk extension of the game as (directed) hypergraphs over the set of players, we assume that each player i can only have two possible types, S i = {s i , s i } with s i < s i . Thus, any message from player i to player j is either fully revealing or non-revealing, and a communication link is formed from i to j when i's message to j is fully revealing. Without further loss of generality, we can restrict ourselves to binary messages spaces, M i = {m, m}. In addition, to get explicit and tractable equilibrium characterizations, we assume that the state of nature is additive in types: θ(s) = i∈N s i . These assumptions on the number of types and the additivity of the state of nature will be relaxed in Section 4.2 when focusing on the conditions for complete information revelation (by all players to all the other players) with certifiable types.

The difference between the two possible signals of player i, s is i , is the value of the private information for player i. When this value is high, player i's private information has a large impact on the fundamentals.

Second-Stage Equilibrium Characterization

With only two possible types for each player, every communication strategy profile (σ i ) i∈N can be characterized by a communication network (R i ) i∈N , where, for every player i,

R i ≡ {j ∈ N \{i} : σ j i (s i ) = σ j i (s i )},
is the set of players player i completely reveals his type to. Let r i = |R i | be the number of players who learn player i's type in the communication stage.

Given a profile of types (s i ) i∈N and a communication strategy profile characterized by (R i ) i∈N , the second stage equilibrium action of each player i ∈ N is uniquely given by (see Appendix 6.1),

a i = j∈I i α(n -1 -r j )E(s j ) + (1 -α)(n -1)s j n -1 -αr j + j∈I i E(s j ) + [(n -1) -(n -2)α]b i + α j =i b j n + α -1 , (2)
where Hence, i's optimal action has three components. The first component is a weighted sum of j's actual type, s j , and the expected value of j's type, E(s j ), for each player j whose type is known by player i (including himself). Note that more relative weight is put on the actual type of player j when the coordination motive, α, is low and when the number of players who know j's type, r j , is high. The second component corresponds to the sum of the expected values of j's type for each player j whose type is unknown by player i. The last component adjusts the action of player i with respect to the bias profile. It is increasing with all players biases, with more relative weight being put on player i's own bias, b i , when the coordination motive decreases. This explicit characterization of second-stage equilibrium actions as a function of the information structure and players' preference allows us to characterize in a very tractable way the efficient and equilibrium communication strategy profiles, as shown in the following subsections.

I i = {k : i ∈ R k } ∪

Efficient Networks

The next proposition compares players' ex ante expected payoffs when the communication network that arises from the communication stage expands, assuming that equilibrium actions are played in the second-stage game.7 While an increase in the set of receivers who learn player i's type is always strictly beneficial for player i and for these receivers, such an increase makes players who don't learn player i's type always strictly worse off.

Proposition 1 Consider two communication networks R = (R i , R -i ) and R ′ = (R ′ i , R -i ) such that R i R ′ i . i) Player i is strictly better off, ex-ante, with the communication network R ′ than with the communication network R; ii) Every player j ∈ R ′ i (with j ∈ R i or j / ∈ R i ) is strictly better off, ex-ante, with the communication network R ′ than with the communication network R; iii) Every player j ∈ N \({i}∪R ′ i
) is strictly worse off, ex-ante, with the communication network R ′ than with the communication network R.

Proof. See Appendix 6.2. This result implies that, in general, communication networks cannot be ordered in the sense of Pareto, except the complete communication network (R i = N \{i} for all i ∈ N ) that Pareto dominates every other network.

The next proposition shows, however, that the overall effect of an increase in information transmission is positive. We define the social welfare as the sum of individual utilities, i∈N u i (a; θ), and consider that a communication network

R ′ = (R ′ i ) i∈N is larger than a communication network R = (R i ) i∈N when R i ⊆ R ′
i for all i ∈ N (with at least one strict inclusion).

Proposition 2 The welfare is always strictly larger, ex-ante, with the communication network R ′ than with the communication network R if R ′ is larger than R.

Proof. See Appendix 6.3.

Equilibrium Networks

The next proposition provides a complete characterization of communication networks that may arise as a perfect Bayesian equilibrium outcome of the cheap talk extension of the game. The proposition tells us that, in equilibrium, player i transmits his information to a set of players R i ⊆ N \{i} if player i's bias is close enough to the average bias of the players who belong to every subset of players in R i .

Proposition 3 There exists a perfect Bayesian equilibrium in which every player i completely reveals his private information to every player in R i ⊆ N \{i} iff for all i ∈ N and R ′ i ⊆ R i , with

|R ′ i | = r ′ i , we have b i - j∈R ′ i b j r ′ i ≤ (n -1 + α)(n -1 -αr ′ i ) 2(n -1)(n -1 -αr i ) (s i -s i ).
(3)

Proof. See Appendix 6.4.

The condition for player i to transmit his information to players in R i does not depend on the communication strategies used by players different from i, which highly simplifies the analysis. As it can be seen from the threshold on the RHS of Inequality (3), this condition for information transmission is weaker when the weight on coordination motives, α, increases, 8 when the value of player i's private information, s i -s i , increases, and when the total number of players, n, decreases. 9

From the previous proposition, a corollary is deduced that gives the necessary and sufficient condition for the efficient communication network, the complete one, to be an equilibrium of the communication game.

Corollary 1 There is a fully revealing equilibrium, characterized by the complete communication network, if and only if for all i ∈ N and

R i ⊆ N \{i}, b i - j∈R i b j r i ≤ (n -1 + α)(n -1 -αr i ) 2(n -1) 2 (1 -α) (s i -s i ). ( 4
)
As an illustration, consider a game with n = 4 players and α = 1/2. To start with, we examine the incentives for information transmission of some player i ∈ N whose value of private information is s is i = 12×3 7 , with a null bias b i = 0. Then, the RHS of Equation ( 3) in Proposition 3 simplifies to 3 6-r ′ i 6-r i . It follows that player i reveals his type to all the other players if for all k, l ∈ N \{i},

j =i b j 3 ≤ 3, b k + b l 2 ≤ 4, and |b k | ≤ 5. (5)
Similarly, player i reveals his type only to players in {j, k} N \{i} if

b j + b k 2 ≤ 3, and |b j |, |b k | ≤ 3.75. (6)
Finally, player i reveals his type only to player

j = i if |b j | ≤ 3.
Consider the bias profile b = (-4.1, 0, 3.8, 4.1). Then, the only informative equilibrium strategy for player i = 2 is to reveal his type to players in R 2 = {1, 3, 4}. This illustrates that there may be no equilibrium in which player 2 transmits his information to any strict subset of R 2 only, but there may be an equilibrium in which player 2 transmits his information to all players in R 2 . More generally, we observe an effect that we call mutual discipline of coordination, reflecting the fact that information transmission from some player to another one depends on whether the former's information is also transmitted to some other players or not. Indeed, the conditions of Proposition 3

8 The RHS of Equation (3) is increasing in α because ∂ ∂α (n-1-αr ′ i ) (n-1-αr i ) = (n-1)(r i -r ′ i ) (n-1-αr i ) 2 ≥ 0. 9 The RHS of Equation (3) is decreasing in n since the sign of its derivative with respect to n is 2α(n -1)r ′ i - α 2 rir ′ i -(n -1) 2 (ri + 1 -r ′ i ), which is always negative.
on the proximity between i's bias and the average bias of the strict subsets R ′ i R i of receivers become weaker as the set of all receivers, R i , increases. This effect is absent in [START_REF] Farrell | Cheap Talk with Two Audiences[END_REF] where the payoff of each decisionmaker only depends on his own action, while in our model players want to coordinate their actions.

Despite this positive effect on information transmission to larger sets of receivers, a maximal equilibrium communication network may not exist. To see this, consider the bias profile b ′ = (0, 2.2, 3.2, 3.7). Then, there is an equilibrium in which player i = 1 reveals his type to R 1 = {2, 3} and an equilibrium in which he reveals his type to R1 = {2, 4}, but there is no equilibrium in which player 1 reveals his type to players in R 1 ∪ R1 = {2, 3, 4}. More generally, when there is an equilibrium in which some player i transmits his information to R i and an equilibrium in which he transmits his information to Ri , a sufficient condition to get an equilibrium in which player i also transmits his information to R i ∪ Ri is that R i and Ri do not overlap.

By looking at the overlapping of the conditions under which a communication strategy is an equilibrium one, further observations can be deduced from Proposition 3 on the way agents can be connected at equilibrium. For instance, assuming without loss of generality that players are numbered so that their biases are arranged in increasing order, the condition that must be satisfied so that a player completely reveals his private information to a single other player enables to deduce what follows: if there exists an equilibrium in which player i transmits his information to a single player k, then there is also an equilibrium in which player i transmits his information to R i , for all R i ⊆ {i + 1 . . . , k -1, k} with k > i. One could also note that the directed connection that is built from player i to player k, when player i transmits his private information to player k, can be reciprocal, in the sense that there also exists an equilibrium in which player k reveals his information to player i, if players i and k both have the same value of information, i.e., si -

s i = sk -s k .
It is also easy to examine the whole strategic communication networks that can arise in the previous 4-player example. In the two strategic communication networks represented in Figure 1 below, every arrow departing from a player i corresponds to player i's unique equilibrium communication strategy that is informative. Players who belong to the same set of receivers appear in the same dotted box (for clarity, singletons are not drawn). Information revelation from player i to a set of receivers R i is represented by an arrow from i to R i . In the example of Figure 1 (a), the value of private information is the same for every player i ∈ N and equal to s i -s i = 12×3 7 . Then, with the bias profile b, the most informative strategic communication network is R = (∅, {1, 3, 4}, {4}, {3}). In this network, players 3 and 4 form a completely linked pair of players for who information revelation in reciprocal whereas player 1 has no informative equilibrium communication strategy. Interestingly, player 2 reveals his type to the set of all other players but there is no player who transmits his information to him in equilibrium. In the example of Figure 1

(b), the values of pri- vate information are s 1 -s 1 = 12×12 7×5 , s 2 -s 2 = 12×6 7×5 , s 3 -s 3 = 12×3 7×5 and s 4 -s 4 = 12 7×5 .
Then, with the bias profile b ′ , the most informative strategic communication network is R ′ = ({2}, {3}, {4}, ∅). In this network, information revelation is never reciprocal which is due to the differences in the players' values of private information.

Extensions

Since larger communication networks are always beneficial in terms of welfare, we now investigate how other types of (strategic and decentralized) communication extensions of the game may allow more effective information transmission than private cheap talk. In this section we first show how communication can be improved by considering group communication, where each player i is required to send the same message to all players in a group Ri ⊆ N \{i}. This includes as a particular case public communication, where Ri = N \{i} for all i ∈ N . In the second subsection we show that, even when the conditions for a fully revealing equilibrium to exist in the private or public cheap talk case are not satisfied, complete information revelation becomes possible whatever the bias profile and the communication protocol (public or private) when players are able to completely certify their types. When types can only be partially certified, the condition for complete information revelation depends again on the communication protocol and the bias profile, but not on the magnitude of players' biases.

Public and Group Communication

It is well known since [START_REF] Farrell | Cheap Talk with Two Audiences[END_REF] that the credibility of a sender's claim may radically depend on whether this claim has been made publicly or privately. In our model, in order to investigate how players' incentives to transmit their information is affected by the communication protocol, we consider group communication games in which each player is required to send the same message to a fixed subset of players.

Formally, in the public communication game, each player i's communication strategy is simply a mapping σ i : S i → M i , where σ i (s i ) is the message publicly observed by all players in N \{i} when player i's type is s i . When there is more than two potential audiences, each player i may also be required to send the same message to a subset Ri of players in N \{i}, for i = 1, . . . , n. This communication extension of the game is called the group R-communication game, where R = ( Ri ) i∈N . In this game, each player i's communication strategy is a mapping σ i :

S i → M i ,
where each player i is required to send the same message σ i (s i ) = m i ∈ M i to all players in Ri .

The definition of a perfect Bayesian equilibrium for the group (and public) communication games is similar to the definition for the private communication game. When focusing on equilibrium outcomes in which each player i fully reveals his type to the subset of players Ri , the only difference between the group and private communication protocols is that the informational incentive constraints are weaker in the former one: the only possible deviation from a common message sent to players in Ri is to jointly lie to all of them, while player i can choose to lie to any subset of Ri when the messages are private.

Proposition 4 In the group R-communication game, there exists a perfect Bayesian equilibrium in which every player i completely reveals his private information to every player in Ri ⊆ N \{i} iff for all i ∈ N , Inequality (3) holds for all i ∈ N , with R ′ i = Ri .

Proof. See Appendix 6.4.

In particular, in the public communication game, there is a fully revealing equilibrium if and only if for all i ∈ N ,

b i - j =i b j n -1 ≤ (n -1 + α) 2(n -1) (s i -s i ). ( 7
)
Notice that if there is an equilibrium in which player i transmits his information to players in R i in the private communication game, then there is also a group communication game with an equilibrium in which player i transmits his information to players in R i . In particular, the set of all private strategic communication networks is included (and may be strictly included) in the set of all group strategic communication networks. This is a generalization of the mutual discipline effect of public communication observed by Farrell and Gibbons (1989, Proposition 1).

As an illustration, consider again the 4-player example of Section 3, with s i -s i = 12×3

7 and b i = 0. We have seen that player i reveals his type to all the other players in private if for all k, l ∈ N \{i}, all conditions of Inequality (5) are satisfied. On the contrary, under public communication, only the first inequality of (5) is required. Similarly, in the group {j, k}-communication game, player i reveals his type to players in {j, k} whenever the first inequality of Equation ( 6) is satisfied. Hence, there is a mutual discipline effect of group communication if, e.g., b 1 ∈ (-9, -5) and b 3 = b 4 = -b 1 , since in that case there is no informative equilibrium from player i = 2 in private, while under group communication there are equilibria in which player 2 reveals his type to players in R 2 = {1, 3, 4}, R 2 = {1, 3} or R 2 = {1, 4} (but not R 2 = {3, 4}). Finally, using the same example as under private communication, it can be seen that a maximal equilibrium communication network may not exist even under group communication.

Certifiable Information

In this section we extent our communication game by allowing the set of messages available to each player to depend on his private information. Following the terminology of [START_REF] Grossman | The Informational Role of Warranties and Private Disclosure about Product Quality[END_REF], [START_REF] Milgrom | Good News and Bad News: Representation Theorems and Applications[END_REF], [START_REF] Green | Partially Verifiable Information and Mechanism Design[END_REF], [START_REF] Okuno-Fujiwara | Strategic Information Revelation[END_REF], [START_REF] Bull | Evidence Disclosure and Verifiability[END_REF], [START_REF] Forges | Communication Equilibria with Partially Verifiable Types[END_REF] or [START_REF] Giovannoni | Secrecy, Two-Sided Bias and the Value of Evidence[END_REF], this means that players are able to provide hard, verifiable, or certifiable information about their type.

Formally, the model is equivalent to the cheap talk model analyzed in section 2, except that each player i can send messages in M i (s i ), where M i (s i ) = ∅ is a type dependent set of messages. In this subsection, the set of types S i of player i is any finite set, and the function θ(s) is not required to be additive in types anymore; we only assume that it is weakly increasing with s i for all i ∈ N . Without further loss of generality, assume that types in S i ⊂ R are increasingly ordered.

The communication game and perfect Bayesian equilibria are defined as in Section 2 except that the belief consistency condition (iii) on page 6 is stronger. For all i, j ∈ N , i = j, and for all s j ∈ S j , we have the following additional condition:

µ j i (s j | m i j ) = 0 if m i j / ∈ M j (s j ).
In addition, in the public communication game,10 condition B(iv) in Fudenberg and Tirole (1991, p. 334) requires common belief even off the equilibrium path, i.e., µ j i (•) = µ j (•) for all i, j ∈ N , i = j. We say that type s i ∈ S i is certifiable if there exists a message c i (s

i ) ∈ M i ≡ t i ∈S i M i (t i ) such that M -1 i (c i (s i )) ≡ {t i ∈ S i : c i (s i ) ∈ M i (t i )} = {s i }.
The following proposition shows that whatever the communication protocol (public or private), if every player can certify his type, then there exists a fully revealing equilibrium in which all players reveal their type to all the other players.

Proposition 5 Whatever the communication protocol (public or private) and the bias profile, (b i ) i∈N , if each type of each player is certifiable, then the communication game has a perfect Bayesian equilibrium which is fully revealing.

Proof. See Appendix 6.5. This proposition extends the results of the literature in several aspects. First, in [START_REF] Okuno-Fujiwara | Strategic Information Revelation[END_REF], the class of n-person games with n > 2 is restricted to the following class of linear-quadratic utility functions for player i:

11 a i [β i (s) -d j =i a j -a i ], (8)
where d ∈ (0, 2) and β i (s 1 , . . . , s n ) is increasing with s i and decreasing with s -i . Developing the utility function in our model (see Equation ( 1)) we get instead (minus a constant):

a i [2(1 -α)(θ(s) + b i ) + 2α n -1 j =i a j -a i ] - α n -1 j =i (a j ) 2 . (9)
Equation ( 9) cannot be rewritten as Equation ( 8) for three important reasons:

1. In our model, β i (s) = 2(1α)(θ(s) + b i ), which is increasing with s j for all j ∈ N ; 2. Our model involves strategic complementarities because d = -2α n-1 is negative, while Okuno-Fujiwara et al. ( 1990) assume strategic substitutes (d > 0); 3. Equation ( 9) contains the additional term -α n-1 j =i (a j ) 2 which is absent from Equation (8). 12 Second, Van Zandt and Vives (2007) also prove the existence of a fully revealing equilibrium in a class of games with strategic complementarities, but they assume that each player's utility function is increasing in the actions of the other players. This assumption of positive externalities in actions is clearly not satisfied in our model. Third, our proposition shows that full revelation of information holds in the public and private communication games while [START_REF] Okuno-Fujiwara | Strategic Information Revelation[END_REF] and [START_REF] Van Zandt | Monotone Equilibria in Bayesian Games of Strategic Complementarities[END_REF] only consider public communication.

Finally, with the exception of some sender-receiver games considered, e.g, by [START_REF] Seidmann | Strategic Information Transmission with Verifiable Messages[END_REF], fully revealing equilibria found in the literature are usually robust to a simple inference that either always puts probability one on the lowest type consistent with the sender's report, or always puts probability one on the highest type. Here, as shown in Appendix 6.5, to support full revelation of information, the form of players' beliefs off the equilibrium path depends on the parameters of the game (the profile of biases (b 1 , . . . , b n )), on the player who deviates, and on the players who observe this deviation (which depends on whether the communication game is public or private). More precisely, in the private communication game, when player j receives a private message m j i from player i and his bias is higher than player i's bias (b j ≥ b i ), then his belief off the equilibrium path consists in believing that player i's type is the highest type compatible with i's message (i.e., player j believes that player i's type is max{t i ∈ S i : m j i ∈ M i (t i )}). On the contrary, when player j's bias is lower than player i's bias, then he believes the lowest type compatible with i's message. In the public communication game, players' inferences depend on whether the bias of the player who deviates is lower or higher than the average bias b = i∈N b i /n. When b ≥ b i , players in N \{i} believe the highest type compatible with player i's report, and when b ≤ b i they believe the lowest type.

The last observation has implications on the certifiability requirements for complete information revelation.

Proposition 6 (Fully revealing equilibrium with partially certifiable types)

In the public communication game, if each player i with a lower bias than the average bias (i.e., b i ≤ b) can certify whatever his actual type s i that his type is at most s i (i.e., there exists m i ∈ M i such that s i = max M -1 i (m i )), and if each player i with a higher bias than the average bias (i.e., b i ≥ b) can certify whatever his actual type s i that his type is at least s i (i.e., there exists m i ∈ M i such that s i = min M -1 i (m i )), then there is a perfect Bayesian equilibrium which is fully revealing. In the private communication game, if each player i with the lowest bias (i.e., b i ≤ b j for all j ∈ N ) can certify whatever his actual type s i that his type is at most s i , if each player i with the highest bias (i.e., b i ≥ b j for all j ∈ N ) can certify whatever his actual type s i that his type is at least s i , and the other players can completely certify their types, then there is a perfect Bayesian equilibrium which is fully revealing.

Hence, as in the cheap talk case, the sufficient conditions for full information revelation are stronger in the private than in the public communication game. As in the cheap talk game, this is because in the public communication game, less deviations in the communication stage are possible. It is however important to notice that, on the other hand, when communication is private, different receivers can make different inferences from the same deviation, while in the public communication game belief consistency requires all receivers to make the same inferences. These two differences between public and private communication are exactly to potential sources of "mutual discipline" (full information revelation in public but not in private) in the cheap talk game of [START_REF] Farrell | Cheap Talk with Two Audiences[END_REF] and of "mutual subversion" (full information revelation in private but not in public) in the corresponding information certification game of [START_REF] Koessler | Lobbying with Two Audiences: Public vs Private Certification[END_REF]. Here, mutual subversion is never possible, and when types of completely certifiable, mutual discipline is also impossible since full revelation of information always occurs in both the public and private cases. However, mutual discipline is again possible with partially certifiable information, as shown in the next example. More precisely, the example gives a simple instance where Proposition 6 applies for the public communication game but there is no fully revealing equilibrium in the private one.

Consider indeed a 3-player game in which only player 1 knows the state θ ∈ {θ 1 , θ 2 , θ 3 }, players' biases satisfy b 2 ≤ b 1 ≤ b 3 and b 1 ≤ b 2 +b 3 2 , and the messages available to player 1 depending on the state are:

M (θ 1 ) = {m 1 , m 2 , m 3 }, M (θ 2 ) = {m 2 , m 3 }, M (θ 3 ) = {m 3 }.
By Proposition 6, these assumptions imply that there exists a fully revealing equilibrium in the public communication. Hence, in equilibrium, players' actions in state θ are given by

a i (θ) = θ + 3b i + j =i b j 5 .
Consider now the fully revealing communication strategy in the private communication game.

When the real state is θ 1 and player 1 deviates by sending message m 2 instead of m 1 to player 2 (without deviating towards player 3), his best response is to choose action

a ′ 1 = 2(θ 1 +b 1 )+a 2 (θ 2 )+a 3 (θ 1 ) 4 = 3θ 1 +θ 2 4
+ 3b 1 +b 2 +b 3

5

. After some simplifications, the condition for this deviation to be profitable for player 1 is

b 1 -b 2 > 15(θ 2 -θ 1 ) 16 .
Hence, under this condition there is no fully revealing equilibrium in the private communication game, while a fully revealing equilibrium exists in the public one whatever the distance between the possible fundamentals and the distance between player 1 and player 2's biases (as long as b 1 ≤ b 2 +b 3 2 ).

Conclusion

In our cheap talk game, information on a common state of nature is dispersed among some players. These players must choose an action by balancing the benefit of choosing it close to their "ideal action", depending on the state and on an idiosyncratic bias, with that of choosing actions close to each other. In such a setting, we investigate the way individuals' heterogeneity affects strategic information transmission that takes place during a cheap talk stage offered to players before they take a decision. We first show that expected social welfare always increases when communication expands but that communication networks cannot be ordered in the sense of Pareto even at the ex ante stage. Next, we provide conditions on the proximity of players' biases to get every possible communication structure as an equilibrium of the cheap talk game and extend results to commu-nication protocols that enable larger networks to emerge under weaker conditions, namely the use of group communication and/or certifiable messages.

In this paper, the game through which networks are built is completely different from usual non-cooperative network formation games starting with [START_REF] Jackson | A Strategic Model of Social and Economic Networks[END_REF]. 13 In such games, every player's strategy consists in listing wished contacts whereas we derive connections from the equilibrium strategy profiles of different communication games. A common point of both network formation approaches lies in the fact that nodes are players whose payoff depends on the communication network effectively formed. Since it is now largely admitted that much of the information useful to economic and social decision making (information about job opportunities, state of the market, environment of the firms, . . . ) is exchanged via networks of relationships, gains associated to network structures are often interpreted in term of information. In network formation games à la [START_REF] Jackson | A Strategic Model of Social and Economic Networks[END_REF], agents create links to maximize their utility with the informational costs and benefits of direct and indirect connections being usually exogenous. On the contrary, we explicitly formalize the information structure and the way a player benefits from informing or being informed is endogenously given by the equilibrium outcome of the decision stage. When information flows through networks of relationships instead of via centralized institutions, it makes sense to assume that agents cannot commit to private information revelation. Whether information will effectively be transmitted once the link is formed is therefore not ensured. Our model makes it possible to analyze the incentives to misrepresent or hide information that circulates through network links. We consider effective strategic communication between players as a central feature when examining the creation of communication links between them.

During the past decade, the theory of network formation has been a very active area of research and most of the existing literature focuses on homogeneous player models.14 Only recently, the role of ex-ante asymmetries among the players in shaping the architecture of networks has been investigated. Arguing that such asymmetries appear in many natural contexts, [START_REF] Galeotti | Network Formation with Heterogeneous Players[END_REF] propose a general model of network formation in which players are heterogeneous with respect to their benefits and costs of forming links. We also approach network formation considering heterogeneous players and provide a tractable link between players' heterogeneity and the emerging network structures. Precisely, in our framework, one main result is that agents are more prone to communicate (or, equivalently, to link) when their ideal actions present some alignments. It is interesting to note that, in social sciences, homophily is a well-documented tendency of individuals to associate with similar others (see [START_REF] Mcpherson | Birds of a Feather: Homophily in Social Networks[END_REF] for an extensive review paper). In particular, this pervasive social fact implies that communication is more likely to take place between agents whose individual characteristics are related, in the sense that their goals are also similar.

Appendix

Second-Stage Equilibrium Characterization

First, we characterize the unique equilibrium action profile under complete information. The best response of each player i to a -i solves ∂u i (a i ,a -i ;θ)

∂a i = 0, i.e., a i (a -i ; θ) = (1 -α)(θ + b i ) + α n -1 j =i a j . ( 10
)
If a i is a best response to a -i , then it follows from Equations ( 9) and ( 10) that player i's utility takes the following simple form (minus a constant):

u i (a i (a -i ; θ), a -i ; θ) = (a i (a -i ; θ)) 2 - α n -1 j =i (a j ) 2 .
(11

)
The system of equations formed by Equation ( 10) leads to:

      a 1 . . . . . . a n       =        1 -α (n-1) • • • -α (n-1) -α (n-1) . . . . . . . . . . . . . . . . . . -α (n-1) -α (n-1) • • • -α (n-1) 1        I -1       (1 -α 1 )(θ + b 1 ) . . . . . . (1 -α n )(θ + b n )       .
Simple algebra yields:

I -1 = 1 (n -1) -(n -2)α -α 2       (n -1) -(n -2)α α • • • α α . . . . . . . . . . . . . . . . . . α α • • • α (n -1) -(n -2)α       .
Therefore, when every player knows the state of nature, the equilibrium actions are given by:

a i (θ) = θ + [(n -1) -(n -2)α]b i + α j =i b j n + α -1 ≡ θ + B i , for every i ∈ N . ( 12
)
Since players' best responses are linear, exactly the same algebra shows that, under incomplete information and whatever the information structure generated by the communication strategy profile, expected equilibrium actions are uniquely characterized by

E(a i ) = E(θ) + B i , for every i ∈ N , (13)
so that an equilibrium strategy for player i is always linear with respect to the signals {s j } j∈I i known by player i after the communication stage, for i ∈ N . Uniqueness of the Bayes-Nash equilibrium can be proved exactly as in the proof of Theorem 1 in Calvó-Armengol and Martí (2007b, pp. 25-26), that uses a sufficient condition for uniqueness in [START_REF] Radner | Team Decision Problems[END_REF], with the potential

V (a 1 , ..., a n ) = -(1 -α) n i=1 (a i -θ -b i ) 2 - α 2(n -1) n i=1 j =i (a i -a j ) 2 , (14)
and exactly the same matrix of cross derivatives that does not depend on the bias profile.

By explicitly solving some particular incomplete information situations as above, it is possible to guess the general form of the unique second-stage equilibrium actions. To check that the solution given by Equation ( 2) is indeed the equilibrium when the communication strategy profile is characterized by (R i ) i∈N , fix some player l ∈ N and suppose that the second stage equilibrium action of every player i = l is given by Equation (2). We show that player l's best response to the profile of second stage actions (a i ) i =l also takes the form of Equation (2).

After the communication stage, for all i ∈ N , recall that The expected payoff of player l after the communication stage takes the following form:

I i = {k : i ∈ R k } ∪
-(1 -α)E l (a l - j∈N s j -b l ) 2 - α n -1 j =l E l (a l -a j ) 2 , (15)
so his best-response is given by:

a l = (1 -α)   j∈I l s j + j∈I l E(s j ) + b l   + α n -1 j =l E l (a j ). ( 16
)
Using Equation (2) for i = l, player l's conditional expectation of player i's action is given by:

E l (a i ) = j∈I i α(n -1 -r j )E(s j ) n -1 -αr j + j∈I i ∩I l (1 -α)(n -1)s j n -1 -αr j + j∈I i ∩I l (1 -α)(n -1)E(s j ) n -1 -αr j + j∈I i E(s j ) + B i .
Summing over all agents different from l, we can write:

i =l E l (a i ) = = i =l j∈I i ∩I l α(n -1 -r j )E(s j ) n -1 -αr j + i =l j∈I i ∩I l α(n -1 -r j )E(s j ) n -1 -αr j + i =l j∈I i ∩I l (1 -α)(n -1)s j n -1 -αr j + i =l j∈I i ∩I l (1 -α)(n -1)E(s j ) n -1 -αr j + i =l j∈I i ∩I l E(s j ) + i =l j∈I i ∩I l E(s j ) + i =l B i . (17)
Every signal s j known by player l is known by r j players different from l and unknown by n -1r j players different from l; every signal s j unknown by player l is known by r j + 1 players different from l and unknown by n -2r j players different from l. This enables to deduce:

i =l E l (a i ) = j∈I l r j α(n -1 -r j )E(s j ) n -1 -αr j + j∈I l (r j + 1) α(n -1 -r j )E(s j ) n -1 -αr j + j∈I l r j (1 -α)(n -1)s j n -1 -αr j + j∈I l (r j + 1) (1 -α)(n -1)E(s j ) n -1 -αr j + j∈I l (n -1 -r j ) E(s j ) + j∈I l (n -2 -r j ) E(s j ) + i =l B i . (18)
In addition, we have:

i =l B i = α(n -1)b l + (n -1) i =l b i n + α -1 . (19)
Plugging ( 19) and ( 18) into ( 16) and simplifying, we get player l's optimal action, which takes exactly the same form as in Equation (2).

Proof of Proposition 1

The ex ante equilibrium payoff of player j ∈ N is given by:

U j = -(1 -α)V (a j - i∈N s i -b j ) -(1 -α)[E(a j - i∈N s i -b j )] 2 - α n -1 m =j V (a j -a m ) - α n -1 m =j [E(a j -a m )] 2 .
It follows from ( 13) that E(a j ) = i∈N E(s i ) + B j , so we get:

U j = -(1 -α)V (a j - i∈N s i ) - α n -1 m =j V (a j -a m ) -(1 -α)[B j -b j ] 2 - α n -1 m =j [B j -B m ] 2 . We consider two communication networks R = (R k ) k∈N and R ′ = (R ′ k ) k∈N such that R i = R ′ i \{t} and R k = R ′ k for all k ∈ N \{i}.
That is, R and R ′ are the same except that player i has one additional receiver (player t) in R ′ . Players i and t are fixed throughout the analysis. We denote

|R i | = r i and |R ′ i | = r ′ i = r i + 1.
The ex ante equilibrium payoff of every player j ∈ N with the communication network R (R ′ , resp.) is denoted U j (U ′ j , resp.). Given the communication network R (R ′ , resp.), the second stage equilibrium action of every player j ∈ N is denoted a j (a ′ j , resp.). For all j ∈ N , given a strategic communication network R (R ′ , resp.), let

I j = {k : j ∈ R k } ∪ {j} (I ′ j = {k ∈ N : j ∈ R ′ k } ∪ {j}, resp.
) denote the set of players whose signals are known by player j, and

I j = {k : j ∈ R k }\{j} (I ′ j = {k : j ∈ R ′ k }\{j}, resp.
) the set of players whose signals are unknown by player j.

For every player j ∈ N , we have:

U j -U ′ j = (1-α) V (a ′ j - i∈N s i ) -V (a j - i∈N s i ) + α n -1   m =j V (a ′ j -a ′ m ) - m =j V (a j -a m )   . (20)
The second-stage equilibrium action a j given by (2) enables to write:

V (a j - i∈N s i ) = V   l∈I j α(n -1 -r l )[E(s l ) -s l ] n -1 -αr l + l∈I j [E(s l ) -s l ] + B j   .
The independence of signals yields:

V (a j - i∈N s i ) = l∈I j V α(n -1 -r l )s l n -1 -αr l + l∈I j V (s l ) = l∈I j α(n -1 -r l ) n -1 -αr l 2 V (s l ) + l∈I j V (s l ) = l∈I j \{i} α(n -1 -r l ) n -1 -αr l 2 V (s l ) + l∈I j \{i} V (s l ) + V (s i ) 1[i ∈ I j ] α(n -1 -r i ) n -1 -αr i 2 + 1[i ∈ I j ] ,
where 1[i ∈ I j ] is the indicator function that equals 1 when player j knows the signal s i and 1[i ∈ I j ] is the indicator function that equals 1 when player j does not know the signal s i . A similar equation holds for V (a ′ j -i∈N s i ), when the communication network is R ′ . The two communication networks R and R ′ that we consider are such that I j \{i} = I ′ j \{i} and I j \{i} = I ′ j \{i}, so for all j ∈ N , we have:

V (a ′ j - i∈N s i ) -V (a j - i∈N s i ) = V (s i ) 1[i ∈ I ′ j ] α(n -1 -r ′ i ) n -1 -αr ′ i 2 + 1[i ∈ I ′ j ] -1[i ∈ I j ] α(n -1 -r i ) n -1 -αr i 2 -1[i ∈ I j ] . (21)
When the communication network is R, for all j ∈ N and m = j, we have, from (2):

V (a j -a m ) = l∈I j ∩Im (1 -α)(n -1) n -1 -αr l 2 V (s l ) + l∈I j ∩Im (1 -α)(n -1) n -1 -αr l 2 V (s l ) = l∈(I j ∩Im)\{i} (1 -α)(n -1) n -1 -αr l 2 V (s l ) + l∈(I j ∩Im)\{i} (1 -α)(n -1) n -1 -αr l 2 V (s l ) + (1 -α)(n -1) n -1 -αr i 2 V (s i ) 1[i ∈ I j ∩ I m ] + 1[i ∈ I j ∩ I m ] .
A similar equation holds for V (a ′ ja ′ m ), when the communication network is R ′ . The two communication networks R and R ′ are such that (I j ∩ I m )\{i} = (I ′ j ∩ I ′ m )\{i} and (I j ∩ I m )\{i} = (I ′ j ∩ I ′ m )\{i}, so for all j ∈ N and m = j we have:

V (a ′ j -a ′ m ) -V (a j -a m ) = ((1 -α)(n -1)) 2 V (s i ) 1[i ∈ I ′ j ∩ I ′ m ] + 1[i ∈ I ′ j ∩ I ′ m ] (n -1 -αr ′ i ) 2 - 1[i ∈ I j ∩ I m ] + 1[i ∈ I j ∩ I m ] (n -1 -αr i ) 2 . ( 22
)
Plugging ( 21) and ( 22) into (20), we get:

U j -U ′ j = (1 -α)V (s i ) 1[i ∈ I ′ j ] α(n -1 -r ′ i ) n -1 -αr ′ i 2 + 1[i ∈ I ′ j ] -1[i ∈ I j ] α(n -1 -r i ) n -1 -αr i 2 -1[i ∈ I j ] + α(1 -α)(n -1) m =j 1[i ∈ I ′ j ∩ I ′ m ] + 1[i ∈ I ′ j ∩ I ′ m ] (n -1 -αr ′ i ) 2 - 1[i ∈ I j ∩ I m ] + 1[i ∈ I j ∩ I m ] (n -1 -αr i ) 2   . (23)
To evaluate the sign of U j -U ′ j in order to know who is better off and who is worse off depending on the communication network, we distinguish four types of players:

• (i): Players who belong both to R i and to R ′ i . For every such player j ∈ R i = R ′ i \{t}, we have i ∈ I j and i ∈ I ′ j .

• (ii): Players different from player i who belong neither to R i nor to R ′ i . For every such player j ∈ N \(R ′ i ∪ {i}) = N \(R i ∪ {i, t}), we have i ∈ I j and i ∈ I ′ j .

• (iii): Player t who belongs to R ′ i but not to R i . For this player we have i ∈ I ′ t and i ∈ I t .

• (iv): Player i, for whom we have i ∈ I i and i ∈ I ′ i .

(i) For every player j ∈ R ′ i \{t}, the set of players different from j can be divided into three sets of players: {i} ∪ (R ′ i \{j, t}), N \(R ′ i ∪ {i}) and {t}. We have:

• for every player m ∈ {i} ∪ (R ′ i \{j, t}), i ∈ I m and i ∈ I ′ m ,

• for every player m ∈ N \(R ′ i ∪ {i}), i ∈ I m and i ∈ I ′ m ,

• for player t, i ∈ I t but i ∈ I ′ t .

Since i ∈ I j , i ∈ I ′ j , and 23) simplifies to:

|N \(R ′ i ∪ {i})| = (n -1 -r ′ i ), Equation (
U j -U ′ j = α(1 -α)V (s i ) (n -1 -r ′ i ) (n -1 -αr ′ i ) - (n -1 -r i ) (n -1 -αr i ) . ( 24
) Using r ′ i = r i + 1, we get U j -U ′ j = - α(1-α) 2 (n-1) (n-1-αr ′ i )(n-1-αr i ) V (s i ) < 0. Hence, for all j ∈ R ′ i \{t}, we have U j < U ′ j .
(ii) For every player j ∈ N \(R ′ i ∪ {i}), the set of players different from j can be divided into three sets of players: {i} ∪ (R ′ i \{t}), N \(R ′ i ∪ {i, j}) and {t}. We have:

• for every player m ∈ {i} ∪ (R ′ i \{t}), i ∈ I m and i ∈ I ′ m ,

• for every player m ∈ N \(R ′ i ∪ {i, j}), i ∈ I m and i ∈ I ′ m ,

• for player t, i ∈ I t but i ∈ I ′ t .

Since i ∈ I j , i ∈ I ′ j , and |{i} ∪ (R ′ i \{t})| = r ′ i , Equation ( 23) gives:

U j -U ′ j = α(1 -α) 2 (n -1)V (s i ) r ′ i + 1 (n -1 -αr ′ i ) 2 - r ′ i (n -1 -αr i ) 2 . ( 25
)
Since r i = r ′ i -1, we have

r ′ i +1 (n-1-αr ′ i ) 2 - r ′ i (n-1-αr i ) 2 > 0. Hence, for all j ∈ N \(R ′ i ∪ {i}), we have U j > U ′ j .
(iii) The set of players different from t can be divided into two sets of players: {i} ∪ (R ′ i \{t}) and N \(R ′ i ∪ {i}). We have:

• for every player m ∈ {i} ∪ (R ′ i \{t}), i ∈ I m and i ∈ I ′ m ,

• for every player m 23) gives:

∈ N \(R ′ i ∪ {i}), i ∈ I m and i ∈ I ′ m , Since i ∈ I t , i ∈ I ′ t , |{i} ∪ (R ′ i \{t})| = r ′ i , and |N \(R ′ i ∪ {i})| = n -1 -r ′ i , Equation (
U t -U ′ t = -(1 -α) 2 (n -1)V (s i ) 1 n -1 -αr ′ i + α r ′ i (n -1 -αr i ) 2 < 0. (26)
Hence, for player t who belongs to R ′ i but not to R i we have U t < U ′ t .

(iv) The set of players different from i can be divided into three sets of players: R ′ i \{t}, N \(R ′ i ∪ {i}) and {t}. We have:

• for every player m ∈ R ′ i \{t}, i ∈ I m and i ∈ I ′ m ,

• for every player m ∈ N \(R ′ i ∪ {i}), i ∈ I m and i ∈ I ′ m ,

• for player t, i ∈ I t but i ∈ I ′ t .

Since i ∈ I i , i ∈ I ′ i , and |N \(R ′ i ∪{i})| = (n-1-r ′ i ), Equation ( 23) gives exactly the same difference as in Equation ( 24). Hence, for player i such that R i = R ′ i \{t}, we have U i < U ′ i . This completes the proof of Proposition 1.

Proof of Proposition 2

As in the proof of Proposition 1, we consider two communication networks R = (R i , R -i ) and

R ′ = (R ′ i , R -i ) such that R i = R ′ i \{t}. Again, player t is such that t ∈ R ′ i but t ∈ R i .
Ex ante expected welfare is the sum of ex ante expected utilities. When the communication network is R ′ , it is given by:

W ′ = j∈R ′ i \{t} U ′ j + j∈N \(R ′ i ∪{i}) U ′ j + U ′ t + U ′ i .
When the communication network is R, it is given by:

W = j∈R i U j + j∈N \(R i ∪{i,t}) U j + U t + U i . Using R ′ i \{t} = R i , N \(R ′ i ∪ {i}) = N \(R i ∪ {i, t})
, and the fact that for all j ∈ R ′ i \{t}, U j -U ′ j = U i -U ′ i , the difference W -W ′ can be written as follows:

W -W ′ = j∈{i}∪(R ′ i \{t}) [U j -U ′ j ] + j∈N \(R ′ i ∪{i}) [U j -U ′ j ] + [U t -U ′ t ]. We have |{i} ∪ (R ′ i \{t})| = r ′ i and |N \(R ′ i ∪ {i})| = n -1 -r ′ i .
Using equations ( 24), ( 25) and ( 26), we get:

W -W ′ = α(1 -α)r ′ i (n -1 -r ′ i ) (n -1 -αr ′ i ) - (n -1 -r i ) (n -1 -αr i ) V [s i ] + α(1 -α) 2 (n -1)(n -1 -r ′ i ) r ′ i + 1 (n -1 -αr ′ i ) 2 - r ′ i (n -1 -αr i ) 2 V [s i ] -(1 -α) 2 (n -1) 1 n -1 -αr ′ i + α r ′ i (n -1 -αr i ) 2 V [s i ].
After some simplifications and using the fact that r ′ i = r i + 1, we get:

W -W ′ = - (1 -α) 3 (n -1) 2 V [s i ] (n -1 -αr ′ i ) 2 (n -1 -αr i ) 2 <0 [α 2 (1 -r ′ i -r ′ i 2 ) + 2α(n -1) + (n -1) 2 ]
x .

Solving x = 0 in α gives the following discriminant: 4(n -1) 2 (r ′ i + r ′ i 2 ) ≥ 0. We have x ≥ 0 if and

only if α ∈ [α 1 , α 2 ], with α 1 = (n-1)(1-r ′ i +r ′ i 2 ) r ′ i +r ′ i 2 -1 and α 2 = (n-1)(1+ r ′ i +r ′ i 2 ) r ′ i +r ′ i 2 -1
. From r ′ i ≥ 1, we deduce that α 1 < 0. From r ′ i ≤ n -1 and the fact that α 2 is decreasing in r ′ i , we deduce that α 2 > 1. Since α ∈ (0, 1), x is always strictly positive. Hence, W < W ′ .

Proof of Propositions 3 and 4

Consider an equilibrium of the private communication game in which each player i reveals his type to players in R i ⊆ N \{i}. Without loss of generality, assume that player i sends to every player j ∈ R i the message m j i = m when his type is s i and the message m j i = m when his type is s i , and sends the same message whatever his type to players outside R i . Given (R i ) i∈N , second stage equilibrium actions are given by (2).

Without loss of generality, we look for the conditions under which player 1 does not deviate from his equilibrium communication strategy above. First, assume that player 1's true type is s 1 = s 1 . In equilibrium, using Equation (2), the second-stage action of every player i ∈ R 1 ∪ {1} is given by

a i = j∈I i \{1} α(n -r j -1)E(s j ) + (1 -α)(n -1)s j n -1 -αr j + j∈I i E(s j ) + B i + α(n -r 1 -1)E(s 1 ) + (1 -α)(n -1)s 1 n -1 -αr 1 , (27)
and the second-stage action of every player i / ∈ R 1 ∪ {1} is given by

a i = j∈I i α(n -r j -1)E(s j ) + (1 -α)(n -1)s j n -1 -αr j + j∈I i \{1} E(s j ) + B i + E(s 1 ). ( 28
)
The relevant deviations for player 1 in the communication stage consist in lying to a subset of players M ⊆ R 1 , i.e., sending message m instead of m to players in M (and not deviating towards the other players). Let m = |M |, and denote by (a ′ i ) i∈N the profile of players' actions after this deviation. Every player i ∈ M chooses action a ′ i = a i , which is given by ( 27) by replacing s 1 by s 1 . The action a ′ i of every player i ∈ N \(M ∪ {1}) is the same as in the original equilibrium. Player 1's optimal action in the second stage is obtained from the best response of Equation ( 16) to (a ′ i ) i =1 , and takes the following form:

a ′ 1 = (1 -α)   j∈I 1 \{1} s j + s 1 + j∈I 1 E(s j ) + b 1   + α n -1 i =1 E 1 (a ′ i ). (29)
Using the same reasoning as the one used to get expression (18), we get:

i =1 E 1 (a ′ i ) = j∈I 1 r j α(n -r j -1)E(s j ) n -1 -αr j + j∈I 1 (r j + 1) α(n -r j -1)E(s j ) n -1 -αr j + j∈I 1 \{1} r j (1 -α)(n -1)s j n -1 -αr j + m(1 -α)(n -1)s 1 n -1 -αr 1 + (r 1 -m)(1 -α)(n -1)s 1 n -1 -αr 1 + j∈I 1 (r j + 1) (1 -α)(n -1)E(s j ) n -1 -αr j + j∈I 1 (n -1 -r j )E(s j ) + j∈I 1 (n -2 -r j ) E(s j ) + i =1 B i . (30)
Plugging ( 30) into (29), using ( 19) and simplifying, we get:

a ′ 1 = j∈I 1 \{1} α(n -r j -1)E(s j ) + (1 -α)(n -1)s j n -1 -αr j + j∈I 1 E(s j ) + B 1 + αm(1 -α)s 1 + (n -1 -αm)(1 -α)s 1 + α(n -r 1 -1)E(s 1 ) n -1 -αr 1 . (31)
We denote by V 1 the expected payoff of player 1 conditional to signal s 1 under the original equilibrium, and V ′ 1 his expected payoff conditional to signal s 1 when he deviates by lying to players in M (and thus plays action a ′ 1 in the second-stage game. Player 1 does not deviate by lying to players in M if V ′ 1 -V 1 ≤ 0. We have:

V ′ 1 -V 1 = (1 -α)E (a 1 - i∈N s i -b 1 ) 2 -(a ′ 1 - i∈N s i -b 1 ) 2 | s 1 + α n -1 i∈M E (a 1 -a i ) 2 -(a ′ 1 -a i ) 2 | s 1 + i∈R 1 \M E (a 1 -a i ) 2 -(a ′ 1 -a i ) 2 | s 1 + i∈N \(R 1 ∪{1}) E (a 1 -a i ) 2 -(a ′ 1 -a i ) 2 | s 1   .
For the sake of simplicity, we examine separately the elements of the difference V ′ 1 -V 1 and use the following notation for i = 1:

z i = j∈(I 1 ∩I i )\{1} (1 -α)(n -1)(s j -E(s j )) n -1 -αr j + j∈(I 1 ∩I i )\{1} (1 -α)(n -1)(E(s j ) -s j ) n -1 -αr j + B 1 -B i .
Using ( 27), ( 28) and ( 31) and the fact that

E[z i | s 1 ] = B 1 -B i , we get: i∈M E (a 1 -a i ) 2 -(a ′ 1 -a i ) 2 | s 1 = i∈M E z 2 i -z i + (1 -α)(n -1 -αm)(s 1 -s 1 ) n -1 -αr 1 2 s 1 = -2 (1 -α)(n -1 -αm)(s 1 -s 1 ) n -1 -αr 1 i∈M (B 1 -B i ) -m (1 -α)(n -1 -αm)(s 1 -s 1 ) n -1 -αr 1 2 .( 32
) i∈R 1 \M E (a 1 -a i ) 2 -(a ′ 1 -a i ) 2 | s 1 = i∈R 1 \M E z 2 i -z i - (1 -α)αm(s 1 -s 1 ) n -1 -αr 1 2 s 1 = 2 (1 -α)αm(s 1 -s 1 ) n -1 -αr 1 i∈R 1 \M (B 1 -B i ) -(r 1 -m) (1 -α)αm(s 1 -s 1 ) n -1 -αr 1 2 . ( 33
) i∈N \(R 1 ∪{1}) E (a 1 -a i ) 2 -(a ′ 1 -a ′ i ) 2 | s 1 = i∈N \(R 1 ∪{1}) E z i + (1 -α)(n -1)(s 1 -E(s 1 )) n -1 -αr 1 2 -z i + (1 -α)αms 1 + (1 -α)(n -1 -αm)s 1 -(1 -α)(n -1)E(s 1 ) n -1 -αr 1 2 s 1 = 2 (1 -α)αm(s 1 -s 1 ) n -1 -αr 1 i∈N \(R 1 ∪{1}) (B 1 -B i ) + (n -1 -r 1 ) (1 -α)(n -1)(s 1 -E(s 1 )) n -1 -αr 1 2 -(n -1 -r 1 ) (1 -α)αms 1 + (1 -α)(n -1 -αm)s 1 -(1 -α)(n -1)E(s 1 ) n -1 -αr 1 2 . ( 34
)
In addition, using

ā1 -a ′ 1 = (1 -α)αm(s 1 -s 1 ) n -1 -αr 1 , and ā2 1 -a ′ 1 2 = α(n -r 1 -1)E(s 1 ) + (1 -α)(n -1)s 1 n -1 -αr l 2 - αm(1 -α)s 1 + (n -1 -αm)(1 -α)s 1 + α(n -r 1 -1)E(s 1 ) n -1 -αr 1 2 + 2   j∈I 1 \{1}
α(nr j -1)E(s j ) + (1α)(n -1)s j n -1αr j +

j∈I 1

E(s j ) + B 1   (1 -α)αm(s 1 -s 1 ) n -1 -r 1 ,
we get:

E (ā 1 - i∈N s i -b 1 ) 2 -(a ′ 1 - i∈N s i -b 1 ) 2 s 1 = E ā2 1 -a ′ 1 2 s 1 -2E   (ā 1 -a ′ 1 )( i∈N \{1} s i + s 1 + b 1 ) s 1   = α(n -r 1 -1)E(s 1 ) + (1 -α)(n -1)s 1 n -1 -αr 1 2 + 2 (B 1 -b 1 -s 1 ) (1 -α)αm(s 1 -s 1 ) n -1 -αr 1 - αm(1 -α)s 1 + (n -1 -αm)(1 -α)s 1 + α(n -r 1 -1)E(s 1 ) n -1 -αr 1 2 . ( 35
)
Next, we plug (32), ( 33), ( 34) and ( 35) into V ′ 1 -V 1 and simplify. To simplify the part of the difference V ′ 1 -V 1 that deals with biases, one should note that:

B 1 -B i = (1 -α)(n -1)(b 1 -b i ) n + α -1 and B 1 -b 1 = -α(n -1)b 1 + j =1 b j n + α -1 .
Finally, we get:

V ′ 1 -V 1 = 2α(1 -α) 2 (n -1)(s 1 -s 1 ) (n + α -1)(n -1 -αr 1 ) i∈M b i -mb 1 - α(1 -α) 2 m(n -1 -αm)(s 1 -s 1 ) 2 (n -1 -αr 1 ) 2 .
Hence, in the private communication game, player 1 of type s 1 = s 1 does not deviate by lying to players in

M ⊆ R 1 if V ′ 1 -V 1 ≤ 0, i.e., -b 1 -i∈M b i m ≤ (n -1 + α)(n -1 -αm) 2(n -1)(n -1 -αr 1 ) (s 1 -s 1 ). ( 36
)
Applying the same reasoning, player 1 of type s 1 = s 1 has no profitable deviation if, for all M ⊆ R 1 , the following condition holds:

b 1 -i∈M b i m ≤ (n -1 + α)(n -1 -αm)
2(n -1)(n -1αr 1 ) (s 1s 1 ). ( 37)

Condition ( 3) is obtained from ( 36) and (37).

In a group R-communication game, every player i is required to send the same message to all players in Ri . Consider an equilibrium in which player 1 sends to all the players in R1 the message m 1 = m when his type is s 1 and the message m 1 = m when his type is s 1 . The only possible deviation for player 1 in the communication stage consists in lying to all the players in R1 , i.e., sending the message m instead of m to all the players in R1 . Therefore, condition (3) for R ′ 1 = R 1 = R1 is the condition under which player 1 does not deviate from his equilibrium communication strategy above in the group R-communication game.

Proof of Proposition 5

When types are fully certifiable, the simplest way to support a fully revealing equilibrium is to consider the communication strategy profile in which every player completely certifies his type to all the other players whatever his type. That is, σ i (s i ) = c i (s i ) for all i ∈ N and s i ∈ S i in the public communication game, and σ j i (s i ) = c i (s i ) for all i ∈ N , j = i and s i ∈ S i in the private communication game, where c i (s i ) ∈ M i is such that M -1 i (c i (s i )) = {s i }. When such strategies are used in the first stage, then each player knows the state θ in the decision stage, so the second stage equilibrium actions are given by Equation ( 12).

We prove the existence of a fully revealing equilibrium in the public and private communication games separately.

• Public Communication. We start from a fully revealing communication strategy profile σ i (s i ) = c i (s i ) for all i ∈ N and s i ∈ S i , and consider a deviation by player i to a message m i = c i (s i ) when his type is s i . To support this equilibrium, we consider the degenerate common belief µ i j (m i ) = µ i (m i ) = max{t i ∈ S i : m i ∈ M i (t i )} for every j = i when b i ≤ b, and µ i j (m i ) = µ i (m i ) = min{t i ∈ S i : m i ∈ M i (t i )} for every j = i when b i ≥ b. By Equation (11), a sufficient condition for player i's deviation not to be profitable is that for all s -i ∈ S -i , a i (a -i (θ(µ i (m i ), s -i )); θ(s))

2 -α n -1

j =i a j (θ(µ i (m i ), s -i )) 2 ≤ [a i (a -i (θ(s)); θ(s))] 2 - α n -1 j =i [a j (θ(s))] 2 . ( 38
)
Given player i's best response (10), this is equivalent to

(1 -α)(θ + b i ) + α n -1 j =i a j (θ(µ i (m i ), s -i )) 2 - α n -1 j =i a j (θ(µ i (m i ), s -i )) 2 ≤ (1 -α)(θ + b i ) + α n -1 j =i a j (θ(s)) 2 - α n -1 j =i [a j (θ(s))] 2 .
(39) By replacing the equilibrium action of every player j = i given by Equation ( 12) in the last inequality we get (after some simplifications): θ(µ i (m i ), s -i )θ(s) θ(s)θ(µ i (m i ), s -i ) + 2 (n -1)b ij =i b j n + α -1 ≤ 0.

(40) Since θ(s) is increasing in s i , a sufficient condition for this inequality to be satisfied is µ i (m i ) = max{t i ∈ S i : m i ∈ M i (t i )} when b i ≤ b, and µ i (m i ) = min{t i ∈ S i : m i ∈ M i (t i )} when b i ≥ b.

• Private Communication. We start from a fully revealing communication strategy profile σ j i (s i ) = c i (s i ) for all i ∈ N , j = i and s i ∈ S i , and consider a deviation by player i to a vector of messages m i = (c i (s i ), . . . , c i (s i )) when his type is s i . To support this equilibrium, we consider the degenerate private beliefs µ i j (m j i ) = max{t i ∈ S i : m j i ∈ M i (t i )} when b i ≤ b j , and µ i j (m j i ) = min{t i ∈ S i : m j i ∈ M i (t i )} when b i ≥ b j . The analogue of Equation ( 39) for the private communication game is: (41) i.e., by ( 12),

(1 -α)(θ + b i ) + α n -1 j =i a j (θ(µ i j (m j i ), s -i )) 2 - α n -1 j =i a j (θ(µ i j (m j i ), s -i )) 2 ≤ (1 -α)(θ + b i ) + α n -1 j =i a j (θ(s)) 2 - α n -1 j =i [a j (θ(s))] 2 ,
(1 -α)θ + α n -1 j =i θ(µ i j (m j i ), s -i ) + B i 2 -θ + B i 2 + α n -1 j =i
θ 2 -(θ(µ i j (m j i ), s -i ))

2 + 2B j (θθ(µ i j (m j i ), s -i )) ≤ 0.

Letting

T ≡ α n -1 2 j =i θ(µ i j (m j i ), s -i ) 2 + 2α(1 -α) n -1 θ j =i θ(µ i j (m j i ), s -i ) - α n -1 j =i [θ(µ i j (m j i ), s -i )] 2 -α(1 -α)θ 2 ,
the condition further simplifies to

T + 2αθ j =i b j + αb i n + α -1 -B i + 2α n -1 j =i θ(µ i j (m j i ), s -i )(B i -B j ) ≤ 0 ⇔ T + 2α(1 -α) n + α -1 j =i b i -b j θ(µ i j (m j i ), s -i ) -θ ≤ 0.
By the construction of players' beliefs, and since θ(s) is increasing in s i , we have b ib j θ(µ i j (m j i ), s -i )θ ≤ 0, for all j = i.

  {i} is the set of signals which are known by player i after the communication stage, and I i = {k : i ∈ R k }\{i} is the set of signals which are unknown by player i after the communication stage.

Figure 1 :

 1 Figure 1: Most informative strategic communication networks in two examples.

  {i} is the set of players whose signals are known by player i, I i = {k : i ∈ R k }\{i} the set of players whose signals are unknown by player i, and let E i (•) = E(• | {s l : l ∈ I i }) be player i's expectation operator conditional to the set of signals that he knows.

[START_REF] Bernheim | A Theory of Conformity[END_REF] and[START_REF] Akerlof | Social Distance and Social Decisions[END_REF] consider such examples.

[START_REF] Desgranges | Conformism, Public News and Market Efficiency[END_REF] develop this example.

The framework of firm internal organization is adopted by[START_REF] Alonso | When Does Coordination Require Centralization?[END_REF].

See, among others, Battaglini (2002),[START_REF] Krishna | A Model of Expertise[END_REF], and[START_REF] Ambrus | Multi-sender cheap talk with restricted state space[END_REF].

Of course, several game theoretical, but more abstract, research papers deal with general cheap talk games, but the focus is mainly on characterizing conditions under which a (mediated) communication equilibrium can be decentralized with multilateral and multistage communication (see, e.g., the references in[START_REF] Forges | Correlated Equilibrium and Communication in Games[END_REF].

See Fudenberg and Tirole (1991). Notice that, in the cheap talk game, this definition yields the same equilibrium outcomes as the Nash equilibrium definition, but we already require sequential rationality and belief consistency here for consistency with the solution concept used in the case of certifiable information in Section 4.2.

As in[START_REF] Crawford | Strategic Information Transmission[END_REF], it is not possible to compare players' expected payoffs at the interim stage.

We do not consider the other group communication games here since we obtain full revelation of information (a complete communication network) in both the private and public settings.

Like us, they consider finite sets of types and assume that players' types are independent.

This term does not modify the second stage equilibrium strategies but may affect players' incentives to communicate.

See Jacskon (2007) for an extensive survey of such models.

See, among others,[START_REF] Bala | A Non-cooperative Model of Network Formation[END_REF],[START_REF] Dutta | The Stability and Efficiency of Directed Communication Networks[END_REF] and[START_REF] Watts | A Dynamic Model of Network Formation[END_REF].

By the property (x1+ • • • + xm) 2 ≤ m((x1) 2 + • • • + (xm) 2 ) ⇔ (m -1)((x1) 2 + • • • + (xm) 2 )i =j xixj ≥ 0 ⇔ i =j (xixj) 2 ≥0 for all (x1, . . . , xm) ∈ R m and m ∈ N+.

Finally, to show that the condition for no deviation is satisfied, it suffices to remark that T is always negative. Indeed, solving T = 0 in θ gives the following discriminant:

which can be checked to be always negative. 15