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Strategic Communication Networks

JEANNE HAGENBACH University of Mannheim

and

FRÉDÉRIC KOESSLER Paris School of Economics and CNRS

First version received March 2008; final version accepted October 2009 (Eds.)

We consider situations in which every agent would like to take an action that is coordinated with those of others, as well as close to a common state of nature, with the ideal proximity to that state varying across agents. Before this coordination game is played, agents decide to whom they reveal their private information about the state. The information transmission occurring in the cheap-talk communication stage is characterized by a strategic communication network whose links represent truthful information transmission. In equilibrium, whether communication takes place between two agents depends not only on the conflict of interest between these agents, but also on the number and preferences of the other agents with whom they communicate. In particular, communication to a large group of recipients may be feasible even though communication to a small subset of that group may not be. We show that agents who are more central in terms of preference tend to communicate more and to have a greater impact on decisions.

1. INTRODUCTION

Many economic situations involve agents who share an interest in coordinating their actions as well as in adapting them to an unknown state of the world. The analysis presented here considers this type of context but departs from the typical assumption that agents agree on the state-contingent profile of decisions. Because their tastes may differ, we let the interacting agents vary in their ideal proximity to the underlying fundamentals. For example, the different divisions of an organization should coordinate their actions, as well as adjust them to the environment of the firm. But, for a number of reasons, ranging from local adaptation costs to career concerns, it is likely that idiosyncratic considerations will influence each division’s actions.1 Similarly, when advocating policies, members of a political party will wish to best suit the situation, but also to be in line with the announcements made by other members to ensure the cohesion of the party. At the same time, activists may have heterogenous preferences regarding the right policy to implement.2 We here consider this type of coordination game of incomplete information in which every player incurs losses from any mismatch between his

1. A multi-divisional organization in which decisions must be adapted to local conditions and information but also coordinated between divisions is considered in Alonso et al. (2008) and Rantakari (2008).

2. See, for example, Dewan and Myatt (2008).

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action and both others’ actions and his own “ideal action”. Every ideal action depends on the state and on a systematic positive or negative bias, as in the cheap-talk or delegation models of Crawford and Sobel (1982) and Dessein (2002). These biases vary across agents, and the profile of biases in the population is a measure of the conflict of interest that they face.

The aim of this paper is to analyse how agents strategically transmit to each other the signals they privately hold about the fundamentals in these types of situation. Indeed, before taking their payoff-relevant actions, we offer players the opportunity to send costless, non- verifiable, and private messages about their information. Within this stylized framework, the communication stage consists of a cheap-talk game in which every player is, at the same time, a sender and a receiver, and we address the question of who truthfully speaks with whom. Our precise focus is on how agents’ heterogeneity in ideal actions affects decentralized and strategic communication between them. We propose to characterize the transmission of private information by what we call a communication network, described by a set of receivers for every player. A player is said to be a receiver of another player if the latter truthfully reveals his private information to the former. A complete characterization of the information transmission occurring in equilibrium is provided, which roughly boils down to the intuitive statement that agents are more prone to communicate when their ideal actions are more similar, and as the need for coordination becomes larger.

Informational incentive constraints require that no player has an interest in lying about his type to his (endogenous) set of receivers. As in standard cheap-talk games (e.g. Crawford and Sobel, 1982), this condition can be formulated as a condition on the proximity between the sender’s and the receivers’ biases. In existing models extending communication to multiple but strategically independent decision makers (see, e.g. Farrell and Gibbons, 1989; Goltsman and Pavlov, 2009; Galeotti et al., 2009), one only has to check that the sender has no incentive to lie to any single receiver. In our model, the informational incentive constraints are more sophisticated than in these games since all of the agents want to coordinate their actions. Due to the strategic interaction between receivers, how each receiver reacts to a sender’s signal depends not only on this signal but also on (his expectation over) the total number of receivers of this signal. At the same time, since the sender also wants to coordinate his action with the receivers, any deviation by the sender in the communication stage induces coordination costs that depend on both the total number of his receivers and the number of receivers he lies to. Combined with the assumption that loss functions are quadratic, our informational incentive constraints require that the sender’s bias be close enough to the average bias of every subset of receivers. Exactly how close biases should be is determined by some threshold that depends on both the total number of receivers and the respective subsets of receivers the sender could lie to.

This feature reveals a key insight of our paper: communication between two agents depends not only on the conflict of interest between them, but also on the preferences and the size of all the agents with whom they communicate. In particular, one main result is that communication to a large group of recipients may occur in equilibrium even though communication only to a strict subset of that group may not. To understand the intuition, consider a simplified three- player situation in which there is a unique informed agent (the sender): the sender and one uninformed agent both want to choose an action exactly adapted to the true state of the nature, and another uninformed agent is positively biased, i.e. wants to choose an action higher than the true state. Assume that every player also wants to coordinate his action with that of the two others. When the sender communicates only to the biased agent, he has a strong incentive to under-report his type in order to decrease this agent’s expectation about the state so that his action gets closer to the sender’s ideal action. On the contrary, when the sender communicates to both the biased and unbiased agents, he may have no incentive to jointly lie to both of

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them because their average bias is small. He may also have no incentive to misrepresent his information only to the biased agent. Indeed, both agents are now more responsive to the sender’s message than when he communicates only to one agent, so this deviation would increase the dispersion of players’ actions and thus induce large coordination losses. It is worth noting that the disciplinary effect that the coordination of multiple audiences has on communication is different from the disciplinary effect of public communication identified by Farrell and Gibbons (1989). They show that communication to two independent decision makers may occur in equilibrium when communication is public, whereas information is revealed to neither decision maker when communication is private. Our disciplinary effect of coordination appears even though communication is not public and relies on the fact that the receivers we consider are not independent decision makers.

We provide sharp predictions regarding equilibrium communication networks for several configurations of preferences. First, when players’ biases are uniformly distributed, we show that a player’s tendency to communicate increases with the proximity of his bias to the average bias in the population. Communication is therefore typically not symmetric: centrists tend to influence the decisions of the other players more because they transmit their information truthfully to more distant players than do extremists, with this effect becoming stronger as the need for coordination increases. When the coordination motive is very strong, middle- biased players may communicate to all of the other players even with a wide dispersion of preferences, while other players may never truthfully report their private information. Second, when players are arranged in groups with the same preferences, we again show that information transmission across groups is typically asymmetric: members from the larger group tend to communicate more easily to members of other groups than do members of a smaller group. That is, large groups of agents tend to influence the decisions of small groups by credibly reporting information, while there is less truthful communication from small to large groups.

The paper is organized as follows. The model is presented in Section 2, which also shows that strategic communication networks cannot be completely Pareto-ranked. Equilibrium communication networks are analysed and illustrated in Section 3, and Section 4 concludes. Most of the proofs are relegated to the Appendix.

2. MODEL

2.1. A class of coordination games with incomplete information

Let N = {1, ..., n} be a finite set of agents, with n ≥ 2. Each agent chooses an action ai ∈ Ai = R. The action profile is denoted a = (a1, . . . , an). 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 si ∈ Si = {si , si } about θ , where si < si . We assume that agents’ types are independent and denote by qi ∈ �(Si ) the prior probability distribution over agent i’s set of types, for every i ∈ N . When the type profile is s = (s1, ..., sn), the underlying state of nature is θ(s) ∈ R.

Agent i’s payoff function is given by

ui (a1, . . . , an; θ(s)) = −(1 − α)(ai − θ(s) − bi )2 − α

n − 1 ∑ j �=i

(ai − aj )2. (1)

The first component of agent i’s utility function is a quadratic loss in the distance between his action ai and his ideal action θ (s) + bi , where bi ∈ R. We allow the bias parameter bi to vary across individuals to reflect agents’ conflicts of interest with respect to their ideal actions. The second component is a miscoordination quadratic loss which increases in the average

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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. Without loss of generality, players are indexed in increasing order of their biases: b1 ≤ · · · ≤ bn.

We assume that the state of nature is the aggregated term θ (s) = ∑i∈N si .3 The sum of players’ private signals is a good approximation of the payoff-relevant state in many situations. In an organizational setting for instance, a signal si for division i may represent division i’s time, budget, or expected benefit from a joint project (which is private information), and the state that matters for the whole organization may be the total time, budget, or expected benefit of the project. More broadly, considering a state of nature which is additive in types is a simplifying standard assumption in common-value environments, especially in auction theory (see, amongst many others, Bulow and Klemperer, 2002; Mares and Harstad, 2003; Levin, 2004), in some models of lobbying with multiple experts (e.g. Wolinsky, 2002) or in organization theory (e.g. Jehiel, 1999). From a theoretical point of view, assuming an additive state and independent types implies that the impact of player i’s signal on the fundamentals, si − si , which can be interpreted as the value of player i’s private information, is independent of others’ signals. It follows that in our analysis we abstract from any effects that the correlation and the degree of complementarity between players’ signals may have on informational incentive constraints. We focus instead on the effect of players’ coordination motives and preference heterogeneity. The robustness of our results to this independence property is discussed and related to the literature at the end of Section 3.1.

2.2. Communication game

Before the coordination game described above 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

i ∈ Mi

to every other player j �= i, with Mi denoting the (non-empty) set of messages available to player i. Let mi = (mji )j �=i ∈ (Mi )n−1 be the vector of messages sent by player i, and mi = (mi

j )j �=i ∈

∏ j �=i Mj ≡ M−i the vector of messages received by player i.

The information transmission occurring during the cheap-talk extension of the game will be characterized by a communication network, whose directed links represent revelation of private information from one player to another. In order to focus on the presence or absence of such information transmission links between the agents, we restrict the analysis to pure communication strategies and abstract from the partial transmission of information generated by random strategies.4 As we consider only two possible types for each player, it follows that any message from player i to player j will either be fully revealing or non-informative. We consider that a communication link is formed from i to j when i’s message to j is fully revealing. Without loss of generality, we assume that message spaces are binary: Mi = {m, m}.

Player i’s communication strategy is a profile σ i = (σ ji )j �=i with σ

i : Si → Mi .

3. Note that the state can be any weighted sum of players’ types (since we do not make any assumptions about the two possible values of each signal), and players’ signals are not assumed to be i.i.d. (they are only assumed to be independent).

4. We do not exclude the existence of non-trivial mixed equilibria, as in the discrete quadratic version of Crawford and Sobel (1982), but the full characterization of such equilibria is quite difficult since we have to consider the possibility that any combination of players randomize over their messages, for any possible combination of receivers.

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Let σ j

i (m

i | si ) be the probability (0 or 1) that player i send the message mji to player j

according to his strategy σ i when his type is si . Since each player i’s utility function is strictly concave with respect to ai , his best response

is necessarily unique, so we can consider without loss of generality pure second-stage strategies. Player i’s second-stage strategy is a mapping

τ i : Si × (Mi)n−1 × M−i → Ai , where τ i (si , mi , m

i ) is the action chosen by player i when his type is si ∈ Si , the vector of private messages mi = (mji )j �=i ∈ (Mi )n−1 was sent, and the vector of private messages mi = (mi

j )j �=i ∈ M−i was received. Let τ (s, (mi )i∈N ) = (τ i (si , mi , mi ))i∈N be the corresponding

action profile. As is usual in cheap-talk games, the set of Nash equilibrium outcomes in our model

coincides with the set of sequential equilibrium outcomes because messages off the equilibrium path can simply be treated as synonyms of equilibrium messages. Hence we do not have to specify a complete belief system: an equilibrium of the communication game is simply a strategy profile (σ , τ ) = ((σ i )i∈N , (τ i )i∈N ) satisfying the following properties:

(i) For all i ∈ N , and si ∈ Si ,

(σ j

i (si ))j �=i ∈ arg max

mi ∈Mn−1i

∑ s−i ∈S−i

q−i (s−i ) ui (τ (s, (σ −i (s−i ), mi )); θ(s)) ,

where q−i(s−i ) = ∏

j �=i qj (sj ). (ii) For all i ∈ N , mi ∈ (Mi)n−1 and mi ∈ supp[σ i−i ], τ i (si , mi , mi ) belongs to

arg max ai ∈Ai

∑ s−i ∈S−i

μi (s−i | mi ) ui ( (τ j (sj , (σ

−i j

(sj ), m i j ), (σ

−i (s−i ), m j

i )))j �=i, ai ; θ (s)

) ,

where μi (s−i | mi ) = ∏

j �=i σ i

j (mi

j |sj )qj (sj )∑

tj ∈Sj σ i j (mi

j |tj )qj (tj )

A communication network that characterizes a communication strategy profile (σ i )i∈N is denoted by (Ri )i∈N , where, for every player i, the set of receivers

Ri ≡ {j ∈ N \{i} : σ ji (si ) �= σ j

i (si )},

is the set of individuals to whom player i truthfully reveals his type. Let |Ri | be the number of individuals who learn player i’s type in the communication stage. Using the terminology of network theory, Ri corresponds to player i’s (out)neighbourhood and |Ri | to player i’s (out)degree.

2.3. Second-stage equilibrium characterization

The quadratic formulation of players’ utility functions, together with the independence of players’ types, enables us to obtain a unique and tractable second-stage equilibrium characterization whatever the information structure generated by the communication stage. Indeed, as in Calvó-Armengol and Martı́ (2009), who consider the same utility functions but without heterogeneity in biases, it can be shown that our payoffs admit a potential that represents common interests for all players. The corresponding common-interest game satisfies the sufficient conditions in Marschak and Radner’s (1972) team theory for the equilibrium to be unique and linear. More precisely, we show in Appendix A.1 that, given a profile of

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types (si )i∈N and a communication strategy profile characterized by (Ri )i∈N , the second-stage equilibrium strategy of each player i ∈ N is uniquely given by:

ai = ∑ j ∈Ii

α(n − 1 − |Rj |)E(sj ) + (1 − α)(n − 1)sj n − 1 − α|Rj |

+ ∑ j ∈I i

E(sj ) + Bi , (2)

where Ii = {k : i ∈ Rk} ∪ {i} is the set of signals which are known by player i after the communication stage, I i = {k : i �∈ Rk}\{i} is the set of signals which are not known by player i after the communication stage, and

Bi = [(n − 1) − (n − 2)α]bi + α

∑ j �=i bj

n + α − 1 . (3)

Player i’s second-stage equilibrium action has three components. The first component is a weighted sum of j ’s actual type, sj , and the expected value of j ’s type, E(sj ), for each player j whose type is known by player i (including himself). The weight put on the actual type of player j increases with the number of players who know j ’s type, |Rj |. This is because a player who wants to be coordinated with the others has a greater incentive to act according to a signal when many other players act according to the same signal. In other words, the larger the set of receivers, the more the sender and those receivers choose an action which is responsive to the sender’s private information. This is one of the key effects that will drive our results regarding efficient and equilibrium communication networks. The second component of player i’s equilibrium action corresponds to the sum of the expected values of j ’s type for each player j whose type is not known by player i. The last component adjusts the action of player i with respect to the bias profile. This increases in all players’ biases, with more weight being put on player i’s own bias, bi , as the coordination motive decreases.

2.4. Efficient communication networks

Before characterizing the networks that arise as equilibria of the communication game, we consider the efficiency of communication networks. The following proposition compares players’ ex ante expected payoffs as the communication network expands, assuming that equilibrium actions are played in the second-stage game.5 While an increase in the size of player i’s set of receivers is always strictly beneficial for player i and for these receivers, this increase always makes the players who do not learn player i’s type strictly worse off.

Proposition 1. Consider two communication networks R = (Ri , R−i ) and R′ = (R′i , R−i ) such that |Ri | < |R′i |. (i) Every player j ∈ R′

i ∪ {i} is strictly better off, ex ante, with the communication network R′

than with the communication network R; (ii) 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 A.2. ‖

The intuition of this result is the following. Consider one sender’s private signal. As we have observed in the previous subsection (see equation (2)), when the number of players

5. As in Crawford and Sobel (1982), it is not possible to compare players’ expected payoffs at the interim stage.

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informed about this signal increases, they become more responsive to it. While this increase benefits informed players whose actions are now better coordinated and adapted to the state, uninformed players suffer larger miscoordination losses, and are therefore worse off.

This result implies that, in general, communication networks cannot be ranked in the sense of Pareto. In particular, defining a communication network R′ = (R′

i )i∈N as more

informative than R = (Ri )i∈N if Ri ⊆ R′i for every i ∈ N (with at least one strict inequality), a more informative communication network does not Pareto-dominate, in general, a less informative communication network. However, using Proposition 1 iteratively, we obtain that the complete communication network (Ri = N \{i} for all i ∈ N ) Pareto-dominates every other communication network.

3. EQUILIBRIUM COMMUNICATION NETWORKS

In this section we provide a full characterization and the general qualitative features of equilibrium communication networks, and derive a number of comparative-static results. We examine how a large number of receivers with strong coordination motives can discipline communication, which is one main novelty of the paper. In particular, we show that there may exist an equilibrium in which a sender reveals his information to a large group of recipients, whereas there is no equilibrium in which he does so to a strict subset of that group only. We also show that players who are more central in terms of preferences communicate more and have a greater impact on the decisions taken. These features are illustrated in two major configurations of preferences. When players’ biases are uniformly distributed, we show that an individual communicates more and to more distant individuals as the proximity of his preference to the average preference of the population increases. When preferences are the same within groups but differ across groups, the impact of group size on communication again produces an interesting result: information transmission across groups is typically asymmetric, since players from the large group communicate more easily to members of the small group than the reverse.

3.1. Full characterization

Our main theorem provides a full characterization of the communication networks that arise as equilibrium outcomes of the cheap-talk stage of the game. In short, the theorem states that a player truthfully reveals his information to a group of players if his taste is not too different from the average taste of every subset of that group. More precisely, there exists an equilibrium network in which player i’s set of receivers is Ri ⊆ N \{i} if and only if, for every subset of players in Ri , the average bias of the players in the subset is close enough to player i’s bias.

Theorem 1. There exists an equilibrium network in which player i’s set of receivers is Ri ⊆ N \{i} iff for all R′i ⊆ Ri we have

∣∣∣∣∣bi − ∑

k∈R′ i bk

|R′ i |

∣∣∣∣∣ ≤ (n − 1 + α)(n − 1 − α|R ′ i |)

2(n − 1)(n − 1 − α|Ri |) (si − si ). (4)

Proof. See Appendix A.3. ‖

To understand the intuition of this characterization, observe that when a sender’s bias is close to the average bias of the receivers, a lie about his type may move these receivers’ actions

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too far from the sender’s ideal point. On the contrary, if the distance between the sender’s bias and the average bias of the receivers is substantial, then the sender has an incentive to over- report or under-report his type so that the receivers’ actions become closer to his ideal action. Since private communication allows the sender to lie to any subset R′

i of the set Ri of receivers,

informational incentive constraints require that the sender’s bias be close enough to the average bias of every subset R′

i ⊆ Ri .

The exact proximity between the sender’s and the receivers’ biases required for truthful communication depends on the threshold given by the right-hand side (RHS) of equation (4). It is worth noting that this threshold depends on the numbers of receivers |Ri | and |R′i |. In existing models of cheap talk to multiple audiences who do not interact strategically in the decision stage (e.g. Farrell and Gibbons, 1989; Goltsman and Pavlov, 2009; Galeotti et al., 2009), the proximity of players’ biases required for truthful communication depends only on the information structure and on some game parameters (here, si − si , α and n). Hence, with independent decision makers, a necessary condition for truthful information transmission from a sender to a set of receivers is that, for every member of this set, there exists an equilibrium in which the sender transmits his information truthfully to this member only. In contrast, in our model, whether communication from a sender to a given receiver can be sustained in equilibrium depends on the whole set of players to whom the sender truthfully reveals his information. In particular, the incentive to communicate to a receiver not only depends on the conflict of interest between the sender and this receiver, but also on the number and the preferences of all the receivers to whom the sender sends a truthful message.

To understand why the RHS of equation (4) depends on |Ri | and |R′i |, observe two differences between a cheap-talk model with independent decision makers and ours. First, since the equilibrium actions of players in Ri depend on i’s signal and others’ actions, they react differently to i’s signal depending on the total number of receivers |Ri |. Second, the sender also wants to coordinate his action with his receivers’ actions. Hence, when he deviates and sends the wrong signal to a subset of receivers R′

i ⊆ Ri (this deviation cannot be observed

by any player different from player i himself), the coordination costs induced by that lie depend on the size of R′

i . Precisely, the threshold of the RHS of equation (4) is decreasing in |R′

i |,

meaning that it is less costly for the sender to lie to large subsets R′ i

of Ri than to small ones. As an illustration, and for future reference, consider a game with n = 4 players, α = 1/2,

and assume that every player has the same value of private information, given by si − si = 12×3

7 . The RHS of equation (4) in Theorem 1 simplifies to 3 6−|R′

i |

6−|Ri | . It follows that player i reveals his type to all the other players if for all k, l ∈ N \{i},∣∣∣∣∣bi −

∑ j �=i bj 3

∣∣∣∣∣ ≤ 3, ∣∣∣∣bi − bk + bl2

∣∣∣∣ ≤ 4, and |bi − bk| ≤ 5. (5) Similarly, player i reveals his type only to players in {j, k} � N \{i} if∣∣∣∣bi − bj + bk2

6. For example, with the bias profile b = (b1, b2, b3, b4) = (0, 3.8, 4.8, 9), (Ri )i∈N = (∅, {3}, {1, 2, 4}, ∅) is the most informative equilibrium communication network.

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of receivers to which j may belong. Given a set of receivers, we can also see that the thresholds on the RHS decrease with the size of the subset considered.

The origin of one main insight of our work is given by the following observation: given |R′

i |, the RHS of equation (4) is increasing in |Ri |. That is, the conditions given by Theorem 1

on the proximity between i’s bias and the average bias of the strict subsets R′ i � Ri of receivers

become weaker as the set of all receivers, Ri , is larger. The intuition is that, as we had already seen in the second-stage equilibrium characterization (equation (2)), as |Ri | increases, receivers are more responsive to whatever the sender is revealing to them. But the more responsive receivers are to a message by the sender, the less the latter has an incentive to over-report or under-report his information as it may affect the actions of the fixed set of receivers R′

i too

much. This feature implies that when the informational incentive constraints are satisfied for

information transmission to a set of receivers, these constraints are not necessarily satisfied for information transmission to a strict subset of these receivers only. In particular, one key effect revealed by our model is that there may exist an equilibrium in which an individual reveals his true type to a group of players whereas there is no equilibrium in which he reveals it only to strict subsets of this group. As an example, consider the bias profile b = (−4.1, 0, 3.8, 4.1) in the previous four-player game. There is then an equilibrium communication network in which player 2’s set of receivers is R2 = {1, 3, 4}, but there is no equilibrium communication network in which player 2’s set of receivers is a strict subset of {1, 3, 4}.

The fact that communication to a large group of recipients may occur in equilibrium, even though communication to a small subset of that group may not, relies on the receivers’ need for coordination. This feature, therefore called the disciplinary effect of coordination, will be illustrated for particular configurations of biases in Section 3.4. Note that this disciplinary effect of coordination differs from the disciplinary effect of public communication identified by Farrell and Gibbons (1989) and further analysed recently by Goltsman and Pavlov (2009). Indeed, our effect does not rely on the public nature of a sender’s message and appears even when communication is private.

To see how coordination motives make large equilibrium communication networks feasible when intermediate communication networks are not, it is instructive to look at extreme situations. When there is almost no coordination motive (α → 0), incentive constraints are as in a model without strategic interactions in the decision stage: the condition for Ri to be an equilibrium set of receivers for player i reduces to

∣∣bi − bj ∣∣ ≤ si − si 2

, for all j ∈ Ri .

That is, there is an equilibrium in which player i truthfully reveals his information to the players in Ri if and only if, for every agent in Ri , there is an equilibrium in which player i truthfully communicates with that agent only. This is because when the need for coordination vanishes, the responsiveness of the receivers’ actions to i’s signal and the sender’s coordination costs mentioned above no longer depend on the number of receivers.

Consider now the opposite situation in which coordination costs are extremely high (α → 1) and let player i transmit his information to all the other players (Ri = N \{i}) so that the responsiveness of players’ actions to player i’s signal is maximal. In that case, the incentive compatibility constraints of Theorem 1 for player i reduce to∣∣∣∣∣bi −

∑ j �=i bj

n − 1

HAGENBACH & KOESSLER COMMUNICATION NETWORKS 1081

That is, the incentive compatibility conditions ensuring that player i does not misrepresent his information to strict subsets of receivers are irrelevant. In particular, full revelation of information from player i to all the other players is possible whenever i’s bias bi is close

enough to the average bias of the other players, ∑

j �=i bj n−1 , whatever the distribution of players’

biases. The intuition is that, with extremely high coordination costs, player i never wants to lie about his type to only a subset of the other players, as, if he does so, his action cannot be perfectly coordinated with both the players to whom he lies and the players to whom he reveals his true type. This also means that as the weight of the coordination motive tends to 1, the conditions for full information revelation from any player are equivalent to the conditions for full information revelation were communication to be public. Indeed, if communication were public, informational incentive constraints would by definition be weaker than under private communication: the only possible deviation from a message sent publicly would be to jointly lie to the whole audience of players, while private communication enables the sender to lie to any subset of these players.

As shown in Theorem 1, each player’s equilibrium communication strategy does not depend on other players’ communication strategies. To obtain the intuition for this independence property, consider a strategy profile in which some player (say, player 1) reveals his type to players in R1. This strategy is optimal if, whatever his type, player 1 has no incentive to lie to some or all players in R1. Now consider the (unobservable) deviation that consists of player 1 lying to all players in R1 (the intuition is exactly the same when he lies only to a subset of the players in R1). Player 1’s deviation affects player 1’s expected utility by changing (i) the second-stage actions of players in R1, who now act in believing player 1’s wrong type instead of the true one and (ii) player 1’s best reply to the latter actions. As can be seen from the (linear and additive) form of second-stage equilibrium actions given by equation (2), the change in the actions of the players in R1 is independent of what they learn about the types of players other than 1. This relies on our specific utility functions, on the additivity of the state, and also on the independence of players’ types.7 Consequently, player 1’s best reply to other players’ actions and player 1’s expected utility are affected by the change in the actions of players in R1 independently of the information transmitted by other senders. In contrast, Austen-Smith (1993, Proposition 1) shows that the incentive for an expert to reveal truthfully his type to a decision maker depends on the communication strategy of the other expert. While we assume that types are independent and the state is additive in types, in Austen-Smith (1993) players’ signals are independent conditional on the state. Hence, in his model, the effect of the message of an expert on the beliefs and action of the decision maker depends on how well informed the decision maker is. More precisely, if the decision maker is well informed, the expert’s message affects his action only slightly, whereas if the decision maker is poorly informed, the message will affect his action significantly. This implies that communication from a given expert to the decision maker is more difficult when the other expert communicates with him.8

7. As an extreme example consider the situation in which some player j �= 1 is almost perfectly informed about player 1’s type, i.e. s1 and sj are strongly correlated. Then, when players in R1 know player j ’s signal, i.e. players in R1 also belong to Rj , player 1’s message only has a limited impact on players’ actions (and thus, on player 1’s expected payoff), whereas if player j does not reveal his type to players in R1 , then player 1’s signal has a stronger impact on the actions of players in R1 .

8. A similar statistical structure is used in, e.g. Morgan and Stocken (2008) and Galeotti et al. (2009), where the willingness of a sender to communicate with a player also declines in the number of senders communicating with that player.

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3.2. Comparative statics

The next corollary, easily deduced from Theorem 1, details the effect of the disparity of players’ preferences, the coordination motive, and the information structure on equilibrium sets of receivers.

Corollary 1 (comparative statics) . The conditions under which information is transmitted from any player i to any set of receivers are relaxed as:

(i) All biases are reduced by the same factor: (b1, . . . , bn) → r(b1, . . . , bn), where r ∈ [0, 1). (ii) The weight on coordination motives, α, increases.

(iii) The value of private information, si − si , increases.

Proof. Reducing the absolute values of the biases as in (i) clearly decreases the left-hand side (LHS) of Inequality (4). The RHS of Inequality (4) is also clearly increasing in si − si . Finally, it is increasing in α because ∂

∂α

(n−1−α|R′ i |)

(n−1−α|Ri |) = (n−1)(|Ri |−|R′i |) (n−1−α|Ri |)2

≥ 0. ‖

The intuition for (i) is quite clear. As all the biases become more similar to each other, the conflict of interest between all players falls, so informational incentive constraints become weaker as in existing cheap-talk models.

Point (ii) is not as direct as (i) since, in our framework, it is the need for coordination that itself results in incentive conflicts between players. Indeed, were there no need for coordination (α = 0), an equilibrium with perfect communication would always exist because a sender would be indifferent between revealing truthfully his information or not. When α is positive, his message has an impact on his payoff through the modification of others’ actions, and misrepresenting his type may be beneficial when his bias is significantly different from the receivers’ biases. However, the higher is α, the more costly it is for the sender to coordinate his action to the actions of the receivers he lies to. The effect of the need for coordination on strategic information transmission is also analysed in Alonso et al. (2008) who consider a two-division organization in which the decisions of the divisions are both responsive to local conditions and coordinated with each other.9 Decision makers’ payoffs are similar to those we consider, but the conflict of interest regarding decisions is modelled differently. In Alonso et al. (2008), 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 that of the other division. Under decentralization, they also show that an increase in the need for coordination facilitates communication between the two divisions. On the contrary, under centralization, when a benevolent principal makes all decisions by relying on cheap-talk statements from the divisions, an increase in the need for coordination worsens communication.

Finally, the intuition of (iii) is standard. As we already observed, the value of player i’s information, si − si , measures the impact of his message on the receivers’ belief about the state. So, when si − si is small, his influence on the receivers’ actions is also small and his incentive to lie about his type is greater. In the extreme case in which si − si tends to zero, the incentive constraints of player i would be similar to the condition for full revelation of information in a model with a continuum of types, as in Crawford and Sobel (1982), which is never satisfied except when players’ preferences exactly coincide.

9. See also Rantakari (2008).

HAGENBACH & KOESSLER COMMUNICATION NETWORKS 1083

We can also note that, for a given sender and a given set of receivers, increasing the total number of players, n, strengthens the conditions on the proximity between the sender’s and receivers’ biases, since the RHS of equation (4) is decreasing in n.10 The intuition of this effect is similar to the intuition of (iii) in the previous corollary: as the total number of players rises, the influence of the actions of a fixed set of receivers is smaller, so the sender’s incentive to misrepresent his type is greater. To account for variations in the size of the population, we could describe the state as the average of players’ signals, instead of the sum (this is irrelevant when n is fixed, since we impose no restrictions on si and si ). Equivalently, we can replace each signal si of every player i by

si n

. In this case, the RHS of Inequality (4) always tends to zero as n increases, whatever the set of receivers, so that information transmission becomes impossible between any pair of players who do not have the same preferences. This effect is similar to that observed by Morgan and Stocken (2008, Proposition 1) who show that truthful reporting is never an equilibrium for a sufficiently large sample of constituents.

3.3. General properties

By taking a closer look at the way in which the informational incentive constraints given in Theorem 1 intersect, some general properties of the equilibrium sets of receivers can be established. First, we can always construct larger equilibrium sets of receivers by adding agents whose biases are closer to the sender’s bias than to those of any of his receivers.

Corollary 2. If there exists an equilibrium network in which player i’s set of receivers is Ri , then there also exists an equilibrium network in which player i’s set of receivers is Ri ∪ {j } for every player j whose bias is closer to i’s bias than to those of any player in Ri , i.e.

|bi − bj | ≤ |bi − bk|, ∀ k ∈ Ri . (7)

Proof. See Appendix A.4. ‖

In particular, the above corollary implies that there always exists an equilibrium communica- tion network in which, for every player i, his set of receivers Ri includes all the players with the same bias bi . Note that if condition (7) holds for only some players in Ri , but not for all of them, then the result may not hold. To see this, consider again the four-player example introduced in Section 3.1 with the bias profile b = (0, 2.2, 3.2, 3.7). In this case, there is an equilibrium in which player 1’s set of receivers is R1 = {2, 4} but no equilibrium in which it is {2, 3, 4}.

Second, it is obvious from Theorem 1 that the existence of an equilibrium network in which player i’s set of receivers is {j } implies the existence of an equilibrium in which his set of receivers is {k}, for every k whose bias is between bi and bj . Applying the previous corollary, this yields:

Corollary 3. If there exists an equilibrium network in which player i’s set of receivers is {j }, j > i, then there also exists an equilibrium network in which player i’s set of receivers is Ri for every Ri ⊆ {i + 1, . . . , j }.

Proof. Directly from Theorem 1 and Corollary 2. ‖

10. The sign of the derivative with respect to n is 2α(n − 1)|R′ i | − α2|Ri ||R′i | − (n − 1)2(|Ri | + 1 − |R′i |), which

is always negative.

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Note that if players i and j have the same value of private information, i.e. si − si = sj − sj , and if there is an equilibrium network in which player i’s set of receivers is {j }, then there is also an equilibrium network in which player j ’s set of receivers is {i}. Combined with the previous corollary, this reciprocity property implies that if all players in {i, . . . , j } have the same value of private information, then any communication network among these players is an equilibrium network. In particular, full revelation of information between all the players in {i, . . . , j } is an equilibrium.

Under some conditions, larger communication networks can also be constructed by forming the union of existing equilibrium networks. As stated in the following corollary, combining two equilibrium sets of receivers Ri and R̃i for player i yields an equilibrium set of receivers Ri ∪ R̃i for player i if Ri and R̃i do not overlap, i.e. Ri ∩ R̃i = ∅.

Corollary 4. If there is an equilibrium network in which player i’s set of receivers is Ri , and an equilibrium network in which player i’s set of receivers is R̃i , and if Ri and R̃i do not overlap, then there is also an equilibrium network in which player i’s set of receivers is Ri ∪ R̃i .

Proof. See Appendix A.4. ‖

More generally, the proof of the previous corollary reveals that a sufficient condition for the result to hold is that the distance between i’s bias and the average bias in Ri ∪ R̃i is smaller than the distance between i’s bias and the average bias in Ri or R̃i . When this condition does not hold, the union of the two equilibrium receiver sets does not necessarily yield another equilibrium receiver set. To see this, consider once more our four-player example with the bias profile b = (0, 2.2, 3.2, 3.7). Here, there is an equilibrium network in which player 1’s set of receivers is R1 = {2, 3} and an equilibrium network in which it is R̃1 = {2, 4}, but no equilibrium network in which player 1’s set of receivers is R1 ∪ R̃1 = {2, 3, 4}. This implies that, in general, there may not exist a “maximal” equilibrium communication network which is more informative than all the other communication networks.

Necessary and sufficient conditions for the complete, Pareto-dominant, communication network to be an equilibrium are easily deduced from Theorem 1. More precisely:

Corollary 5 (complete network) . The complete communication network is an equilibrium network if and only if for all i ∈ N and Ri ⊆ N \{i},∣∣∣∣∣bi −

∑ j ∈Ri bj |Ri |

∣∣∣∣∣ ≤ (n − 1 + α)(n − 1 − α|Ri |)2(n − 1)2(1 − α) (si − si ). (8) Proof. Directly from Theorem 1. ‖

Note that when all players have the same value of private information, the set of these conditions is reduced, since we only have to check that the incentive constraints are satisfied for the two extreme players (player 1 and player n). In that case, a simple sufficient condition for the complete network to be an equilibrium is that there exist an equilibrium network in which player 1’s set of receivers is R1 = {n} or player n’s set of receivers is Rn = {1}. © 2009 The Review of Economic Studies Limited

HAGENBACH & KOESSLER COMMUNICATION NETWORKS 1085

3.4. Illustrations

In this subsection, we analyse two particular configurations of biases and obtain further results regarding the structure of equilibrium communication networks. Our aim is also to illustrate the disciplinary effect of coordination identified after Theorem 1. First, we consider uniformly distributed biases, assuming that players’ biases are equidistant: bi+1 − bi = β ≥ 0 for every i ∈ N . Second, we consider two-spike biases situations, in which the players are partitioned into two groups, L = {1, . . . , l} and M = {l + 1, . . . , n}. Players in the first group have a bias bL and players in the second group have a bias bM , with bM − bL = β > 0. For both configurations of biases, we know from Corollary 1 (i) that in equilibrium the maximal number of receivers of every player falls with the distance β.

To focus on the impact of players’ positions on their communication behaviour, we assume from now on that they all have the same value of private information: si − si = � for all i. In addition, we restrict our attention to equilibrium communication networks R which are maximal (i.e. such that there exists no equilibrium communication network more informative than R) and such that, for every i, players in {i} ∪ Ri are consecutive.11

3.4.1. Uniformly distributed biases. When biases are uniformly distributed, observe that, for any size |Ri |, the distance between i’s bias and the average bias of the |Ri | players who are the closest to i in the population falls with the proximity of i’s bias to the average bias in the whole population. Hence, from the equilibrium characterization of Theorem 1 we have:

Corollary 6. If players’ biases are uniformly distributed, then the maximal number of equilibrium receivers increases with the proximity of a sender’s bias to the average bias in the population.

After stating Corollary 5, we noted that full revelation of information between all players is possible whenever the two extreme players reveal their information to all the other players. With uniformly distributed biases, the previous corollary further asserts that middle-biased players (i.e. players whose biases are close to the average bias in the population) communicate more than extremists (i.e. players whose biases are far from the average bias in the population).

The impact of players’ position on their communication behaviour is even stronger than stated in Corollary 6. If every player i, whatever his position, communicates to all the players whose biases are less distant than some threshold d > 0, i.e. Ri = {k �= i : |bi − bk| ≤ d}, then it is clear that central players communicate more than extremists since |{k �= j : |bj − bk| ≤ d}| ≥ |{k �= i : |bi − bk| ≤ d}| whenever j is more central than i. The next corollary shows that not only the number of receivers increases with the sender’s centrality, but so does the distance between the sender’s bias and his receivers’ biases. In other words, central players can truthfully communicate with agents with whom they have higher conflicts of interest than less central players can do.

Corollary 7. If players’ biases are uniformly distributed and Ri = {k �= i : |bi − bk| ≤ d} is an equilibrium set of receivers for some player i and some distance d ≥ 0, then Rj = {k �= j : |bj − bk| ≤ d} is also an equilibrium set of receivers for every player j who is more central than i. In general, the reverse is not true.

11. By consecutive we mean that there is no player in N\({i} ∪ Ri ) whose bias lies between the biases of any two players in {i} ∪ Ri . From Theorem 1, it is easy to show that such an equilibrium always exists.

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0 1

2 3

4 5

6 7

Players

Number of receivers

0,1 0,3

0,5 0,7

0,9 Alpha

Figure 1

Number of equilibrium receivers with n = 7 players, Uniformly distributed biases, β = 0.6 and � = 2

These results are illustrated by Figure 1, which plots the number of receivers |Ri | as a function of the coordination motive α and player i’s position, when n = 7, � = 2, and β = 0.6. The figure shows how the number of receivers increases with players’ centrality whatever the value of α, and with α whatever the players’ position. When α is high enough (in this figure, when α ≥ 0.3), we also see that more central senders communicate to more distant receivers. For example, when α = 0.5, the most central player (player 4) communicates to all the other players, while players 1 and 7 only communicate to a single player.

To understand better the role of the coordination motive on the impact of players’ position on their incentive to communicate, consider again extreme situations. When α → 0, we have already observed that the equilibrium condition for Ri to be an equilibrium set of receivers for player i reduces to maxj ∈Ri

∣∣bi − bj ∣∣ ≤ �2 whatever player i’s position. So, in that case, only the distance between a sender and one receiver matters for incentive compatibility, and the reverse of Corollary 7 is true. More generally, when α > 0, the equilibrium condition for a single receiver, namely

β ≤ n − 1 + α 2(n − 1) �,

is the same for every sender, but will depend on the sender’s position for more than two receivers. To see this, notice that incentive compatibility constraints for the less central players (player 1 and player n) to communicate to ri ∈ {1, . . . , n − 1} receivers can be written as:

β ≤ (n − 1 + α)(n − 1 − αr ′ i )

(n − 1)(2ri − r′i + 1)(n − 1 − αri ) �, ∀ r′i ∈ {1, . . . , ri }.

HAGENBACH & KOESSLER COMMUNICATION NETWORKS 1087

It can be shown12 that for every α ∈ (0, 1) this incentive compatibility condition becomes strictly stronger as ri increases. That is, it is always more difficult for extremists to communicate to a larger group than to a smaller group, even when the coordination motive is very strong. On the contrary, for strong enough coordination motives, incentive compatibility conditions for more central players are not necessarily monotonic in the number of receivers. That is, large receiver sets may be equilibrium outcomes for more central players, while small receiver sets are not. For example, when n is odd, the incentive compatibility condition for the central player i = n+12 is always strictly weaker for ri + 1 receivers than for ri receivers when ri ≤ n − 1 is odd. As α tends to 1, the condition always holds for ri = n − 1 receivers, but continues to represent a constraint for ri < n − 1 receivers. In particular, a central player may reveal his information to all players, while an extreme player may transmit his information to none.

3.4.2. Two-spike biases. When players are partitioned into two groups, full revelation of information amongst all the players in the same group is always an equilibrium. In addition, Corollary 2 implies that if a player i ∈ L (i ∈ M, resp.) transmits his information to players in Ri in equilibrium, then there is also an equilibrium in which i reveals his information to players in Ri ∪ (L\{i}) (Ri ∪ (M\{i}), resp.). Hence, there is a unique maximal equilibrium network such that the set of receivers of each player i ∈ L includes all players in L\{i} and the set of receivers of each player i ∈ M includes all players in M\{i}.

Since it is always possible for players to communicate to players with the same bias, and since the RHS of the informational incentive constraint (4) is increasing in the total number of receivers, a player’s informational incentive constraints are relaxed as the relative number of players with the same bias increases. More precisely:

Corollary 8. In the maximal equilibrium network with two-spike biases, a player’s set of receivers increases, and includes more players from the other group, as the relative number of players in his own group increases.

Proof. Directly from Theorem 1 and the observation above. ‖

In particular, this corollary implies that intergroup information transmission is higher for players in the larger group than for players in the smaller group. As a simple example, consider the situation in which α → 1. Then, there is complete communication from players in group L if and only if β ≤ n2(n−l) �, which becomes easier to satisfy as the size of this group, l, increases.

This property does not extend to more than two groups of players. For example, in the four-player example, when α = 0.9 and si − si = 1 for all i, the condition for player i to reveal his information to all players is∣∣∣∣∣bi −

∑ j �=i bj 3

∣∣∣∣∣ ≤ 0.65, ∣∣∣∣bi − bk + bl2

∣∣∣∣ ≤ 2.6, and |bi − bk| ≤ 4.55. (9) Hence, when b = (−3, 0, 0, 3) there is an equilibrium in which players with zero bias transmit their information to all the other players, but the first inequality does not hold when b = (0, 0, 0, 3). Actually, when there are more than two groups of players with the same bias, the corollary above only applies, in general, for players in the two extreme groups, i.e. the group with bias b1 and that with bias bn.

12. A formal proof is available from the authors upon request.

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4. CONCLUSION

In this paper, we consider a class of economically relevant coordination games in which information about a common state of nature is distributed among the players. Each of these players chooses an action by trading off the benefit of it being close to his own “ideal action”, which depends both on the state and on an idiosyncratic bias, with that of being close to the other players’ decisions. Before taking such actions, the players are offered the opportunity to communicate with each other in a decentralized and strategic manner. In this setting, our focus is on the way heterogeneity in preferences shapes strategic information transmission. We provide explicit conditions on the proximity of players’ biases for information to be revealed by any sender to any group of receivers. Precisely, we show that an agent reveals his information to a group as long as this group is large enough and his ideal action is close enough to the average ideal action of every subset of agents in this group.

Similar coordination games with incomplete information have already been analysed in the literature, but under the assumption that there is no conflict of interest between agents regarding the ideal state-contingent action profile (see, for example, Morris and Shin, 2002; Angeletos and Pavan, 2007). When agents’ goals are aligned, but there are physical or cost constraints on the number of communication links between agents, others have identified the most efficient communication structures; see, among others, Marschak and Radner (1972), Radner (1993), Jehiel (1999), Chwe (2000), Calvó-Armengol and Martı́ (2007, 2009), and Morris and Shin (2007). In these papers, efficient networks are characterized under physical communication constraints. On the contrary, our approach studies the equilibrium communication networks that arise under strategic communication constraints. To that extent, our work mainly borrows from the literature on strategic information transmission based on Crawford and Sobel (1982) but then proposes a framework in which every player is at the same time a sender and a receiver.

One key insight that stems from our characterization of equilibrium communication networks is that large networks may be easier to sustain in equilibrium that smaller ones. In other words, we show that the need for coordination of multiple interacting audiences can discipline communication, in the sense that truthful communication may be feasible in a large group but not in strict subsets of this group. A similar phenomenon is obtained in a team-theoretic framework in Dessein and Santos (2006), which also considers a coordination- adaptation situation with quadratic costs, but where communication is non-strategic.13 Another main result is that agents who are more central in terms of preferences communicate more and have a greater impact on the decisions taken. Note that such a prediction contrasts with those of models of costly communication (e.g. Banerjee and Somanathan, 2001) where there is a tendency for extremists to express more voice.14

The way in which communication links have been constructed in the current analysis completely departs from usual non-cooperative network-formation games in a number of ways.15 In typical games of this type, players’ strategies mainly consist in listing desired contacts, given the exogenous costs and benefits of direct and indirect connections. In addition, since it is commonly admitted that much of the information required for economic decision making is exchanged via networks of relationships, the value of these connections is often

13. In that model, a communication link requires “bundling of tasks”, which is assumed to be costly. Bundling a few tasks together may reduce profits relative to stand-alone tasks (no communication links at all), while bundling a lot of them together may actually improve profitability.

14. In their model, individuals only differ in terms of beliefs about a binary state of nature. Centrists are those who put relatively equal weight on the two states of the world, while extremists firmly believe in one or the other of the states.

15. See Jackson (2008) for an extensive survey of such models.

HAGENBACH & KOESSLER COMMUNICATION NETWORKS 1089

interpreted as being informational. However, whether agents have an effective interest in transmitting information once a link exists has not yet been investigated to the best of our knowledge. By way of contrast, we explicitly model agents’ informational frameworks and derive the equilibrium links directly from the informativeness of agents’ communication strategies. Given that the connection conveys truthful information, the benefits from linking are then endogenously determined by the way in which the information transmitted is used in the decision stage.

APPENDIX

A.1. Second-stage equilibrium characterization

We first characterize the unique equilibrium action profile under complete information. The utility function of player i (see equation (1)) can be rewritten as (minus a constant):

⎡ ⎣2(1 − α)(θ + bi ) + 2α

n − 1 ∑ j �=i

aj − ai ⎤ ⎦ − α

n − 1 ∑ j �=i

(aj ) 2 . (A.1)

The best response of each player i to a−i is given by:

ai (a−i ; θ ) = (1 − α)(θ + bi ) + α

n − 1 ∑ j �=i

aj . (A.2)

If ai is a best response to a−i , then it follows from equations (A.1) and (A.2) that player i’s utility takes the following simple form (minus a constant):

ui (ai (a−i ; θ), a−i ; θ ) = (ai (a−i ; θ))2 − α

n − 1 ∑ j �=i

(aj ) 2 . (A.3)

The system of equations formed by equation (A.2) can be written as:

⎛ ⎜⎜⎜⎜⎜⎝

a1 . . .

⎞ ⎟⎟⎟⎟⎟⎠ =

⎛ ⎜⎜⎜⎜⎜⎜⎝

1 − α (n−1) · · · − α(n−1)

− α (n−1)

. . . . . .

. . . .

. . . − α (n−1)

− α (n−1) · · · − α(n−1) 1

⎞ ⎟⎟⎟⎟⎟⎟⎠

︸︷︷︸ I

−1 ⎛ ⎜⎜⎜⎜⎜⎝

(1 − α)(θ + b1) . . .

(1 − α)(θ + bn)

⎞ ⎟⎟⎟⎟⎟⎠ .

Simple algebra yields:

I −1 = 1

(n − 1) − (n − 2)α − α2

⎛ ⎜⎜⎜⎜⎜⎝

(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, equilibrium actions are given by:

ai (θ) = θ + [(n − 1) − (n − 2)α]bi + α

∑ j �=i bj

n + α − 1 ≡ θ + Bi , for every i ∈ N . (A.4)

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(ai ) = E(θ) + Bi , for every i ∈ N. (A.5)

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The uniqueness of the linear equilibrium identified in equation (2) is proved as in Calvó-Armengol and Martı́ (2009, Theorem 1). We define the following payoff function:

v(a1, ..., an; s) = −(1 − α) ∑ i∈N

(ai − θ (s) − bi )2 − α

2(n − 1) ∑ i∈N

∑ j �=i

(ai − aj )2. (A.6)

The set of equilibria of our second-stage coordination game is the same as that in the corresponding Bayesian game with identical agent preferences in which every player’s payoff function is given by (A.6), as the best responses are identical in both games.

Theorem 4 of Marschak and Radner (1972)[167–168] provides a sufficient condition for the equilibrium of a Bayesian game with identical agent preferences to be determined uniquely by a system of linear equations when the set of states of the world is finite and payoff functions are given by:

λ(s) + 2 ∑ i∈N

μi (s)ai − ∑

i,j ∈N vij (s)ai aj , (A.7)

where the λ, μi and vij are all real-valued functions of the state of the world, s ∈ S. It is easily checked that the payoff function (A.6) can be written as (A.7), with

λ(s) = −(1 − α) ∑ i∈N

(θ(s) + bi )2,

μi (s) = (1 − α)(θ(s) + bi ), vii (s) = vii = 1, vij (s) = vij = −

n − 1 .

The sufficient condition in Theorem 4 of Marschak and Radner (1972) then boils down to the n-square matrix [vij ]i,j ∈N being positive definite. The determinant of [vij ]i,j ∈N is:∣∣∣∣∣∣∣∣∣∣∣

1 − α n−1 · · · − αn−1

− α n−1

. . . . . .

. . . .

. . . − α n−1

− α n−1 · · · − αn−1 1

∣∣∣∣∣∣∣∣∣∣∣ = (1 − α)

∣∣∣∣∣∣∣∣∣∣∣

1 − α n−1 · · · − αn−1

1 . . .

. . . . . .

. . . .

. . . − α n−1

1 · · · − α n−1 1

∣∣∣∣∣∣∣∣∣∣∣

= (1 − α)

∣∣∣∣∣∣∣∣∣∣∣

1 − α n−1 · · · − αn−1

0 1 + α n−1 0

. . . .

. . . 0 0 · · · 0 1 + α

n−1

∣∣∣∣∣∣∣∣∣∣∣ = (1 − α)

( 1 + α

n − 1 )n−1

The first equality results from the replacement of the elements in the first column by the row sum, and then taking out the common factor (1 − α). The second equality is obtained by subtracting the first row from every other row. We are left with an upper triangular matrix whose determinant is just the product of the diagonal term, which is positive. Similarly, we deduce that the leading principal minors of [vij ]i,j ∈N are positive. The matrix [vij ]i,j ∈N is thus positive definite.

Next, 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 (Ri )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 then show that player l’s best response to this profile of second-stage actions (ai )i �=l is also of the form of equation (2).

After the communication stage, for all i ∈ N , recall that Ii = {k : i ∈ Rk } ∪ {i} is the set of players whose signals are known by player i, I i = {k : i �∈ Rk }\{i} the set of players whose signals are not known by player i, and let Ei (·) = E(· | {sk : k ∈ Ii }) be player i’s expectation operator conditional on the set of signals that he knows.

The expected payoff of player l after the communication stage is as follows:

−(1 − α)El ⎡ ⎣(al − ∑

j ∈N sj − bl )2

⎤ ⎦ − α

n − 1 ∑ i �=l

El [ (al − ai )2

] , (A.8)

HAGENBACH & KOESSLER COMMUNICATION NETWORKS 1091

so his best response is given by:

al = (1 − α) ⎛ ⎝∑

j ∈Il sj +

∑ j ∈I l

E(sj ) + bl ⎞ ⎠ + α

n − 1 ∑ i �=l

El (ai ). (A.9)

From now on and for every i ∈ N , we use the notation ri for |Ri |. Using equation (2) for i �= l, player l’s conditional expectation of player i’s action is given by:

El (ai ) = ∑ j ∈Ii

α(n − 1 − rj )E(sj ) n − 1 − αrj

+ ∑

j ∈Ii ∩Il

(1 − α)(n − 1)sj n − 1 − αrj

+ ∑

j ∈Ii ∩I l

(1 − α)(n − 1)E(sj ) n − 1 − αrj

+ ∑ j ∈I i

E(sj ) + Bi .

Summing over all agents other than l, we can write:

∑ i �=l

El (ai ) = ∑ i �=l

∑ j ∈Ii ∩Il

α(n − 1 − rj )E(sj ) n − 1 − αrj

+ ∑ i �=l

∑ j ∈Ii ∩I l

α(n − 1 − rj )E(sj ) n − 1 − αrj

+ ∑ i �=l

∑ j ∈Ii ∩Il

(1 − α)(n − 1)sj n − 1 − αrj

+ ∑ i �=l

∑ j ∈Ii ∩I l

(1 − α)(n − 1)E(sj ) n − 1 − αrj

+ ∑ i �=l

∑ j ∈I i ∩Il

E(sj ) + ∑ i �=l

∑ j ∈I i ∩I l

E(sj ) + ∑ i �=l

Bi . (A.10)

Every signal sj known by player l is known by rj players other than l and not known by n − 1 − rj players different from l; every signal sj not known by player l is known by rj + 1 players other than l and not known by n − 2 − rj players other than l. This enables us to deduce that:

∑ i �=l

El (ai ) = ∑ j ∈Il

rj α(n − 1 − rj )E(sj )

n − 1 − αrj + ∑ j ∈I l

(rj + 1) α(n − 1 − rj )E(sj )

n − 1 − αrj

+ ∑ j ∈Il

rj (1 − α)(n − 1)sj

n − 1 − αrj +

∑ j ∈I l

(rj + 1) (1 − α)(n − 1)E(sj )

n − 1 − αrj

+ ∑ j ∈Il

(n − 1 − rj ) E(sj ) + ∑ j ∈I l

(n − 2 − rj ) E(sj ) + ∑ i �=l

Bi .

= ∑ j ∈Il

rj (1 − α)(n − 1)sj + (n − 1)(n − 1 − rj )E(sj ) n − 1 − αrj

+ ∑ j ∈I l

(n − 1) E(sj ) + ∑ i �=l

Bi . (A.11)

In addition, we have:

∑ i �=l

Bi = α(n − 1)bl + (n − 1)

∑ i �=l bi

n + α − 1 . (A.12)

Plugging equations (A.12) and (A.11) into equation (A.9) and simplifying, we obtain player l’s optimal action, which takes exactly the same form as that in equation (2).

A.2. Proof of Proposition 1

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

Uj = −(1 − α)V ar (

aj − ∑ i∈N

si − bj )

− (1 − α) [ E

( aj −

∑ i∈N

si − bj )]2

− α n − 1

∑ m�=j

V ar(aj − am) − α

n − 1 ∑ m�=j

[E(aj − am)]2.

1092 REVIEW OF ECONOMIC STUDIES

It follows from equation (A.5) that E(aj ) = ∑

i∈N E(si ) + Bj , so we have:

Uj = −(1 − α)V ar (

aj − ∑ i∈N

) − α

n − 1 ∑ m�=j

V ar(aj − am) − (1 − α)[Bj − bj ]2 − α

n − 1 ∑ m�=j

[Bj − Bm]2.

We consider two communication networks R = (Rk)k∈N and R′ = (R′k)k∈N such that Ri �= R′i and Rk = R′k for all k ∈ N\{i}. That is, R and R′ are identical except that player i has a different set of receivers in R′. Player i is fixed throughout the analysis. The ex ante equilibrium payoff of every player j ∈ N with the communication network R (R′, resp.) is denoted by Uj (U ′j , resp.). Given the communication network R (R

′, resp.), the second-stage equilibrium action of every player j ∈ N is denoted by aj (a′j , resp.). For all j ∈ N , given a strategic communication network R (R′, resp.), let Ij = {k : j ∈ Rk } ∪ {j } (I ′j = {k : j ∈ R′k } ∪ {j }, resp.) denote the set of players whose signals are known by player j , and I j = {k : j �∈ Rk }\{j } (I ′j = {k : j �∈ R′k }\{j }, resp.) the set of players whose signals are not known by player j .

For every player j ∈ N , we have:

Uj − U ′j = (1 − α) (

V ar

( a

′ j −

∑ i∈N

) − V ar

( aj −

∑ i∈N

))

+ α n − 1

⎛ ⎝∑

m�=j V ar

( a

′ j − a′m

) − ∑ m�=j

V ar(aj − am) ⎞ ⎠ .

(A.13)

The second-stage equilibrium action aj given by equation (2) enables us to write:

V ar

( aj −

∑ i∈N

) = V ar

⎛ ⎜⎝∑

l∈Ij

α(n − 1 − rl )[E(sl) − sl ] n − 1 − αrl

+ ∑ l∈I j

[E(sl ) − sl ] + Bj

⎞ ⎟⎠ .

The independence of signals yields:

V ar

( aj −

∑ i∈N

) = ∑ l∈Ij

V ar

( α(n − 1 − rl )sl

n − 1 − αrl

) + ∑ l∈I j

V ar(sl )

= ∑ l∈Ij

( α(n − 1 − rl ) n − 1 − αrl

)2 V ar(sl) +

∑ l∈I j

V ar(sl)

= ∑

l∈Ij \{i}

( α(n − 1 − rl ) n − 1 − αrl

)2 V ar(sl )

+ ∑

l∈I j \{i} V ar(sl ) + 1[i ∈ Ij ]

( α(n − 1 − ri ) n − 1 − αri

)2 V ar(si )

+ 1[i ∈ I j ] V ar(si ),

where 1[i ∈ Ij ] is an indicator function that equals 1 when player j knows the signal si , and 1[i ∈ I j ] is an indicator function that equals 1 when player j does not know the signal si . A similar equation holds for V ar(a

′ j

− ∑i∈N si ), when the communication network is R′.

The two communication networks R and R′ that we consider are such that Ij \{i} = I ′j \{i} and I j \{i} = I ′j \{i}, so that for all j ∈ N we have:

V ar

( a

′ j −

∑ i∈N

) − V ar

( aj −

∑ i∈N

) = V ar(si )[

1[i ∈ I ′j ] (

α(n − 1 − r′ i )

n − 1 − αr′ i

)2 + 1[i ∈ I ′j ] − 1[i ∈ Ij ]

( α(n − 1 − ri ) n − 1 − αri

)2 − 1[i ∈ I j ]

] .

(A.14)

HAGENBACH & KOESSLER COMMUNICATION NETWORKS 1093

When the communication network is R, for all j ∈ N and m �= j , we have, from equation (2):

V ar(aj − am) = ∑

l∈Ij ∩I m

( (1 − α)(n − 1) n − 1 − αrl

)2 V ar(sl ) +

∑ l∈I j ∩Im

( (1 − α)(n − 1) n − 1 − αrl

)2 V ar(sl )

= ∑

l∈(Ij ∩I m)\{i}

( (1 − α)(n − 1) n − 1 − αrl

)2 V ar(sl )

+ ∑

l∈(I j ∩Im)\{i}

( (1 − α)(n − 1) n − 1 − αrl

)2 V ar(sl)

+ (

(1 − α)(n − 1) n − 1 − αri

)2 [ 1[i ∈ Ij ∩ I m] + 1[i ∈ I j ∩ Im]

] V ar(si ).

A similar equation holds for V ar(a′ j

− a′m) when the communication network is R′. The two communication networks R and R′ are such that (Ij ∩ I m)\{i} = (I ′j ∩ I ′m)\{i} and (I j ∩ Im)\{i} =

(I ′j ∩ I ′m)\{i}, so for all j ∈ N and m �= j we have:

V ar(a ′ j − a′m) − V ar(aj − am)

= ((1 − α)(n − 1))2 [

1[i ∈ I ′ j

∩ I ′m] + 1[i ∈ I ′j ∩ I ′m] (n − 1 − αr′

i )2

− 1[i ∈ Ij ∩ I m] + 1[i ∈ I j ∩ Im] (n − 1 − αri )2

] V ar(si ).

(A.15)

Plugging (A.14) and (A.15) into (A.13), we obtain: Uj − U ′j =

(1 − α) [

1[i ∈ I ′j ] (

α(n − 1 − r′ i )

n − 1 − αr′ i

)2 + 1[i ∈ I ′j ] − 1[i ∈ Ij ]

( α(n − 1 − ri ) n − 1 − αri

)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 ∈ Ij ∩ I m] + 1[i ∈ I j ∩ Im] (n − 1 − αri )2

)⎤ ⎦ V ar(si ).

(A.16)

We next focus on the particular case in which |Ri | < |R′i | and let L = Ri ∩ R′i be the set of agents who belong both to Ri and R

′ i . The set L is fixed throughout the analysis, and |L| = l. Let |Ri | = ri , |R′i | = r′i , |Ri \l| = ri − l

and |R′ i \l| = r′

i − l. To evaluate the sign of Uj − U ′j , in order to establish who is better off and who is worse off

under the networks R and R′, the set N of players is divided into the following five types:

• (i) Players who belong to R′ i

and also to Ri . For every such player j ∈ L, we have i ∈ Ij and i ∈ I ′j . • (ii) Players who belong to R′

i but not to Ri . For every such player j ∈ R′i \L, we have i ∈ I j and i ∈ I ′j .

• (iii) Players other than player i who belong neither to R′ i

nor to Ri . For every such player j ∈ N\(Ri ∪ R′i ∪ {i}), we have i ∈ I j and i ∈ I ′j .

• (iv) Players who do not belong to R′ i

but belong to Ri . For every such player j ∈ Ri \L, we have i ∈ Ij and i ∈ I ′j .

• (v) Player i, for whom we have i ∈ Ii and i ∈ I ′i .

(i) For every player j ∈ L, the set of players other than j can be divided into four disjoint sets of players: {i} ∪ L\{j }, N\(Ri ∪ R′i ∪ {i}), R′i \L, and Ri \L. We have:

• for every player m ∈ {i} ∪ L\{j }, i ∈ Im and i ∈ I ′m; • for every player m ∈ N\(Ri ∪ R′i ∪ {i}), i ∈ I m and i ∈ I ′m, and we have |N\(Ri ∪ R′i ∪ {i})| = n − 1 − ri −

r′ i + l;

• for every player m ∈ R′ i \L, i ∈ I m but i ∈ I ′m;

• for every player m ∈ Ri \L, i ∈ Im but i ∈ I ′m.

Since i ∈ Ij and i ∈ I ′j , equation (A.16) simplifies to:

Uj − U ′j = α(1 − α) (

n − 1 − r′ i

n − 1 − αr′ i

− n − 1 − ri n − 1 − αri

) V ar(si ). (A.17)

1094 REVIEW OF ECONOMIC STUDIES

Using r′ i > ri , we obtain Uj − U ′j = −

( α(1−α)2(n−1)(r′

i −ri )

(n−1−αr′ i )(n−1−αri )

) V ar(si ) < 0. Hence, for all j ∈ L, we have Uj < U ′j .

(ii) For every player j ∈ R′ i \L, the set of players other than j can be divided into four disjoint sets of players:

{i} ∪ L, N\(Ri ∪ R′i ∪ {i}), R′i \(L ∪ {j }) and Ri \L. We have:

• for every player m ∈ {i} ∪ L, i ∈ Im and i ∈ I ′m, and we have |{i} ∪ L| = l + 1; • for every player m ∈ N\(Ri ∪ R′i ∪ {i}), i ∈ I m and i ∈ I ′m, and we have |N\(Ri ∪ R′i ∪ {i})| = n − 1 − ri −

r′ i + l;

• for every player m ∈ R′ i \(L ∪ {j }), i ∈ I m but i ∈ I ′m;

• for every player m ∈ Ri \L, i ∈ Im but i ∈ I ′m.

Since i ∈ I j and i ∈ I ′j , equation (A.16) simplifies to:

Uj − U ′j = −(1 − α)2(n − 1) (

n − 1 − αr′ i

+ α (ri + 1) (n − 1 − αri )2

) V ar(si ) < 0. (A.18)

Hence, for all players j ∈ R′ i \L, we have Uj < U ′j .

(iii) For every player j ∈ N\(Ri ∪ R′i ∪ {i}), the set of players other than j can be divided into four disjoint sets of players: {i} ∪ L, N\(Ri ∪ R′i ∪ {i, j }), R′i \L, and Ri \L. We have:

• for every player m ∈ {i} ∪ L, i ∈ Im and i ∈ I ′m, and we have |{i} ∪ L| = l + 1; • for every player m ∈ N\(Ri ∪ R′i ∪ {i, j }), i ∈ I m and i ∈ I ′m; • for every player m ∈ R′

i \L, i ∈ I m but i ∈ I ′m;

• for every player m ∈ Ri \L, i ∈ Im but i ∈ I ′m.

Since i ∈ I j and i ∈ I ′j , equation (A.16) simplifies to:

Uj − U ′j = α(1 − α)2(n − 1) (

r′ i + 1

(n − 1 − αr′ i )2

− ri + 1 (n − 1 − αri )2

) V ar(si ). (A.19)

Using r′ i > ri , we have

[ r′ i +1

(n−1−αr′ i )2

− ri +1 (n−1−αri )2

] > 0. Hence, for all N\(Ri ∪ R′i ∪ {i}), we have Uj > U ′j .

(iv) For every player j ∈ Ri \L, the set of players other than j can be divided into four disjoint sets of players: {i} ∪ L, N\(Ri ∪ R′i ∪ {i}), R′i \L, and Ri \(L ∪ {j }). We have:

r′ i + l;

• for every player m ∈ R′ i \L, i ∈ I m but i ∈ I ′m;

• for every player m ∈ Ri \(L ∪ {j }), i ∈ Im but i ∈ I ′m.

Since i ∈ Ij and i ∈ I ′j , equation (A.16) simplifies to:

Uj − U ′j = (1 − α)2(n − 1) (

n − 1 − αri + α (r

′ i + 1)

(n − 1 − αr′ i )2

) V ar(si ) > 0. (A.20)

Hence, for every player j ∈ Ri \L, we have Uj > U ′j . (v) The set of players other than i can be divided into four disjoint sets of players: L, N\(Ri ∪ R′i ∪ {i}), R′i \L,

and Ri \L. We have:

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

i \L, i ∈ I m but i ∈ I ′m

• for every player m ∈ Ri \L, i ∈ Im but i ∈ I ′m.

Since i ∈ Ii and i ∈ I ′i , equation (A.16) yields exactly the same difference as equation (A.17). Hence, for player i such that ri < r

′ i , we have Ui < U

′ i . This completes the proof of Proposition 1.

HAGENBACH & KOESSLER COMMUNICATION NETWORKS 1095

A.3. Proof of Theorem 1

Consider an equilibrium in which each player i reveals his type to players in Ri ⊆ N\{i}. Without loss of generality, assume that each player i sends to every player j ∈ Ri the message mji = m when his type is si and the message m

i = m when his type is si , and sends the same message whatever his type to players outside Ri . Given (Ri )i∈N ,

the second-stage equilibrium actions are given by equation (2). Without loss of generality, we look for the conditions under which player 1 does not deviate from his equilibrium

communication strategy described above. First, assume that player 1’s true type is s1 = s1. In equilibrium, using equation (2), the second-stage action of every player i ∈ R1 ∪ {1} is given by

ai = ∑

j ∈Ii \{1}

α(n − 1 − rj )E(sj ) + (1 − α)(n − 1)sj n − 1 − αrj

+ ∑ j ∈I i

E(sj ) + Bi

+ α(n − 1 − r1)E(s1) + (1 − α)(n − 1)s1 n − 1 − αr1

, (A.21)

and the second-stage action of every player i /∈ R1 ∪ {1} is given by

ai = ∑ j ∈Ii

α(n − 1 − rj )E(sj ) + (1 − α)(n − 1)sj n − 1 − αrj

+ ∑

j ∈I i \{1} E(sj ) + Bi + E(s1). (A.22)

The relevant deviations for player 1 in the communication stage consist in lying to a subset of players M ⊆ R1, i.e. sending message m instead of m to players in M (and not deviating towards the other players).16 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 = ai , which

is given by replacing s1 by s1 in (A.21). The action a ′ i

of every player i ∈ N\(M ∪ {1}) is the same as that in the original equilibrium. Player 1’s optimal action in the second stage is obtained from the best response of equation (A.9) to (a′

i )i �=1, and takes the following form:

a ′ 1 = (1 − α)

⎛ ⎝ ∑

j ∈I1\{1} sj + s1 +

∑ j ∈I 1

E(sj ) + b1 ⎞ ⎠ + α

n − 1 ∑ i �=1

E1(a ′ i ). (A.23)

Using the same reasoning as that used to obtain expression (A.11), we have:

∑ i �=1

E1(a ′ i ) =

∑ j ∈I1

rj α(n − 1 − rj )E(sj )

n − 1 − αrj +

∑ j ∈I 1

(rj + 1) α(n − 1 − rj )E(sj )

n − 1 − αrj

+ ∑

j ∈I1\{1} rj

(1 − α)(n − 1)sj n − 1 − αrj

+ m(1 − α)(n − 1)s1 n − 1 − αr1

+ (r1 − m)(1 − α)(n − 1)s1 n − 1 − αr1

+ ∑ j ∈I 1

(rj + 1) (1 − α)(n − 1)E(sj )

n − 1 − αrj

+ ∑ j ∈I1

(n − 1 − rj )E(sj ) + ∑ j ∈I 1

(n − 2 − rj ) E(sj ) + ∑ i �=1

Bi . (A.24)

Plugging (A.24) into (A.23), using (A.12), and simplifying, we obtain:

a ′ 1 =

∑ j ∈I1\{1}

α(n − 1 − rj )E(sj ) + (1 − α)(n − 1)sj n − 1 − αrj

+ ∑ j ∈I 1

E(sj )

+ αm(1 − α)s1 + (n − 1 − αm)(1 − α)s1 + α(n − 1 − r1)E(s1) n − 1 − αr1

+ B1. (A.25)

We denote by V1 the expected payoff of player 1 conditional on signal s1 under the original equilibrium, and by V ′1 his expected payoff conditional on signal s1 when he deviates by lying to players in M (and thus plays action a

′ 1

16. In equilibrium, any message off of the equilibrium path is interpreted as exactly m or m.

1096 REVIEW OF ECONOMIC STUDIES

in the second-stage game). Player 1 does not deviate by lying to players in M if V ′1 − V1 ≤ 0. We have:

V ′ 1 − V1 = (1 − α)E

[ (a1 −

∑ i∈N

si − b1)2 − (a′1 − ∑ i∈N

si − b1)2 | s1 ]

+ α n − 1

(∑ i∈M

E [ (a1 − ai )2 − (a′1 − ai )2 | s1

]

+ ∑

i∈R1\M E [ (a1 − ai )2 − (a′1 − ai )2 | s1

] +

∑ i∈N \(R1∪{1})

E [ (a1 − ai )2 − (a′1 − ai )2 | s1

]⎞⎠ . For the sake of simplicity, we examine separately the elements of the difference V ′1 − V1 and use the following

notation for i �= 1:

zi = ∑

j ∈(I1∩I i )\{1}

(1 − α)(n − 1)(sj − E(sj )) n − 1 − αrj

+ ∑

j ∈(I 1∩Ii )\{1}

(1 − α)(n − 1)(E(sj ) − sj ) n − 1 − αrj

+ B1 − Bi .

Using (A.21), (A.22), and (A.25) and the fact that E[zi | s1] = B1 − Bi : ∑ i∈M

E [ (a1 − ai )2 − (a′1 − ai )2 | s1

] = ∑ i∈M

[ z

2 i −

( zi +

(1 − α)(n − 1 − αm)(s1 − s1) n − 1 − αr1

)2∣∣∣∣∣ s1 ]

= −2 (

(1 − α)(n − 1 − αm)(s1 − s1) n − 1 − αr1

) ∑ i∈M

(B1 − Bi ) − m (

(1 − α)(n − 1 − αm)(s1 − s1) n − 1 − αr1

)2 . (A.26)

∑ i∈R1\M

E [ (a1 − ai )2 − (a′1 − ai )2 | s1

] =

∑ i∈R1\M

[ z

2 i −

( zi −

(1 − α)αm(s1 − s1) n − 1 − αr1

)2∣∣∣∣∣ s1 ]

= 2 (

(1 − α)αm(s1 − s1) n − 1 − αr1

) ∑ i∈R1\M

(B1 − Bi ) − (r1 − m) (

(1 − α)αm(s1 − s1) n − 1 − αr1

)2 . (A.27)

∑ i∈N \(R1∪{1})

E [ (a1 − ai )2 − (a′1 − ai )2 | s1

] =

∑ i∈N \(R1∪{1})

[( zi +

(1 − α)(n − 1)(s1 − E(s1)) n − 1 − αr1

− (

zi + (1 − α)αms1 + (1 − α)(n − 1 − αm)s1 − (1 − α)(n − 1)E(s1)

n − 1 − αr1

)2∣∣∣∣∣ s1 ]

= 2 (

(1 − α)αm(s1 − s1) n − 1 − αr1

) ∑ i∈N \(R1∪{1})

(B1 − Bi ) + (n − 1 − r1) (

(1 − α)(n − 1)(s1 − E(s1)) n − 1 − αr1

− (n − 1 − r1) (

(1 − α)αms1 + (1 − α)(n − 1 − αm)s1 − (1 − α)(n − 1)E(s1) n − 1 − αr1

)2 . (A.28)

In addition, using

a1 − a′1 = (1 − α)αm(s1 − s1)

n − 1 − αr1 ,

and

a1 2 − a′1

2 = (

α(n − r1 − 1)E(s1) + (1 − α)(n − 1)s1 n − 1 − αrl

− (

αm(1 − α)s1 + (n − 1 − αm)(1 − α)s1 + α(n − r1 − 1)E(s1) n − 1 − αr1

+ 2 ⎛ ⎝ ∑

j ∈I1\{1}

α(n − rj − 1)E(sj ) + (1 − α)(n − 1)sj n − 1 − αrj

+ ∑ j ∈I 1

E(sj ) + B1 ⎞ ⎠ ( (1 − α)αm(s1 − s1)

n − 1 − r1

) ,

HAGENBACH & KOESSLER COMMUNICATION NETWORKS 1097

we obtain:

[ (a1 −

∑ i∈N

si − b1)2 − (a′1 − ∑ i∈N

si − b1)2 ∣∣∣∣∣ s1

]

= E [ a1

2 − a′1 2 ∣∣∣ s1] − 2E

⎡ ⎣ (a1 − a′1)( ∑

i∈N \{1} si + s1 + b1)

∣∣∣∣∣∣ s1 ⎤ ⎦

= (

α(n − r1 − 1)E(s1) + (1 − α)(n − 1)s1 n − 1 − αr1

)2 + 2 (B1 − b1 − s1)

( (1 − α)αm(s1 − s1)

n − 1 − αr1

)

− (

αm(1 − α)s1 + (n − 1 − αm)(1 − α)s1 + α(n − r1 − 1)E(s1) n − 1 − αr1

)2 . (A.29)

Next, we plug (A.26), (A.27), (A.28), and (A.29) into V ′1 − V1 and simplify. To simplify the part of the difference V ′1 − V1 that deals with biases, note that:

B1 − Bi = (1 − α)(n − 1)(b1 − bi )

n + α − 1 and B1 − b1 = −α(n − 1)b1 + α

∑ j �=1 bj

n + α − 1 .

Finally, simple but tedious calculus yields:

V ′ 1 − V1 =

2α(1 − α)2(n − 1)(s1 − s1) (n + α − 1)(n − 1 − αr1)

(∑ i∈M

bi − mb1 )

− α(1 − α) 2m(n − 1 − αm)(s1 − s1)2 (n − 1 − αr1)2

Hence, player 1 of type s1 = s1 does not deviate by lying to players in M ⊆ R1 if V ′1 − V1 ≤ 0, i.e.

− (

b1 − ∑

i∈M bi m

) ≤ (n − 1 + α)(n − 1 − αm)

2(n − 1)(n − 1 − αr1) (s1 − s1). (A.30)

Applying the same reasoning, player 1 of type s1 = s1 has no profitable deviation if, for all M ⊆ R1 , the following condition holds:

b1 − ∑

i∈M bi m

≤ (n − 1 + α)(n − 1 − αm) 2(n − 1)(n − 1 − αr1)

(s1 − s1). (A.31)

Condition (4) is obtained from equations (A.30) and (A.31).

A.4. Other proofs

Proof of Corollary 2. We have to show that for every R′′ i

⊆ Ri ∪ {j } we have:∣∣∣∣∣bi − ∑

k∈R′′ i

r′′ i

∣∣∣∣∣ ≤ (n − 1 + α)(n − 1 − αr ′′ i )

2(n − 1)(n − 1 − α(ri + 1)) (si − si ). (A.32)

If j /∈ R′′ i , then equation (4) clearly implies equation (A.32), because the LHS is the same in both inequalities,

but the RHS is larger in equation (A.32). Now, let R′′ i

= R′ i ∪ {j } for some R′

i ⊆ Ri . By equation (7), the LHS of

equation (A.32) is smaller than the LHS of equation (4). Since r′′ i

= r′ i + 1, it remains for us to check that the RHS

of equation (A.32) is larger than the RHS of equation (4), i.e.

n − 1 − α(r′ i + 1)

n − 1 − α(ri + 1) ≥ n − 1 − αr

′ i

n − 1 − αri ⇐⇒ α2(ri − r′i ) ≥ 0, (A.33)

which is satisfied since ri ≥ r′i .

Proof of Corollary 4. Let Ti = Ri ∪ R̃i . For T ′i ⊆ Ti , let R′i ⊆ Ri and R̃′i ⊆ R̃i be such that T ′i = R′i ∪ R̃′i . Since Ri and R̃i do not overlap, we have:∣∣∣∣∣bi −

∑ k∈T ′

i bk

t ′ i

∣∣∣∣∣ ≤ max {∣∣∣∣∣bi −

∑ k∈R′

i bk

r′ i

∣∣∣∣∣ , ∣∣∣∣∣bi −

∑ k∈R̃′

i bk

r̃′ i

∣∣∣∣∣ }

1098 REVIEW OF ECONOMIC STUDIES

Since Ri and R̃i are equilibrium sets of receivers, Theorem 1 implies:∣∣∣∣∣bi − ∑

k∈T ′ i bk

t ′ i

∣∣∣∣∣ ≤ max {

(n − 1 + α)(n − 1 − αr′ i )

2(n − 1)(n − 1 − αri ) (si − si ),

(n − 1 + α)(n − 1 − αr̃′ i )

2(n − 1)(n − 1 − αr̃i ) (si − si )

}

≤ max {

(n − 1 + α)(n − 1 − α(r′ i + r̃i ))

2(n − 1)(n − 1 − α(ri + r̃i )) (si − si ),

(n − 1 + α)(n − 1 − α(r̃′ i + ri ))

2(n − 1)(n − 1 − α(ri + r̃i )) (si − si )

}

≤ (n − 1 + α)(n − 1 − αt ′i)

2(n − 1)(n − 1 − αti ) (si − si ),

where the last inequality comes from r′ i + r̃i ≥ t ′i and r̃′i + ri ≥ t ′i . Hence, by Theorem 1, Ri ∪ R̃i is an equilibrium

set of receivers for player i.

Proof of Corollary 7. The equilibrium conditions for Ri to be a set of receivers for player i can be written as:

max

{∣∣∣∣∣bi − ∑

k∈R′ i bk

∣∣∣∣∣ : |R′i | = x, R′i ⊆ Ri }

≤ (n − 1 + α)(n − 1 − αx) 2(n − 1)(n − 1 − αri )

�, ∀ x = 1, . . . ri .

If j is more central than i, then for every x = 1, . . . ri ,

max

{∣∣∣∣∣bi − ∑

k∈R′ i bk

∣∣∣∣∣ : |R′i | = x, R′i ⊆ Ri }

= max ⎧⎨ ⎩ ∣∣∣∣∣∣bj −

∑ k∈R′

j bk

∣∣∣∣∣∣ : |R′j | = x, R′j ⊆ Rj ⎫⎬ ⎭ .

From rj ≥ ri we also have (n−1+α)(n−1−αx)2(n−1)(n−1−αri ) � ≤ (n−1+α)(n−1−αx) 2(n−1)(n−1−αrj )

� for every x = 1, . . . ri , so we obtain:

max

⎧⎨ ⎩ ∣∣∣∣∣∣bj −

∑ k∈R′

j bk

∣∣∣∣∣∣ : |R′j | = x, R′j ⊆ Rj ⎫⎬ ⎭ ≤ (n − 1 + α)(n − 1 − αx)2(n − 1)(n − 1 − αrj ) �, ∀ x = 1, . . . ri . (A.34)

Next, for every x = ri , . . . rj , we have:

max

⎧⎨ ⎩ ∣∣∣∣∣∣bj −

∑ k∈R′

j bk

∣∣∣∣∣∣ : |R′j | = x, R′j ⊆ Rj ⎫⎬ ⎭ ≤

∣∣∣∣∣bi − ∑

k∈Ri bk ri

∣∣∣∣∣ ≤ (n − 1 + α)2(n − 1) �. Using (n−1+α)2(n−1) � ≤

(n−1+α)(n−1−αx) 2(n−1)(n−1−αrj )

� for every x = ri , . . . rj we obtain:

max

⎧⎨ ⎩ ∣∣∣∣∣∣bj −

∑ k∈R′

j bk

∣∣∣∣∣∣ : |R′j | = x, R′j ⊆ Rj ⎫⎬ ⎭ ≤ (n − 1 + α)(n − 1 − αx)2(n − 1)(n − 1 − αrj ) �, ∀ x = ri , . . . rj . (A.35)

Finally, from Inequalities (A.34) and (A.35) we deduce that Rj is an equilibrium set of receivers for player j .

Acknowledgements. We are deeply indebted to Bruno Biais and three anonymous referees for very helpful comments and suggestions. We also thank Antoni Calvó-Armengol, Ying Chen, Andrew Clark, Olivier Compte, Wouter Dessein, Françoise Forges, Matthew Jackson, Philippe Jehiel, Navin Kartik, David Martimort, Eduardo Perez, Ludovic Renou, Aggey Semenov, Francesco Squintani, Joel Sobel, Jean-Marc Tallon, Jean-Christophe Vergnaud, and seminar participants at Bristol, Cergy-Pontoise, École Polytechnique, Leicester, Lille, Mannheim, Marseille, Montpellier, Paris 1, PSE, Strasbourg, Toulouse, University of California at Santa Barbara (Southwest Economic Theory), the 2007 workshop on strategic communication and networks in Valencia, the 2008 world congress of the Game Theory Society in Northwestern, the 2009 CESifo area conference on applied microeconomics in Munich, and the 9th SAET conference in Ischia for useful comments and discussions. This research was partly carried out while the first author was at University Paris 1.

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