Hierarchical cheap talk∗

Attila Ambrus, Eduardo Azevedo, and Yuichiro Kamada†

September 9, 2009

Abstract

We investigate situations in which agents can only communicateto each other through a chain of intermediators, for example becausethey have to obey institutionalized communication protocols. We as-sume that all involved in the communication are strategic, and mightwant to influence the action taken by the final receiver. The set ofoutcomes that can be induced in pure strategy perfect Bayesian Nashequilibrium is a subset of the equilibrium outcomes that can be in-duced in direct communication, characterized by Crawford and Sobel(1982). Moreover, the set of supportable outcomes in pure equilibriais monotonic in each intermediator’s bias, and the intermediator withthe largest bias serves as a bottleneck for the information flow. On theother hand, there can be mixed strategy equilibria of intermediatedcommunication that ex ante Pareto-dominate all equilibria in directcommunication, as mixing by an intermediator can relax the incentivecompatibility constraints on the sender. We provide a partial char-acterization of mixed strategy equilibria, and show that the order ofintermediators matters with respect to mixed equilibria, as opposedto pure strategy ones.

∗We thank Shih-En Lu, Marco Ottaviani, and seminar participants at Harvard Uni-versity, at the 20th Stony Brook Game Theory Conference, and at the 2009 Far East andSouth Asia Meetings of the Econometric Society for useful comments.

†Ambrus: Department of Economics, Harvard University, Cambridge, MA 02138, e-mail: ambrus@fas.harvard.edu. Azevedo: Department of Economics, Harvard University,Cambridge, MA 02138, e-mail: azevedo@fas.harvard.edu. Kamada: Department of Eco-nomics, Harvard University, Cambridge, MA 02138, e-mail: ykamada@fas.harvard.edu.

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1 Introduction

In many settings, physical, social, or institutional constraints prevent peoplefrom communicating directly. In the US army, companies report to battal-ions, which in turn report to brigades (companies are not allowed to reportdirectly to brigades). Similarly, in many organizations, there is a rigid hi-erarchical structure for communication flow within the organization. Evenwithout explicit regulations, there are time and resource constraints pre-venting all communication to be direct. The managing director of a largecompany cannot give instructions to all workers of the company directly. In-stead, she typically only talks directly to high level managers, who furthercommunicate with lower level managers, who in turn talk to the workers.Finally, in traditional societies, the social network and various conventionsmight prevent direct communication between two members of the society.For example, a man might not be allowed to talk directly to a non-relativewoman; instead, he has to approach the woman’s parents or husband, andask them to transfer a piece of information.There is a line of literature in organizational economics, starting with Sah

and Stiglitz (1986) and Radner (1992), investigating information transmis-sion within organizations.1 However, all the papers in this literature assumehomogeneity of preferences and hence abstract away from strategic issues incommunication. As opposed to this, in this paper we analyze informationtransmission through agents who are strategic and interested in influencingthe outcome of the communication.To achieve this, we extend the classic model of Crawford and Sobel (1982;

from now on CS), to investigate intermediated communication.2 We inves-tigate communication along a given chain: player 1 privately observes therealization of a continuous random variable and sends a message to player

1Radner (1992), Bolton and Dewatripont (1994), and van Zandt (1999) examine orga-nizations in which different pieces of information have to get to the same member, butany member can potentially process a task once having all pieces of information. Sah andStiglitz (1986) and Visser (2000) study the contrast between the performance of hierar-chic and polyarchic organizations in a related setting. Garricano (2000) and Arenas et al.(2008) consider networks in which individuals specialize to solve certain tasks, and it takesa search procedure (through communication among agents) to find the right individual forthe right problem. Cremer et al. (2007) studies nonstrategic communication in differenthierarchical structures. Alonso et al. (2008) and Rantakari (2008) consider strategic com-munication between a head quarter and organizational divisions, but the communicationis direct (not intermediated) between participants. Outside the organizational economicsframework, Acemoglu et al. (2009), and Golub and Jackson (2009) examine nonstrategiccommunication in social networks.

2For a more general class of sender-receiver games than the CS framework, see Greenand Stokey (2007).

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2, who then sends a message to player 3, and so on, until communicationreaches player n. We refer to player 1 as the sender, player n as the receiver,and players 2, .., n − 1 as the intermediators. The receiver, after receivinga message from the last intermediator, chooses an action on the real line,which affects the well-being of all players. We assume that all intermediatorsare strategic, and have preferences from the same class of preferences thatCS consider for senders.First, we consider pure strategy perfect Bayesian Nash equilibria (PBNE)

of such indirect communication games, and show that any outcome that canbe induced in such equilibria can also be induced in the direct communica-tion game between the sender and the receiver (the equilibria of which arecharacterized in CS). This is a general result that applies to general com-munication networks and protocols, not only the hierarchical communicationprotocol along a chain that we focus on in the paper. Hence, if one restrictsattention to pure strategy equilibria, intermediators can only filter out in-formation, as opposed to facilitating more efficient information transmission.We present a simple condition for checking if a given equilibrium outcome ofthe direct communication game can be achieved with a certain chain of inter-mediators. This condition reveals that the order of intermediators does notmatter in pure strategy PBNE, that is, even if the order of intermediators ischanged, the set of pure PBNE does not change. We also show that the set ofpure strategy PBNE outcomes is monotonic in each of the intermediators’ bi-ases: increasing an intermediator’s bias (in absolute value) weakly decreasesthe set of PBNE outcomes. In the standard context of state-independent bi-ases and symmetric loss functions, only the intermediator with the largestbias (in absolute terms) matters: this intermediator becomes a bottleneck ininformation transmission.More surprisingly, we show that when allowing for mixed strategies, there

can be equilibria of the indirect communication game that can strictly im-prove communication (resulting in higher ex ante expected payoff for boththe sender and the receiver) relative to all equilibria of the game with di-rect communication. This has implications for organizational design, as theresult shows that hierarchical communication protocols can increase infor-mation transmission in the organization, if communication is strategic.At the core of this result is the observation first made by Myerson (1991,

p285-288), that noise can improve communication in sender-receiver games.Myerson provides an example with two states of the world in which thereis no informative equilibrium with noiseless communication. However, whenplayer 1 has access to a messenger pigeon that only reaches its target withprobability 1/2, then there is an equilibrium of the game with communica-tion in which the sender sends the pigeon in one state but not the other

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one, and the receiver takes different actions depending on whether the pi-geon arrives or not. Obviously, the same equilibrium can be induced with astrategic intermediator (instead of a noisy communication device) if condi-tional on the first state, the intermediator happens to be exactly indifferentbetween inducing either of the two equilibrium actions. What we show inthe context of the CS model is that such indifferences, which are necessary toinduce strategic intermediators to randomize, can be created endogenouslyin equilibrium, for an open set of environments. The intuition why such in-duced mixing can improve information revelation by the sender is similar tothe one in Myerson (1991) and in Blume et al. (2007): the induced noisecan relax the incentive compatibility constraints on the sender, by makingcertain messages (low messages for a positively biased sender, high messagesfor a negatively biased sender) relatively more attractive.As a motivation for studying such mixed equilibria, we think that the idea

of purification (Harsányi (1973)) is particularly appealing in communicationgames. In particular, one can view mixed equilibria in indirect communica-tion games in which all players have a fixed known bias function as limits ofpure strategy equilibria of communication games in which players’ ex postpreferences have a small random component. This assumption makes themodel more realistic, as it is typically a strong assumption that the bias ofeach player is perfectly known by others.3

We provide a characterization of all mixed strategy equilibria of indirectcommunication games. In particular, we show that there is a positive lowerbound on how close two actions induced in a PBNE can be to each other,which depends on the last intermediator’s bias. This implies that there is afinite upper bound on the number of actions induced in a PBNE. Further-more, we show that all PBNE are outcome-equivalent to an equilibrium inwhich: (i) the state space is partitioned into a finite number of subintervalssuch that in the interior of any interval, the sender sends a pure message; (ii)after any history the sender and the intermediators mix at most between twomessages; (iii) the receiver plays a pure action after any message; and (iv) foreach player except the receiver, the distribution of actions induced by differ-ent equilibrium messages can be ranked with respect to first-order stochasticdominance. We establish additional results for the case of one intermediator,including that the number of distinct messages sent by the sender is exactlyequal to the number of actions induced in equilibrium. By example we showthat, in contrast to pure strategy PBNE, the order of multiple intermedi-ators can matter with respect to the set of possible mixed strategy PBNEoutcomes.

3See an online supplementary note for formally establishing this point.

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We analytically solve for two types of mixed strategy PBNE in the broadlystudied uniform-quadratic specification of the model, for one intermediator,and characterize the set of bias pairs for which these equilibria exist. Weshow that both of these types of equilibria exist in a full-dimensional set ofparameters. As opposed to the case of pure strategies, the set of equilibria isnot monotonic in the intermediator’s bias: For certain specifications of thesender bias, the only PBNE with an intermediator whose bias is close enoughto 0 is babbling, while there are informative equilibria with intermediatorshaving larger biases. However, once the bias of the intermediator is toolarge, again only the babbling equilibrium prevails. We find PBNE involvingnontrivial mixing both when the intermediator is biased in the opposite andwhen she is biased in the same direction as the sender. Interestingly, thelatter requires the intermediator to be strictly more biased than the sender.We conclude the paper by providing a simple sufficient condition for the

existence of an intermediator that can facilitate nontrivial information trans-mission in cases when the only equilibrium in the direct communication gamebetween the sender and the receiver is babbling.Our paper is complementary to several recent papers. Li (2007) and

Ivanov (2009) consider a setting similar to ours, but with only one intermedia-tor, and focusing on the uniform-quadratic specification of the CS framework.Li only considers pure strategy equilibria and concludes that intermediationcannot improve efficiency. Even within the class of pure strategies, Li doesnot characterize all equilibria. Ivanov recognizes the importance of mixingby the intermediator, but asks a different set of questions than our paper.Instead of investigating the set of equilibria of an indirect communicationgame with exogenously given preferences for all players, Ivanov shows thatwhen the intermediator can be freely selected by the receiver, there existsa strategic mediator and a mixed strategy PBNE of the resulting indirectcommunication game in which the receiver’s ex ante payoff is as high as themaximum payoff attainable by any mechanism. This is a sharp welfare im-provement result, but it only applies to the uniform-quadratic specificationof the model. Finally, Galeotti et al. (2009), a working paper concurrentwith the first version of our paper, examines strategic communication ongeneral networks, but in a setting where the state space and therefore com-munication is much simpler than in the CS model: players receive binarysignals. Moreover, Galeotti et al. (2009) restricts attention to equilibria inpure strategies.There are other recent papers investigating the effect of nonstrategic noise

in communication in the CS framework.4 Blume et al. (2007) examine com-

4There is also a line of literature investigating the possibility that the sender’s prefer-

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munication in the CS model with an exogenously specified noise: with acertain probability the receiver gets not the sender’s original message, but arandom message. Although some of the intuition for improved communica-tion is similar to ours, the equilibria achieving welfare improvement are verydifferent than in our model: some sender types are required to mix among allpossible messages. Goltsman et al. (2008) characterize equilibrium payoffs inthe uniform-quadratic specification of the CS framework if players have ac-cess to an impartial (nonstrategic) mediator.5 They also consider stochasticdelegation, that is when the receiver can ex ante commit to a probabilisticmessage-contingent action plan.6

Also related to our paper is Krishna and Morgan (2004), which shows thatthere exist mixed equilibria in multiple rounds of two-sided communicationthat can improve information transmission relative to the best equilibriumwith only one round of communication.7 In such equilibria high and lowtypes of senders pool at some stage of the communication, while intermediatetypes send a separate message. This type of non-monotonicity cannot occurin equilibrium in our setting.Niehaus (2008) considers chains of communication, as in our paper, but in

a setting with no conflict of interest among agents, and hence non-strategiccommunication. Instead, Niehaus assumes an exogenous cost of communica-tion and examines the welfare loss arising from agents not taking into accountpositive externalities generated by communication.Finally, there are multi-sender extensions of the CS framework in which

communication is not intermediated, instead all senders observe some signaland then talk directly to the final decision-maker(s), either simultaneouslyor sequentially: see Gilligan and Krehbiel (1989), Austen-Smith (1993), Kr-ishna and Morgan (2001a, 2001b), Battaglini (2002), Ottaviani and Sørensen(2006a, 2006b), Ambrus and Takahashi (2008), Alonso et al. (2008), and Esoand Fong (2009). In these models payoff-relevant mixing cannot occur on theequilibrium path.

ences are private information, so that from the point of view of the receiver, the sender’saction is a random variable even conditioning on the state, just like in the case of nosiycommunication. See Olszewski (2004), Chen et al. (2006) and Li and Madarász (2008).

5See Forges (1986) and Myerson (1986) for earlier papers on general communicationdevices.

6See also Kovác and Mylovanov (2007).7See also Aumann and Hart (2003) for cheap talk with multiple rounds of communica-

tion.

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2 The model

Here we formally extend the model in CS to chains of communication. Inparticular, we impose the same assumptions for the preferences of all playersas CS for all players involved in the communication chain.We consider the following sequential-move game with n ≥ 3 players. In

stage 1, player 1 (the sender) observes the realization of a random variableθ ∈ Θ = [0, 1], and sends a message m1 ∈ M1 to player 2 (which onlyplayer 2 observes, not the other players). From now on we will refer toθ as the state. The c.d.f. of θ is F (θ), and we assume it has a densityfunction f that is strictly positive and absolutely continuous on [0, 1]. Instage k ∈ 2, ..., n − 1 player k sends a message mk ∈ Mk to player k + 1(which only player k + 1 observes). Note that the message choice of playerk in stage k can be conditional on the message she received in the previousstage from player k− 1 (but not on the messages sent in earlier stages, sinceshe did not observe those). We assume that Mk is a Borel set that hasthe cardinality of the continuum, for every k ∈ 1, ..., n − 1. We refer toplayers 2, ..., n − 1 as intermediators. In stage n of the game, player n (thereceiver) chooses an action y ∈ R. This action choice can be conditional onthe message she received from player n − 1 in stage n − 1 (but none of themessages sent in earlier stages).More formally, the strategy of player 1 is defined by a probability distribu-

tion μ1 on the Borel-measurable subsets of [0, 1]×M1 for which μ1(A×M1) =RA

f for all measurable sets A, the strategy of player k ∈ 2, ..., n − 1 is a

probability distribution μk on the Borel-measurable subsets of Mk−1 ×Mk,and the strategy of player n is a probability distribution μn on the Borel-measurable subsets of Mn−1 ×R. Given the above probability distributions,there exist regular conditional distributions p1(·|θ) for every θ ∈ Θ, andpk(·|mk−1) for every k ∈ 2, ..., n and mk−1 ∈Mk−1.8

The payoff of player k ∈ 1, ..., n is given by uk (θ, y), which we assume tobe twice continuously differentiable and strictly concave in y. Note that themessages m1, ...,mn−1 sent during the game do not enter the payoff functionsdirectly; hence the communication we assume is cheap talk.We assume that for fixed θ, un (θ, y) reaches its maximum value 0 at

yn(θ) = θ, while uk (θ, y) reaches its maximum value 0 at yk(θ) = θ + bk (θ)for some bk(θ) ∈ R. We refer to yk(θ) as the ideal point of player k at stateθ, and to bk (θ) as the bias of agent k at state θ. Note that we normalize thereceiver’s bias to be 0 in every state.As opposed to the original CS game, intermediators in our model might

8See Loeve (1955, p137-138).

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need to condition their messages on a nondegenerate probability distribu-tion over states. For this reason, it will be convenient for us to extendthe definition of a player’s bias from single states to probability distribu-tions over states. Let Ω be the set of probability distributions over Θ. Letbk (μ) = argmax

y∈R

Rθ∈Θ

uk (θ, y) dμ−argmaxy∈R

Rθ∈Θ

un (θ, y) dμ, for every probabil-

ity distribution μ ∈ Ω, and every k ∈ 1, ..., n − 1. In words, bk (μ) is thedifference between the optimal actions of player k and the receiver, condi-tional on belief μ. Note that the term is well-defined, since our assumptionsimply that both

Rθ∈Θ

uk (θ, y) dμ andR

θ∈Θun (θ, y) dμ are strictly concave in y.

We adopt two more assumptions of CS into our context. The first isthe single-crossing condition ∂2uk(θ,y)

∂θ∂y> 0, for every k ∈ 1, ..., n. This in

particular implies that all players in the game would like to induce a higheraction at a higher state. The second one is that either b1(θ) > 0 at everyθ ∈ Θ or b1(θ) < 0 at every θ ∈ Θ, and that either bk(μ) > 0 at everyμ ∈ Ω or bk(μ) < 0 for every k ∈ 2, ..., n − 1 and μ ∈ Ω. In words,players 1, ..., n − 1 have well-defined directions of biases (either positive ornegative). The condition imposed on the sender is the same as in CS, whilethe condition imposed on the intermediators is stronger in that their biaseswith respect to any belief (as opposed to only point beliefs) are required tobe of the same sign.Finally, we assume that all parameters of the model are commonly known

to the players.We refer to the above game as the indirect communication game. Oc-

casionally we will also refer to the direct communication game between thesender and the receiver. This differs from the above game in that there areonly two stages, and two active players. In stage 1 the sender observes therealization of θ ∈ Θ and sends message m1 ∈ M1 to the receiver. In stage 2the receiver chooses an action y ∈ R.The solution concept we use is perfect Bayesian Nash equilibrium (PBNE).

For the formal definition of PBNE we use in our context, see Appendix A.Both in the context of the indirect and the direct communication game, we

will refer to the probability distribution on Θ×R induced by a PBNE strat-egy profile as the outcome induced by the PBNE. Two PBNE are outcome-equivalent if outcomes induced by them are the same.

3 Pure strategy equilibria

Our first result establishes that every pure strategy PBNE in the game ofindirect communication is outcome-equivalent to a PBNE of the direct com-

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munication game between the sender and the receiver. That is, the set ofpossible pure strategy equilibrium outcomes in any indirect communicationgame is a subset of the set of possible equilibrium outcomes in the direct com-munication game obtained by eliminating the intermediators. This makescharacterizing pure strategy PBNE in indirect communication games fairlystraightforward, since the characterization of PBNE in the direct communi-cation game is well-known from CS. In particular, whenever the sender hasa nonzero bias, there is a finite number of distinct equilibrium outcomes.For the formal proofs of all propositions, see Appendix B.

Proposition 1: For every pure strategy PBNE of the indirect commu-nication game, there is an outcome-equivalent PBNE of the direct commu-nication game.

In the Appendix we actually prove the above result for a much largerclass of games than communication chains. We show that in any commu-nication game in which exactly one player (the sender) observes the stateand in which exactly one player (the receiver) takes an action, no matterwhat the communication network and the communication protocol are, theresulting pure strategy PBNE outcomes are a subset of the PBNE outcomesin the direct communication game between the sender and the receiver. Inparticular, at any stage any player can simultaneously communicate to anyfinite number of other players, intermediators can be arranged along an ar-bitrary communication network instead of a line, and there can be multiplerounds of communication between the same players. The intuition behindthe proof for the particular case of communication chains is that in a purestrategy PBNE, every message of the sender induces a message of the finalintermediator and an action by the receiver deterministically. Hence, thesender can effectively choose which action to induce, among the ones thatcan be induced in equilibrium. To put it differently, the conditions for op-timality of strategies for the sender and receiver in a pure strategy profileare essentially the same in any indirect communication game as in the cor-responding direct communication game. In the direct communication gamethey comprise necessary and sufficient conditions for equilibrium, while inan indirect communication game they are only necessary, since optimality ofthe intermediators’ strategies should also be satisfied.Next we give a necessary and sufficient condition for a given PBNE of

the direct communication game to have an outcome-equivalent PBNE in anindirect communication game, and hence we completely characterize the setof pure strategy equilibrium outcomes.Let Θ(y) be the set of states at which the induced outcome is y, for every

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y ∈ Y . Furthermore, for ease of exposition, we introduce the convention thatwheneverΘ(y) is a singleton consisting of state θ0, then

Rθ∈Θ(y)

uk(y, θ)f(θ)dθ ≥Rθ∈Θ(y)

uk(y0, θ)f(θ)dθ iff uk(y, θ0) ≥ uk(y0, θ0), although formally both integrals

above are 0.

Proposition 2: Fix a PBNE of the direct communication game, and letY be the set of actions induced in equilibrium. Then there is an outcome-equivalent PBNE of the indirect communication game if and only ifZ

θ∈Θ(y)

uk(y, θ)f(θ)dθ ≥Z

θ∈Θ(y)

uk(y0, θ)f(θ)dθ (1)

for every y, y0 ∈ Y and k ∈ 2, ..., n− 1.

In words, the condition in the proposition requires that conditional onthe set of states in which a given equilibrium action is induced, none of theintermediators would strictly prefer any of the other equilibrium actions. Theintuition behind the result is straightforward: if conditional on states in whichequilibrium action y is induced, an intermediator strictly prefers a differentequilibrium action y0, then there has to be at least one equilibrium messageafter which the equilibrium strategy prescribes the intermediator inducingy even though given his conditional belief he prefers y0 - a contradiction.The condition in Proposition 2 is convenient, since it can be checked for allintermediators one by one.As in Proposition 1, the “if" part of Proposition 2 can be generalized

to general networks and protocols, as long as there is at least one line ofcommunication actually reaching the receiver. But we note that, as op-posed to this, the “only if" part of Proposition 2 does not generalize in astraightforward manner to general communication networks and protocols.To provide some intuition, consider an example with four players in theuniform-quadratic specification of the model, with biases for players 1-3 be-ing: b1 = 0.1, b2 = 0.8, and b3 = 0.8. The direct communication gamehas a 2-cell partition equilibrium with cutoff point 0.3, implying equilib-rium actions 0.15 and 0.65. The resulting outcome cannot be supported inthe indirect communication games we consider, namely when communicationhappens along a line sequence. In fact, it could not be supported even if onlyone of the above intermediators were in the chain, as the bias of both of themare large enough so that they would prefer to induce action 0.65 instead ofaction 0.15 when the expectation of the state is 0.15. However, suppose nowthat instead of communicating only to one intermediator, player 1 can send a

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message, simultaneously, to both players 2 and 3. After observing these mes-sages, both players 2 and 3 can send a message to player 4, who then takesan action. In short, the sender communicates to the intermediator throughtwo parallel channels. In this game there is an outcome-equivalent PBNEto the two-cell partition equilibrium above. The reason is that when thereceiver receives inconsistent messages from the two intermediators talkingto him, his (out-of-equilibrium) beliefs can be specified in a way that hurtsboth of the intermediators, for example by specifying a point belief at state0, leading to action 0. In general, if for a given equilibrium of the directcommunication game it holds that for every equilibrium action there existsa point in the state space which is, conditionally on the partition cell wherethe equilibrium action belongs to, worse for both intermediators than theequilibrium action, then an outcome-equivalent PBNE can be constructed inthe indirect communication game with the above structure.9

An immediate corollary of Proposition 2 is that the order of intermedi-ators is irrelevant with respect to the set of pure strategy PBNE outcomes,since the necessary and sufficient condition in Proposition 2 is independentof the sequencing of intermediators.Another corollary of the result, stated formally below, is that the set of

equilibrium outcomes is monotonically decreasing in the bias of any inter-mediator. For the intuition behind this, consider an intermediator with apositive bias (the negative bias case is perfectly symmetric). Conditional onthe set of states inducing equilibrium action y, the set of actions that theintermediator strictly prefers to y is an open interval with the left-endpointat y. Moreover, this interval gets strictly larger if the intermediator’s biasincreases, making it less likely that the condition in Proposition 2 holds fora given PBNE of the direct communication game.In order to define increased bias formally in the general specification of

the model, we need to introduce some new notation (the definition is muchsimpler for state-independent biases, see below). Fix two payoff functions,u and v. We say that v is more positively (resp. negatively) biased than u,if there exist affine transformations of u and v, u∗ and v∗ respectively, suchthat

∂v∗(θ, y)

∂y>

∂u∗(θ, y)

∂y

µresp.

∂v∗(θ, y)

∂y<

∂u∗(θ, y)

∂y

¶(2)

for every θ and y.An example of v being more positively biased than u is when v is obtained

by shifting u to the right, that is if there exists δ > 0 such that u(y, θ) =

9A similar construction is used to construct equilibria in Ambrus and Takahashi (2008),in the context of two-sender cheap talk games.

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v(y + δ, θ) for every y and θ.

Proposition 3: Let k ∈ 2, ..., n−1 and fix the preferences of all playersother than k. Let uk be a payoff function implying positive (respectively,negative) bias. If vk is more positively (resp. negatively) biased than uk,then for every pure strategy PBNE of the indirect communication game inwhich player k’s payoff function is vk, there is an outcome-equivalent pure-strategy PBNE of the indirect communication game in which player k’s payofffunction is uk.

The above results simplify considerably for the case where players havestate-independent biases and symmetric loss functions, that is when thereexist b1, ..., bn−1 ∈ R and l : R → R+ with l(0) = 0 such that uk(θ, y) =−l(|y − θ − bk|) for every k ∈ 1, ..., n. In this context, conditional on theset of states that induce an equilibrium action y, the set of actions player k(for k ∈ 2, ..., n−1) strictly prefers to y is (y, y+2bk). Therefore, the condi-tion in Proposition 2 simplifies to |y−y0| ≥ 2|bk| for every two actions y 6= y0

induced in equilibrium, while Proposition 3 simplifies to stating that the setof outcomes that can be supported in pure strategy PBNE is monotonicallydecreasing in |bk|, for every k ∈ 2, ..., n − 1. Note also that if for somek∗ ∈ 2, ..., n − 1, we have |bk∗| ≥ |bk| for every k ∈ 2, ..., n − 1 then|y − y0| ≥ 2|bk∗| implies that |y − y0| ≥ 2|bk| for every k ∈ 2, ..., n− 1, andhence the condition in Proposition 2 holds. This means that only the inter-mediator with the largest absolute value bias matters in determining whichpure strategy PBNE outcomes of the direct communication game can be sup-ported as a PBNE outcome in indirect communication, as the intermediatorbecomes a bottleneck in the strategic transmission flow of information.We conclude the section with a brief discussion of a limit case of our

model when player 1 is unbiased (from the point of view of the receiver),since this case is not considered explicitly in CS. Propositions 1-3 above caneasily modified to cover this case. In particular, the direct communicationgame in this case has an equilibrium with full information revelation, andfor every m ∈ Z++ at least one partition equilibrium with m partition cells(in the uniform-quadratic specification of CS, the unique such equilibriumpartition is when all cells have length 1

m). With intermediators involved, in

pure strategy PBNE only a subset of the above outcomes can be supported,the ones in which induced equilibrium actions are far enough from each other,relative to the biases of the intermediators.

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4 Mixed strategy equilibria

In this section, we analyze mixed strategy PBNE of indirect communicationgames. We show that for some parameter values there can be mixed equi-libria that ex ante Pareto-dominate all PBNE of the direct communicationgame. That is, intermediators might facilitate better information transmis-sion. Moreover, as opposed to pure strategy PBNE, the set of mixed strategyPBNE is a complicated nonmonotonic function of the intermediators’ biases.In Subsection 4.1, we derive general properties of mixed strategy PBNE.

In Subsection 4.2, we provide two classes of examples of mixed strategy PBNEin the uniform-quadratic specification of the model. In Subsection 4.3, weshow that the order of intermediators matters with respect to the set ofpossible PBNE outcomes. Finally, in Subsection 4.4, we provide a conditionfor an intermediator to be able to facilitate information transmission, in caseswhere no information can be transmitted in a direct communication game.

4.1 General properties of mixed equilibria

Below we show that although there might be many different mixed strat-egy PBNE of an indirect communication game, all of them are outcome-equivalent to some equilibrium in which the following properties hold: (i)the state space is partitioned to a finite number of intervals such that in theinterior of each partition cell, player 1 sends the same (pure) message; (ii) thereceiver plays a pure strategy after any message, and all other players mixbetween at most two distinct messages after any history; (iii) the probabilitydistribution over actions that different messages of a player i ∈ 1, ..., n− 1induce can be ordered with respect to first-order stochastic dominance. More-over, there is a finite upper bound on the number of actions that can beinduced in a PBNE of a given indirect communication game.Before we state the above results formally, we first establish that the

assumptions we imposed on the preferences of players imply that for everyintermediator, there exists a positive minimal bias. That is, there exists abelief over states such that the intermediator’s bias is weakly smaller giventhis belief than given any other belief, and this minimal bias is strictly largerthan zero.

Claim 1: There exists bk > 0 such that minμ∈Ω

|bk(μ)| = bk, for every k ∈2, ..., n− 1.

We refer to bk as the minimum absolute bias of player k. Next we showthat in any PBNE, the receiver plays a pure strategy, and any distinct actions

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induced in equilibrium cannot be closer to each other than the minimumabsolute bias of player n − 1. The first result is an immediate consequenceof the strict concavity of the receiver’s payoff in the action choice, implyingthat for any belief, there is a unique payoff-maximizing action. The intuitionbehind the second result is that given that the receiver always chooses theaction maximizing his expected payoff conditional on the message he receives,if two equilibrium actions y and y0 are closer to each other than the minimumabsolute bias of player n − 1, then along the equilibrium path, it has to bethat player n − 1 either sends a message inducing y although conditionalon his beliefs y0 would yield him a higher expected payoff, or the other wayaround.

Proposition 4: After any message, the receiver plays a pure strategy,and if y, y0 ∈ R are two distinct actions that are induced in a PBNE, thenthe distance between them is strictly larger than bn−1.

The result implies that 1bn−1

serves as an upper bound on the number ofdistinct actions that can be induced in a PBNE. Note also that if player n−1has a state-independent loss function and constant bias bn−1 (as assumed inmost of the literature) then Proposition 4 implies that equilibrium actionshave to be at least |bn−1| away from each other, since in this case bn−1(μ) =bn−1 for every μ ∈ Ω.

The next result shows that just like in a direct communication game ala CS, in every PBNE of an indirect communication game, the state spaceis partitioned to a finite number of intervals such that at all states withinthe interior of an interval, the sender sends essentially the same message.Moreover, the distribution of actions induced by equilibrium messages ofeach i ∈ 1, . . . , n − 1 can be ranked with respect to first-order stochasticdominance. To get an intuition for this result, first note that given the strictconcavity of the receiver’s payoff function, given any belief, he has a uniqueoptimal action choice. Therefore, the distributions of actions induced byequilibrium messages of player n− 1 can be trivially ordered with respect tofirst-order stochastic dominance. Then strict concavity of the payoff functionof player n− 1 implies that in equilibrium he can mix between at most twomessages, and that if he mixes between two different messages, then therecannot be a third message inducing an in-between action. This in turn impliesthat the distributions of actions induced by equilibrium messages of playern− 2 can be ranked with respect to first-order stochastic dominance. By aniterative argument we show that this result extends to players n − 3, ..., 1.Then the single-crossing property holding for player 1’s payoff function can

14

be used to establish that the set of states from which player 1 sends a givenequilibrium message form an interval.

Proposition 5: Every PBNE is outcome-equivalent to a PBNE in whichΘ is partitioned into a finite number of intervals such that in the interiorof any interval, player 1 sends the same message, and in which after anyhistory any player i ∈ 1, ..., n − 1 mixes at most between two messages.Moreover, the distribution of outcomes induced by different messages playeri ∈ 1, ..., n− 1 sends in a PBNE can be ranked with respect to first-orderstochastic dominance.

If there is a single intermediator in the game, then some additional struc-ture has to hold for any PBNE. In particular, the state space is partitionedinto components, where play within a component is connected through mix-ing by the intermediator. More precisely, components are the smallest eventssuch that, along the equilibrium path, if the state is in the interior of a givencomponent, then it is common knowledge among players that the state is inthe closure of the component. We will use this result in the next subsectionto derive specific examples of mixed PBNE.

Proposition 6: If n = 3, then for every PBNE there is an outcome-equivalent PBNE, such that Θ can be partitioned into a finite number ofintervals B1, ..., BK , referred below as components, such that for any com-ponent Bk the following hold: (i) The interior of Bk can be partitioned intoa finite number of intervals Ik1 , ..., I

kjksuch that player 1 sends message mj,k

1

with probability 1 at any θ ∈ int(Ikj ) and message mj,k1 is not sent from

any state θ /∈ cl(Ikj ); (ii) If the intermediator is positively biased, then forj ∈ 1, ..., jk − 1 after message mj,k

1 he mixes between messages mj,k2 and

mj+1,k2 , and after message mjk,k

1 he sends message mjk,k2 with probability 1;

(iii) If the intermediator is negatively biased, then for j ∈ 2, ..., jk aftermessage mj,k

1 he mixes between messages mj,k2 and mj−1,k

2 , and after messagem1,k1 he sends message m1,k

2 with probability 1; (iv) The receiver chooses adifferent action after every message sent in equilibrium.

Proposition 6 implies that in a game with a single intermediator, thenumber of distinct messages (those inducing different distributions of actions)sent in equilibrium, both by the sender and by the intermediator, is equal tothe number of actions induced in equilibrium.Another special feature of an indirect communication game with a single

intermediator is that if the intermediator is biased in the same direction asthe sender, and his bias is more moderate than the sender’s, then all PBNE

15

are essentially in pure strategies. The only caveat is that at a finite numberof states, the sender might mix between two distinct messages. Hence, bythe results obtained in the previous section, such an intermediator cannotimprove the efficiency of communication. This point was also made by Ivanov(2009), in the uniform-quadratic specification of the model.10

Proposition 7: Suppose n = 3 and both the sender and the intermedia-tor have positive (respectively, negative) bias. If the sender is more positively(respectively, negatively) biased than the intermediator, then every PBNE isoutcome-equivalent to a PBNE in which players play pure actions at almostevery state.

4.2 Examples of mixed equilibria

Below we investigate two types of single-component mixed PBNE with a sin-gle intermediator, in the frequently studied uniform-quadratic specificationof the model: the one with two equilibrium actions, and the one with threeequilibrium actions. We show that both types of mixed equilibria exist fora full-dimensional set of biases, hence the existence of nondegenerate mixedequilibria is not a knife-edge case in the indirect communication game. Theexamples also demonstrate that, as opposed to pure-strategy PBNE, the out-comes that can be supported in mixed-strategy PBNE are not monotonic inthe magnitude of the intermediator’s bias.Throughout this subsection, we assume that n = 3. Moreover, we assume

that the state is distributed uniformly on [0, 1], and the payoff functions aregiven by:

ui(θ, y) = −(θ + bi − y)2

for every i ∈ 1, 2, 3 with b3 = 0. In words, players have fixed biases andtheir loss functions are quadratic.Note that due to the quadratic payoff, player 3 must play the conditional

expectation of θ given the message she receives. That is,

y = E(θ|m2).

4.2.1 2-action mixed equilibria

From Proposition 6, we know that there is an outcome-equivalent PBNE inwhich (i) player 1 sends messagem1

1 at any θ ∈ [0, x1) and messagem21 at any

θ ∈ (x1, 1] (at state x1, he can choose any mixture of m11 and m

21); (iii) if the

intermediator is positively biased, then after receiving m11, he mixes between

10See Proposition 1 on page 9.

16

m12 and m2

2, and after receiving m21, he sends m

22; (ii) if the intermediator is

negatively biased, then after receiving m11, he sends m

12, and after receiving

m21, he mixes between m1

2 and m22; (iv) the receiver’s action after receiving

m12 is different from her action after receiving m2

2. Figure 1 depicts such amixed PBNE for b1 = 3

10and b2 = − 2

15.

Figure 1

In Appendix A we characterize the region where such equilibrium existand analytically compute equilibrium strategies. Figure 2 illustrates therange of parameter values for which a 2-action mixed equilibrium exists, forb1 > 0. The horizontal axis represents the sender’s bias, while the verticalaxis represents the intermediator’s bias. 2-action mixed PBNE are uniquefor any given pair of biases.The upper triangular region depicts the cases when the sender and the

intermediator are both positively biased and a 2-action mixed PBNE exists.Note that in all these cases the intermediator is more biased than the sender,consistent with Proposition 7. The lower four-sided region represents thecases when the intermediator’s bias is of the opposite sign to the sender anda 2-action mixed PBNE exists. Recall from CS that if b1 ∈ (0.25, 0.5), thenthe only PBNE in the game of direct communication is babbling, while foreach such b1 there is a range of b2 (in the negative domain) such that thereexists a 2-action mixed PBNE.

17

Notice that for any fixed b1, if b2 is small enough in absolute value, thenthere is no 2-action mixed PBNE. Hence, the set of mixed PBNE outcomes,as opposed to pure strategy ones, is not monotonic in the magnitude of theintermediator’s bias.

Figure 2

4.2.2 3-action mixed equilibria

Here, we show that within a component, mixed strategies can have a morecomplicated structure than in a 2-action mixed equilibria above. In particu-lar, there is a region of parameter values for which there is a single componentmixed PBNE with three actions induced in equilibrium.

18

Figure 3

Suppose that b2 > 0. Then Proposition 6 implies that any 3-action mixedPBNE is outcome-equivalent to a PBNE in which (i) if θ ∈ [0, x1) then thesender sends message m1

1, if θ ∈ (x1, x2) then the sender sends message m21,

while if θ ∈ (x2, 1] then the sender sends message m31; (ii) after receiving

m11, the intermediator mixes between m1

2 and m22, after receiving m2

1, theintermediator mixes between m2

2 and m32, and after receiving m

31, the inter-

mediator sends m32; (iii) the receiver takes three different actions given the

three different messages from the intermediator. Figure 3 illustrates such anequilibrium when the sender’s bias is 1/48, while the intermediator’s bias is1/16.Like 2-action mixed PBNE, 3-action mixed PBNE are also unique for any

given pair of biases. But they are much more complicated to solve for than 2-cell equilibria. The reason is that one of the equilibrium conditions requiresplayer 1 to be indifferent at state x1 between two nontrivial lotteries overactions. Still, we are able to characterize 3-action mixed PBNE. In AppendixA we derive a closed form solution for all the variables of interest describing a3-action mixed equilibrium, as a function of x1. The value x1 is the solutionof a complicated cubic equation. The complexity of analytically solving forthis type of equilibrium, together with the result below that such equilibriumdoes exist for a full-dimensional set of parameter values, suggests that a sharpcharacterization of all mixed PBNE outcomes might be infeasible.

19

Figure 4 illustrates the regions of parameter values for which this type ofequilibrium exists, for b1 > 0.

Figure 4

As opposed to 2-action mixed equilibria, 3-action mixed equilibria canexist when both b1 and b2 are close to 0. Hence, the existence of mixedPBNE with k number of actions for a given pair of biases is not monotonicin k.

4.3 The order of intermediators in mixed equilibrium

In the previous section, we showed that the order of intermediators doesnot matter in pure strategy PBNE. We conclude this section with an exam-ple showing that the order of intermediators does matter in mixed strategyPBNE.Consider 4 players, i, j, k, and h, with biases bi = 3

10, bj = 3

10, bk = − 2

15,

and bh = 0, respectively. We consider two indirect communication games: Ingame A, i = 1, j = 2, k = 3, and h = 4. In game B, i = 1, j = 3, k = 2, andh = 4. Notice that in both cases i is the sender and h is the receiver. Theonly difference is the order of the intermediators, j and k.In game A, it is straightforward to see that the outcome that renders

action 310with probability 1 to states [0, 2

15), while actions 3

10and 17

30with

20

probabilities 752and 45

52respectively to states [ 2

15, 1] is a PBNE outcome. To

see this, notice that i and j have exactly the same bias. Thus, given anybelief, if i were j, he would like to induce whatever probability distributionon the state that j wants to induce for player k. Hence, i fully revealing thestate to j is always compatible with PBNE, and analyzing such equilibriais equivalent to analyzing a 3-player indirect communication game, where jis the sender, k is the intermediator, and h is the receiver. In the previoussubsection, we showed that in this game it is a PBNE that player j partitionsthe states into two components, [0, 2

15), in which case k advises h to play 3

10

with probability 1, and [ 215, 1], in which case k advises h to play 3

10with

probability 752and to play 17

30with probability 45

52.

Now we show that this outcome cannot be achieved in any PBNE ofgame B. Suppose the contrary, i.e. that this outcome is generated by somePBNE in game B. Since

¡310+ 17

30

¢/2− 3

10= 2

15, j advises h to play 3

10with

probability one if the expectation of the state is strictly less than 215, and j

advises h to play 1730with probability one if the expectation of the state is

strictly more than 215. If the expectation of the state is 2

15, j is indifferent

between two choices. But in the PBNE in consideration, action 310is taken

with probability 1 conditional on the state lying in [0, 310). Thus, the only

possibility that j can randomize is when the state is exactly at 215. But this

happens with ex ante probability zero, leading to contradiction.The result that the order of intermediators matters in mixed strategy

PBNE is precisely the consequence of the nonmonotonicity of the set ofPBNE in each intermediator’s bias. Without the nonmonotonicity, the ordercannot matter because all the intermediators have the payoff functions inthe same quadratic class. But as we have seen in Figures 2 and 4, the set ofmixed PBNE is not monotonic in the intermediators’ biases with quadraticpayoff functions. What we have seen in the above example is that this non-monotonicity actually causes the possibility that the order matters.

4.4 When can an intermediator facilitate informationtransmission?

In the previous subsections, we presented examples showing that there canbe nontrivial information transmission in an indirect communication game,improving the ex ante welfare of the receiver, even if in the correspondingdirect communication game, all equilibria involve babbling. A natural ques-tion to ask is when this is the case. The following result provides a simplesufficient condition for the existence of an intermediator being able to facili-tate information transmission in equilibrium. We focus on the case when the

21

sender is positively biased (the case of a negatively biased sender is perfectlysymmetric).Let yba = argmax

R baun(θ, y)f(θ)dθ.

Proposition 8: Let b1(θ) > 0 for every θ ∈ Θ. If u1(a, a) < u1(a, y1a) foreach a ∈ [0, 1) and b1(0) < y10, then all PBNE of the direct communicationgame involve babbling, while there exists an intermediator such that in theresulting indirect communication game, there is a PBNE in which the exante payoff of the receiver is higher than in a babbling PBNE.

It is easy to see that the condition u1(a, a) < u1(a, y1a) for each a ∈ [0, 1)is necessary and sufficient for the direct communication game not to haveany informative equilibria. Condition b1(0) < y10 is an easy to check suffi-cient condition for the existence of an informative equilibrium in an indirectcommunication game, for some intermediator.

4.5 Discussion: Ex ante welfare

At the end of the previous section, we provided a sufficient condition for indi-rect communication to be able to improve the ex ante welfare of the receiver,in cases when direct communication cannot facilitate information transmis-sion. Ivanov (2008) establishes a much stronger welfare-improvement resultfor the uniform-quadratic specification of the model: he shows that wheneverb1 ∈ (0, 12), that is whenever there is a mechanism that can improve the exante welfare of the parties, there is a strategic intermediator and a PBNE ofthe resulting indirect communication game which attains the same ex antewelfare as the optimal mechanism.11 The ex ante welfare gains when usingan intermediator can be quite large, as Ivanov points out.For general preferences and prior distributions on the state, welfare com-

parisons between pure and mixed PBNE in indirect communication games isa hard problem. One difficulty is that unlike in the uniform-quadratic case,the interests of the sender and the receiver are not aligned anymore ex ante.Below we demonstrate this through an example of a mixed PBNE in whichex ante the sender is strictly worse off than in the babbling equilibrium, thatis in the worst pure strategy PBNE.Consider again the example in Figure 1, with the only modification that

the sender’s payoff function is now u1(θ, y) = −(θ+ 310−y)2r, where r ∈ Z. By

symmetry, the very same construction of the strategy profile as in Figure 1constitutes a PBNE. It can be shown that, whenever r ≥ 3, the mixed PBNE11The benchmark that is the maximum welfare that can be attained through a mecha-

nism was derived in the uniform-quadratic case by Goltsman et al. (2007).

22

gives a strictly lower payoff to the sender than in the babbling PBNE. This isbecause in high states, in the mixed PBNE, there is a significant chance thatthe induced action is very far from the ideal point of the sender, relative tothe maximal distance between the sender’s ideal point and the action inducedin babbling equilibrium. If the sender is risk-averse enough, this makes hisover all ex ante welfare worse than in the babbling equilibrium. Hence, in anindirect communication game, the sender can be worse off than in the worstPBNE of the corresponding direct communication game. By contrast, it iseasy to see that the receiver can never be worse off in any PBNE than in ababbling equilibrium: one strategy that is always feasible for the receiver ischoosing the babbling action independently of the messages received.

5 Conclusion

Our analysis of intermediated communication yields simple implications fororganizational design if one restricts attention to pure-strategy equilibria:intermediators cannot facilitate transmission of information that cannot betransmitted in equilibrium in direct communication between a sender and areceiver, but they can invalidate informative equilibria of direct communica-tion. The information loss relative to direct communication is smaller theless intermediators are involved in the chain, and the less biased they arerelative to the receiver. We also show that the order of intermediators doesnot matter for what information can be transmitted through the chain.At the same time, our findings reveal that the implications are much

more complex with respect to mixed strategy equilibria. Different types ofnontrivial mixed equilibria exist for an open set of parameter values of themodel, and the existence of a given type of equilibrium is nonmonotonic inthe intermediators’ biases. By introducing noise in the information trans-mission, intermediators in a mixed strategy equilibrium can improve infor-mation transmission relative to direct communication. This can provide arationale for establishing hierarchical communication protocols in an organi-zation, even if such protocols are not necessitated by capacity constraints.Our investigations in the uniform-quadratic specification of the model sug-gest that involving an intermediator can improve information transmission ifthe intermediator’s bias (relative to the receiver) is more moderate than thesender’s, and it is in the opposite direction.

23

6 Appendix A

6.1 Formal definition of perfect Bayesian Nash equi-librium

In order to define PBNE formally in our context, we need to introduce beliefsof different players at different histories. We define a collection of beliefsthrough a probability distribution βk on the Borel-measurable subsets ofMk−1 × Ω for every k ∈ 2, ..., n, as a collection of regular conditionaldistributions βk(mk−1) for every mk−1 ∈ Mk−1 and k = 2, ..., n that areconsistent with the above probability distributions.

Definition: A strategy profile (pk())k=1,...,n and a collection of beliefs(βk())k=2,...,n constitute a PBNE if:(i) [optimality of strategies given beliefs]For every θ ∈ Θ and m1 ∈ supp(p1(·|θ)), we have:

m1 ∈ argmaxm01∈M1Z

m2∈M2

...

Zmn−1∈Mn−1

Zy∈R

u1(θ, y)dpn(y|mn−1)dpn−1(mn−1|mn−2)...dp

2(m2|m01).

For every k ∈ 2, ..., n − 1, mk−1 ∈ Mk−1 and mk ∈ supp(pk(·|mk−1)),we have:

mk ∈ argmaxm0k∈MkZ

θ∈Θ

E(uk(θ, y)|m0k)dβ

k(θ|mk−1)

where

E(uk(θ, y)|m0k) =Z

mk+1∈Mk+1

...

Zmn−1∈Mn−1

Zy∈R

uk(θ, y)dpn(y|mn−1)dpn−1(mn−1|mn−2)...dp

k+1(mk+1|m0k).

24

And for every mn−1 ∈Mn−1 and y ∈ supp(pn(·|mn−1)), we have:

y ∈ argmaxy0∈RZ

θ∈Θ

un(θ, y0)dβn(θ|mn−1).

(ii) [consistency of beliefs with actions]βk() constitutes a conditional distribution of the probability distribution

on Θ×Mk−1 generated by strategies p1(), ..., pk−1(), for every k ∈ 2, ..., n.(iii) [consistency of beliefs across players]For any k ∈ 2, ..., n − 1, if mk ∈ Mk is sent along some path of

play consistent with (pk())k=1,...,n, then βk+1(mk) is in co(βk(mk−1)|mk−1 ∈cMk−1(mk)), where cMk−1(mk) is the set of messages mk−1 inMk−1 such thatthere is a path of play consistent with (pk())k=1,...,n in which player k − 1sends message mk−1 and player k sends message mk. Similarly, if m1 ∈ M1

is sent along some path of play consistent with (pk())k=1,...,n then β2(m1) isin co(β1(θ)|θ ∈ bΘ(m1)), where bΘ(m1) is the set of states at which player1 sends m1.

We note that condition (iii) trivially follows from condition (ii) for strat-egy profiles in which any message sent along the path of play is sent withpositive probability. Condition (iii) establishes a consistency of beliefs acrossplayers along equilibrium message chains that are sent with probability 0, forexample when the message chain is induced at exactly one state.

6.2 Complete characterization of 2-action and 3-actionsingle-component equilibria in the uniform-quadraticcase

To simplify notation, we will label each message such that

mjk = E(θ|mk = mj

k).

With this notation we have that player 3’s strategy is just y = m2, while theset of messages sent by player 1 correspond to the midpoints of the partitioncells of the given equilibrium.We let ∆ = b2 − b1.

25

6.2.1 2-action mixed equilibria

Without loss of generality, assume that b2 < 0 (the case of b2 > 0 is perfectlysymmetric). It is convenient to do an analysis with a fixed signed b2, becausethe sign of b2 determines after which message(s) player 2 mixes in a giventype of equilibrium. Notice that the example in Figure 1 corresponds to thecase of b2 < 0. When we draw Figure 2, we obtained the region with b1 > 0and b2 > 0 by rotating the region with b1 < 0 and b2 < 0 in a point-symmetricmanner with respect to the origin.By Bayes’ rule, m1 only depends on x1: we must have m1

1 = x1/2 andm21 =

1+x12. Also by Bayes’ rule, m2

2 = m21 =

1+x12. Because player 2 must

be indifferent between her messages after receiving m21, we must also have

m12 = m2

2 + 2b2 = 1+x1

2+ 2b2. Then, for player 1 to be indifferent between

both messages in state x1 we must have

x1 = 1 + 2∆.

So it must be that −12≤ ∆ ≤ 0. Substituting this value of x1 we can

solve for the messages m11 = ∆+ 1

2, m1

2 = 1+∆+2b2, and m21 = m2

2 = 1+∆.For the probability p(m1

2|m21), which we denote simply by p, by Bayes’ rule

we have

p =1

8· (1 + 4b

2)(1 + 2∆)

b2∆.

For this to be feasible, p ≥ 0, so −14≤ b2 (as 1 + 2∆ = x1 has to be

nonnegative). From p ≤ 1 we get ∆ ≤ −2b2 − 12. It is trivial to check that

these conditions together with the condition 0 ≤ x1 ≤ 1 are also sufficient forequilibrium. In terms of b1 and b2 the constraints become max−1

4, b1− 1

2 ≤

b2 ≤ 13b1 − 1

6.

6.2.2 3-action mixed equilibria

Without loss of generality, assume that b2 > 0 (the case of b2 < 0 is perfectlysymmetric). Notice that the example in Figure 3 corresponds to the case ofb2 < 0. Similarly to what we did in Figure 2, when we draw Figure 4, weobtained the region with b1 > 0 and b2 < 0 by rotating the region with b1 < 0and b2 > 0 in a point-symmetric manner with respect to the origin.As in the case of 2-action mixed PBNE, the messages sent in equilibrium

by player 1 are determined by x1 and x2: mj1 = (xj−1 + xj)/2 for every

j ∈ 1, 2, 3 where we let x0 = 0 and x3 = 1. By Bayes’ rulem11 = m1

2 = x1/2.Using player 2’s indifferences between messages in which she mixes, we getthat m2

2 = x1/2 + 2b2, m3

2 = x1/2 + x2.

26

Player 1’s indifference, when the state is x2, is equivalent to

x2 = x1 + 2∆.

Denote the probabilities p(mj+12 |mj

1) by pj. From Bayes’ rule applied tom32, we get

p2 =(1− x2) (1− x2 − x1)

x2∆.

And using Bayes’ rule for m22 we get

p1 =1

4

(x2 − 4b2) (2x2 − 1) (1− x1)

x1x2b2.

These equations determine the equilibrium in terms of x1. Now, to actu-ally calculate x1, it is necessary to work with player 1’s indifferences betweentwo nontrivial lotteries.Assuming that 0 ≤ x1 ≤ x2 we must have that

• p2 ≥ 0 iff (x1 + x2)/2 = x1 +∆ ≤ 1/2.

• p2 ≤ 1 iff x2 = x1 + 2∆ ≥ 1/2.Notice that (assuming x2 ≥ 1/2, which follows from p2 ≤ 1)• p1 ≥ 0 iff x2 ≥ 4b2, or b2 ≤ x2/4 = x1/4 +∆/2.

• p1 ≤ 1 iff2x31 +

¡−3 + 4 b2 − 8 b1

¢x21 +

¡2 b2 + 1 + 10 b1 − 8 b1 b2 + 8 (b1)2

¢x1

(3)

+8 (b2)2 − 2 b2 − 2 b1 − 8 (b1)2 ≥ 0

These equations are complicated, but we will simplify them below.

The final equation we need is player 1’s indifference constraint when hertype is x1. This reduces to

6x31 +¡12 b2 − 9− 24 b1

¢x21 +

¡−24 b1 b2 + 3 + 18 b1 − 6 b2 + 24 (b1)2

¢x

(4)

+8 (b2)2 − 8 (b1)2 − 2 b2 − 2 b1 = 0Unfortunately, the closed form Cardano solution of this equation is very

complicated and not very helpful. But we may use it to simplify the conditionthat p1 ≤ 1 to

(3x1 + 4∆− 1)Σ ≥ 0,where Σ = b1 + b2. Assuming p2 ≤ 1, this reduces to

27

• p2 ≤ 1 iff Σ ≥ 0.

Summing up, there is a solution iff we can find x1 solving equation (4)with:

• 1/2− 2∆ ≤ x1 ≤ 1/2−∆ from 0 ≤ p2 ≤ 1.

• 0 ≤ 2Σ ≤ x1 from from p1 ≤ 1 and p1 ≥ 0.

Note that these imply 0 ≤ x1 ≤ x1 + 2∆ = x2 ≤ 1.Now we have to consider two cases.

28

Case 1: b2 ≤ 1/8

In this case, the binding constraints are:

• 1/2− 2∆ ≤ x1 ≤ 1/2−∆ from 0 ≤ p2 ≤ 1.

• 0 ≤ Σ from from p1 ≤ 1.

Notice that we must have ∆ ≤ 1/4.Let f(x1) be the function defined by the left-hand side of (4). We have

f(1/2− 2∆) = (1− 4∆)(2∆− b2)

f 0(1/2− 2∆) = (6 + 24b2)∆− 3/2 ≤ 0and

f(1/2−∆) = −12(1 + 2∆)(1− 6∆)Σ

f 0(1/2−∆) = −32(1− 2∆)2 ≤ 0.

First notice that the leading coefficient of the cubic f(x) is positive. Be-cause f 0 is negative in both endpoints of the interval, there is no solution inthe interval if f has the same sign in the extremes. So there are two possiblecases:(i) 2∆ − b2 ≤ 0 and 1 − 6∆ ≤ 0. This leads to a contradiction, as

1/6 ≤ ∆ ≤ b2/2 ≤ 1/16.(ii) The other case is 2∆− b2 ≥ 0 and 1− 6∆ ≥ 0. There are equilibria

in this case iff:

• 0 ≤ ∆ ≤ 1/6

• b2 ≤ 1/8

• Σ ≥ 0

• 2∆ ≥ b2 or b1 ≤ b2/2 (notice that this implies 0 ≤ ∆).

29

Case 2: b2 ≥ 1/8

In this case, the binding constraints are:

• 0 ≤ 2Σ ≤ x1 ≤ 1/2−∆.

Note that the binding ones are b2 ≥ 1/8, b1 + b2 ≥ 0 and 3b2 + b1 ≤ 1/2.And this implies that b2 ≤ 1/4 and −1/4 ≤ b1 ≤ 1/8. Also, ∆ ≤ 1/2 andΣ ≤ 1/4. We have

f(2(b1 + b2)) = 4(1 + 24(b2)2 − 10b2 − 2b1)Σ

f 0(2Σ) = 3(1− 4b2)(1− 10b2 − 6b1) < 0.If ∆ ≤ 1/6, then the right endpoint is negative, as we have seen in the

analysis of Case 1. So there will be solutions in the area that satisfies theadditional constraints

• ∆ ≤ 1/6

• 1 + 24(b2)2 − 10b2 − 2b1 ≥ 0

If ∆ ≥ 1/6, f is positive on the right endpoint. It can be shown that theleft endpoint has positive f , so that there can be no solutions.Summarizing, there are two regions where the equilibria under consider-

ation exist. These are:

• The region corresponding to Case 1:

1. b2 ≤ b1 + 1/6.

2. b2 ≤ 1/8.

3. b2 ≥ −b1.

4. b2 ≥ 2b1.

• The region corresponding to Case 2:

1. ∆ ≤ 1/6.

2. 1 + 24(b2)2 − 10b2 − 2b1 ≥ 0.

3. b2 ≥ 1/8.

These regions are depicted in Figure 4, with the point symmetric rotationdiscussed at the beginning of this subsection.

30

7 Appendix B: Proofs

Proof of Proposition 1:We will show the following stronger result:Consider a (k + 2)-stage generalized communication game with players

N ≡ 1, ..., n in which (i) At stage 1 player 1 observes the realization ofθ; (ii) At stage j = 2, ..., k + 1 any player i sends a message simultaneouslyto players N j

i ⊆ N ; (iii) At stage k + 2 player n chooses action y. Thepreferences and information of players satisfy the same assumptions as madein Section 2. Then any pure strategy PBNE of the generalized communicationgame is outcome-equivalent to a PBNE of the direct communication gamebetween player 1 and player n.First, note that generalized communication games indeed encompass the

set of communication games defined in Section 2: they correspond to gen-eralized communication games in which N j

j−1 = j and N ji = ∅ for every

j ∈ 2, ..., k + 1 and i ∈ N \ j − 1.Fix a pure strategy PBNE of the generalized communication game. Let

Sn be the set of possible message sequences that player n can receive duringthe game in this equilibrium. For any s ∈ Sn, let Θ(s) be the set of states atwhich player 1 sends messages that induce message sequence s for player n.Note that Θ(s) is well-defined, since in a pure strategy profile any sequence ofactions by player 1 deterministically lead to action sequences of other players.Construct now the following strategy profile in the direct communication

game: choose a distinct m1(s) ∈ M1 for every s ∈ Sn, and let the sendersend message m1(s) at every θ ∈ Θ(s). Furthermore, let the action choice ofthe receiver afterm1(s) be the same as her action choice after s in the PBNEof the generalized communication game, for every s ∈ Sn. After any othermessage m1 ∈ M1 (which are not associated with any s ∈ Sn), assume thatthe receiver chooses one of the actions along the above-defined play path.To show that this is a PBNE, first we point out that given the receiver’s

strategy, the sender does not have a profitable deviation at any state. Thisis because in the given profile, at any state, the sender can induce the sameaction choices as she can in the PBNE of the indirect communication game.Second, the receiver gets equilibrium messagem1(s) in the above direct com-munication profile at exactly the same states as she receives message sequences in the PBNE of the indirect communication game. Hence, after any mes-sage sent along the induced play path, the action prescribed for the receiveris sequentially rational, given the updated belief of the receiver regarding thestate after receiving m1(s). Finally, conditional on any off-path messages bythe sender, the receiver can have arbitrary beliefs in PBNE, so her choicefrom the actions that are used on the path of the play can be sequentially

31

rational. ¥

Proof of Proposition 2:(“If" part)Supposing that (1) holds, we construct a PBNE of an indirect commu-

nication game that is outcome-equivalent to the original equilibrium in thedirect communication game: For each y and k, choose exactly one messagefromMk,mk(y), so thatmk(y) 6= mk(y

0) if y 6= y0, where y, y0 ∈ Y . Let player1 send message m1(y) conditional on Θ(y) and let player k ∈ 2, . . . , n− 1send message mk(y) conditional on player k − 1’s message mk−1(y). In theoff-path event that player k − 1 sends a message not in

Sy∈Y mk−1(y), let

player k send an arbitrary message inS

y∈Y mk(y). Finally, let player ntake an action y conditional on player n − 1’s message mn−1(y). Again, inthe off-path event that player n− 1 sends a message not in

Sy∈Y mn−1(y),

let player n take an arbitrary action in Y .Note that players 1 and n do not have an incentive to deviate because

the constructed strategy profile specifies the same correspondence of mes-sages/actions to states as in the original PBNE of the direct communicationgame. Moreover, condition (1) implies that players k ∈ 2, . . . , n−1 do nothave an incentive to deviate, given the beliefs induced by the strategy profiledescribed above. This concludes that the strategy profile constructed aboveconstitutes a PBNE.(“Only if" part)Let Mk−1(y) be the set of messages of player k− 1 along the equilibrium

path that induce player k to send a message fromMk that eventually inducesy. Let Θ(mk−1) be the set of states at which message mk−1 is sent by playerk− 1, for every mk−1 ∈Mk−1(y). By optimality of strategies given beliefs inPBNE (see Appendix A for the formal definition of PBNE),Z

θ∈Θ

uk(y, θ)βk(θ|mk−1)dθ ≥Z

θ∈Θ

uk(y0, θ)βk(θ|mk−1)dθ. (5)

By consistency of beliefs with actions in PBNE, βk() constitutes a condi-tional distribution of the probability distribution on Θ×Mk−1 generated bythe PBNE strategies, which together with (5) impliesZ

θ∈Θ(y)

uk(y, θ)f(θ)dθ ≥Z

θ∈Θ(y)

uk(y0, θ)f(θ)dθ. ¥

Proof of Proposition 3: We will provide a proof for the case of positivebiases. The case of negative biases is perfectly symmetric.

32

Let Gu and Gv stand for the games where the payoff function of player kis uk and vk, respectively. Let s∗ constitute a PBNE Gv. Then Proposition 1implies that Θ(y) is an interval (possibly degenerate) for every y ∈ Y , whereY is the set of actions induced by s∗. Proposition 2 implies that there is anoutcome-equivalent PBNE to s∗ in Gu iffZ

θ∈Θ(y)

uk(y, θ)f(θ)dθ ≥Z

θ∈Θ(y)

uk(y0, θ)f(θ)dθ (6)

for every y, y0 ∈ Y (recall our convention for the above inequality if Θ(y) is asingleton). Also by Proposition 2, since s∗ constitutes a PBNE Gv, we have:Z

θ∈Θ(y)

vk(y, θ)f(θ)dθ ≥Z

θ∈Θ(y)

vk(y0, θ)f(θ)dθ (7)

for every y, y0 ∈ Y.Fix now y, y0 ∈ Y. Since uk implies positive bias, (6) holds trivially if

y0 < y. Suppose now that y0 > y.Since vk is more positively biased than uk, condition (2), together with

f(θ) > 0 for every θ ∈ Θ, implies there exists affine transformations of vk

and uk, vk∗ and uk∗ respectively, such that:

∂Rθ∈Θ(y) v

k∗(θ, by)f(θ)dθ∂by >

∂Rθ∈Θ(y) u

k∗(θ, by)f(θ)dθ∂by (8)

for all by ∈ [y, y0]. This implies Rθ∈Θ(y)

vk∗(y, θ)f(θ)dθ −R

θ∈Θ(y)vk∗(y0, θ)f(θ)dθ

is strictly larger thanR

θ∈Θ(y)uk∗(y, θ)f(θ)dθ−

Rθ∈Θ(y)

uk∗(y0, θ)f(θ)dθ. Then (7)

implies (6). ¥

Proof of Claim 1: Without loss of generality, assume that player khas a positive bias (the negative bias case is perfectly symmetric). Thenby assumption bk(μ) > 0 for every μ ∈ Ω, that is argmax

y∈R

Rθ∈Θ

uk(θ, y)dμ >

argmaxy∈R

Rθ∈Θ

un(θ, y)dμ for every μ ∈ Ω. Since uk and un are continuous in

y and θ and,R

θ∈Θuk(θ, y)dμ and

Rθ∈Θ

un(θ, y)dμ are continuous in y and in

μ (the latter with respect to the weak topology), argmaxy∈R

Rθ∈Θ

uk(θ, y)dμ −

argmaxy∈R

Rθ∈Θ

un(θ, y)dμ is continuous in μ. Moreover, Ω is compact, hence

there are μ ∈ Ω and bk > 0 such that

33

argmaxy∈R

Zθ∈Θ

uk(θ, y)dμ− argmaxy∈R

Zθ∈Θ

un(θ, y)dμ ≥

argmaxy∈R

Zθ∈Θ

uk(θ, y)dμ− argmaxy∈R

Zθ∈Θ

un(θ, y)dμ = bk.

¥

Proof of Proposition 4: In PBNE, after any messagemn−1 ∈Mn−1, thereceiver plays a best response to belief βn(mn−1). Since the receiver’s payoffis strictly concave in y and takes its maximum in [0, 1] for every θ ∈ Θ, theexpected payoff is strictly concave in y and takes its maximum in [0, 1] forany belief. Hence, there is a unique best response action in [0, 1] for thereceiver for βn(mn−1).Fix now a PBNE, let mn−1 ∈ Mn−1 be a message sent in equilibrium,

and let y(mn−1) be the action chosen by the receiver after receiving messagemn−1. Without loss of generality, assume that player n − 1 has a positivebias (the negative bias case is perfectly symmetric). Note that by our de-finition of PBNE, βn(mn−1) is a convex combination of beliefs βn−1(mn−2)for which mn−1 ∈ supp(pn−1(mn−2)). It cannot be that for every such beliefβn−1(mn−2),

argmaxy∈R

Zθ∈Θ

un(θ, y)dβn−1(mn−2) < y(mn−1),

since this would imply that argmaxy∈R

Rθ∈Θ

un(θ, y)dβn(mn−1) < y(mn−1), con-

tradicting that y(mn−1) is an optimal choice for player n after receivingmn−1.Therefore, there ismn−2 ∈Mn−2 such that argmax

y∈R

Rθ∈Θ

un(θ, y)dβn−1(mn−2) ≥

y(mn−1). By Claim 1,

argmaxy∈R

Zθ∈Θ

un−1(θ, y)dβn−1(mn−2) >

argmaxy∈R

Zθ∈Θ

un(θ, y)dβn−1(mn−2) + bn−1 ≥ y(mn−1) + bn−1.

Thus, given belief βn−1(mn−2), player n− 1 strictly prefers inducing any ac-tion from (y(mn−1), y(mn−1) + bn−1] to inducing y(mn−1). Therefore, therecannot be anymessagem0

n−1 ∈Mn−1 that induces an action from (y(mn−1), y(mn−1)+

34

bn−1], since this would contradict the optimality of mn−1 given mn−2. Thisimplies that the distance between any two equilibrium actions has to bestrictly greater than bn−1. ¥

Proof of Proposition 5: Fix a PBNE, and consider an outcome-equivalent PBNE in which if two messages mi,m

0i ∈ Mi used in equilibrium

induce the same probability distribution over actions, then mi = m0i, for

every i ∈ 1, ..., n− 1.For ease of exposition, if the probability distribution over actions induced

by mi ∈ Mi first-order stochastically dominates that of m0i ∈ Mi for i ∈

1, ..., n− 1, then we will simply say that mi is higher than m0i.

Proposition 4 implies that every mn−1 induces a pure action by the re-ceiver. Since different equilibrium messages induce different actions, it triv-ially holds that the distribution of outcomes that different messages thatplayer n − 1 sends in PBNE can be ranked with respect to first-order sto-chastic dominance. Moreover, since un−1(θ, y) is strictly concave in y forevery θ ∈ Θ, there can be at most two optimal messages for player n−1, andin this case they have to induce actions such that there is no other equilib-rium action in between them (otherwise inducing the latter action would bestrictly better than inducing the originally considered actions). This estab-lishes that the distribution of actions induced by the equilibrium messages ofplayer n− 2 can be ranked with respect to first-order stochastic dominance:they can be either degenerate distributions corresponding to one of the fi-nite number of actions induced in equilibrium (which in turn corresponds toone of the equilibrium messages of player n − 1), or mixtures between twoneighboring equilibrium actions (which correspond to mixtures between twoequilibriummessages of player n−1). Hence, we can partition the equilibriummessages of player n− 2 into a finite number of sets Sn−2

1 , ..., Sn−2kn−2

such thateach set consists of messages inducing a distribution of actions with the samesupport, and the distribution of actions induced by messages in a set with ahigher index first-order stochastically dominates the distribution of actionsinduced by messages in a set with a lower index. Moreover, the distributionof outcomes induced by messages within set Sn−2

j , for any j ∈ 1, ..., kn−2,can be ranked with respect to first-order stochastic dominance, too.We will now make an inductive argument. Suppose that for some l ∈

2, ..., n − 2, it holds that the equilibrium messages of every player l0 ∈l, ..., n − 2 can be partitioned into a finite number of sets Sl0

1 , ..., Sl0kl0such

that each set consists of messages inducing a distribution of actions with thesame support, the distribution of actions induced by messages in a set witha higher index first-order stochastically dominates the distribution of actionsinduced by messages in a set with a lower index, and the distributions of

35

outcomes induced by messages in each set can be ordered with respect tofirst-order stochastic dominance. Let ml0

j and ml0j stand for the highest and

lowest message from Sl0j , whenever it exists.

Given thatR

θ∈Θul(θ, y)dβ is strictly concave in y for any belief β, at any

history in which player l moves, the set of optimal messages for player lis either: (i) all elements of Sl

j for some j ∈ 1, ..., kl; (ii) mlj for some

j ∈ 1, ..., kl; (iii) mlj for some j ∈ 1, ..., kl; or (iv) ml

j and mlj+1 for some

j ∈ 1, ..., kl − 1. Therefore, any equilibrium message of player l − 1 canbe partitioned into a finite number of sets Sl−1

1 , ..., Sl−1kl−1

such that each setconsists of messages inducing a distribution of actions with the same support,the distribution of actions induced by messages in a set with a higher indexfirst-order stochastically dominates the distribution of actions induced bymessages in a set with a lower index, and the distributions of outcomesinduced by messages in each set can be ordered with respect to first-orderstochastic dominance. By induction, for every l0 ∈ 1, ..., n−2, it holds thatthe equilibriummessages of player l0 can be partitioned into a finite number ofsets Sl0

1 , ..., Sl0kl0such that each set consists of messages inducing a distribution

of messages of player l0+1 with the same support, the distribution of actionsinduced by messages in a set with a higher index first-order stochasticallydominates the distribution of actions induced by messages in a set with alower index, and the distributions of outcomes induced by messages in eachset can be ordered with respect to first-order stochastic dominance.Given the above result, the single-crossing condition imposed on u1 im-

plies that there is a finite set of states θ0,θ1, ..., θt+1 such that θ0 = 0, θt+1 = 1,and θ, θ0 ∈ (θj, θj+1) implies that there is a message m1

j that is uniquely op-timal for player 1 at both θ and θ0, for every j ∈ 0, ..., t.Next we show that the distribution of states conditional on equilibrium

messages can also be ordered with respect to first-order stochastic dominance.Above we established that this holds for messages of player 1. Suppose nowthat the statement hold for players 1, .., l, where l ∈ 1, ..., n − 2. Letml+1 be a message that player l + 1 sends after receiving message ml fromplayer l. Then by the single-crossing condition on preferences, player l + 1cannot send a higher (similarly, lower) message than ml+1 when receivinga message eml conditional on which the distribution of states is first-orderstochastically dominated by (similarly, first-order stochastically dominates)the distribution of states conditional onml. This implies that the distributionof states conditional on equilibrium messages of player l + 1 can be orderedwith respect to first-order stochastic dominance.Finally, we show that in the equilibrium at hand, any player 1, ..., n − 1

can mix between at most two messages. We already showed this for player

36

n− 1. Furthermore, we showed that player 1 can only mix between differentmessages at states θ0,θ1, ..., θt+1. Moreover it is clear that if for j ∈ 1, ..., t itholds that player 1 mixes between distinct messages at θj then the messageshave to be m1

j−1 and m1j . Similarly, if at θ0 = 0 (respectively, at θt+1 =

1) player 1 mixes between distinct messages then it has to be between m11

(respectively, m1t ) and a message associated with θ = 0 (respectively, θ = 1).

This concludes that player 1 mixes between at most two distinct messages atany state. Consider now any player l ∈ 2, ..., n− 2. Above we establishedthat the distribution of states conditional on equilibrium messages of playerl − 1 can be ordered with respect to first-order stochastic dominance. Thisimplies that the support of messages of player l+1 induced by two differentequilibrium messages of player l cannot be the same. But in that case strictquasi-concavity of player l’s utility function implies that player l cannot havemore than two optimal messages to send, at any history. ¥

Proof of Proposition 6: Given a PBNE, construct an outcome-equivalentPBNE as we did in the proof of Proposition 5. This allows us to have a parti-tion of Θ, P = (P1, . . . , PJ). Parts (i) and (iv) are immediate from the proofof Proposition 5 and from Proposition 4, respectively. For now, set m1 = jif θ ∈ Pj, and m2 = y if p3(y|m2) > 0 (so y is given probability 1), withoutloss of generality.Let partition B be such that each cell Bk is minimal with the property

that, if Pj ⊆ Bk and p2(y|j) > 0 where y is a PBNE action taken by player3, then Pj0 ⊆ Bk for all j0 with p2(y|j0) > 0. Fixing k, consider Bk and thepartition of Bk, Ik = (Ik1 , . . . , I

kjk), whose cells are also the cells of the original

partition, P = (P1, . . . , PJ).Now we prove part (iii). The proof for (ii) is perfectly symmetric, so we

provide a proof only for part (iii). Let player 1 send a messagemj1 conditional

on the state lying in Ikj (for natational simplicity, in this proof, we suppressthe superscripts k on messages). Also, let y1, . . . , yjk(, yjk+1) be the PBNEactions induced by states in Bk (We already know that there are jk (or jk+1)actions induced), with yj < yj

0if j < j0. Set mj

2 = yj.Now suppose the contrary, i.e. that after message m1

1 player 2 sends twomessages, m1

2 and m22 with positive probabilities. (We know from the proof

of Proposition 5 that player 2 mixes over at most two messages, that there isno message induced in equilibrium between these two messages, and that m1

2

must have positive probability after m11 by the construction of B; hence it

suffices to rule out the case in consideration.) For this to be an equilibrium,conditional on m1

1, player 2 has to be indifferent between m12 and m2

2:ZIk1

u2(θ,m12)f(θ)dθ =

ZIk1

u2(θ,m22)f(θ)dθ. (9)

37

Next, note that player 3 is maximizing his payoff at m12, which is induced

only by m11, so we have:

m12 = argmax

y

ZIk1

u3(θ, y)f(θ)dθ.

Since player 2 is negatively biased, this implies:

m12 > argmax

y

ZIk1

u2(θ, y)f(θ)dθ.

Hence, the first-order condition, the strict positiveness of f , and the strictconcavity of u2 implies: Z

Ik1

∂u2(θ,m12)

∂yf(θ)dθ < 0.

Now, recall again that u2 is strictly concave. This implies:ZIk1

∂u2(θ, y)

∂yf(θ)dθ < 0 for all y ∈ [m1

2,m22],

which implies: ZIk1

u2(θ,m12)f(θ)dθ >

ZIk1

u2(θ,m22)f(θ)dθ.

This contradicts Equation (9), which completes the proof. ¥

Proof of Proposition 7: Without loss of generality, assume that bothu1 and u2 imply negative bias, and fix a PBNE. Suppose there is a componentBk (as defined in Proposition 6) in which player 2 mixes (Remember thatthis is the only case in which players mix in the PBNE, except player 1’spossible mixing at a finite number of states (the boundaries of the cells) thathas ex ante zero probability (hence never affects the outcome equivalenceresult)). Let (Ik1 , . . . , I

kjk) be the partition of the states within the component

corresponding to distinct messages from player 1.By Proposition 6, after receiving m1

1, player 2 puts probability 1 on mes-sage m1

2. Since jk ≥ 2, the following two claims are true: Conditional on thestate lying in Ik1 , player 1 weakly prefers the action that player 3 plays afterreceiving m1

2, call it y1, to the action that player 3 plays after receiving m22,

call it y2 (Recall from Proposition 4 that the receiver always plays a purestrategy). Also, he weakly prefers y2 to y1 conditional on Ik2 .

38

Thus, the continuity of u1 implies that he is indifferent between y1 andy2 at the state θ ∈ cl(Ik1 ) ∩ cl(Ik2 ):

u1(θ, y1) = u1(θ, y2). (10)

But since player 2 is mixing between m12 and m2

2, he is indifferent betweeny1 and y2 conditional on m2

1:ZIk2

u2(θ, y1)f(θ)dθ =

ZIk2

u2(θ, y2)f(θ)dθ. (11)

By definition, θ < θ for all θ ∈ Ik2 \ θ. Thus the single-crossing condition∂2u2

∂θ∂y> 0 implies that for all θ ∈ Ik2 \ θ,

u2(θ, y2)− u2(θ, y2) > u2(θ, y1)− u2(θ, y1).

This and Equation (11) imply:ZIk2

u2(θ, y1)f(θ)dθ >

ZIk2

u2(θ, y2)f(θ)dθ,

which implies:u2(θ, y1) > u2(θ, y2). (12)

The assumption that player 1 is more negatively biased than player 2 impliesthat there exist affine transformations of u1 and u2, u1∗ and u2∗ respectively,such that

∂u1∗(θ, y)

∂y<

∂u2∗(θ, y)

∂y(13)

for all y. Equation (10) implies:Z y2

y1

∂u1∗(θ, y)

∂ydy = 0.

This and Equation (13) implies:Z y2

y1

∂u2∗(θ, y)

∂ydy > 0,

which implies: u2∗(θ, y1) < u2∗(θ, y2), hence u2(θ, y1) < u2(θ, y2). But thiscontradicts condition (12). This concludes that there cannot be an equi-librium component with more than 1 induced actions, which completes theproof. ¥

39

Proof of Proposition 8: First, note that, by definition, y10 is the optimalchoice of player 3 in the babbling PBNE. Next, note that the only purestrategy PBNE is babbling. To see this, suppose that there are exactly twoactions induced in a pure PBNE. Then there exists x ∈ (0, 1) such that thesetwo actions are yx0 and y1x, and u1(x, yx0 ) = u1(x, y1x). Thus, strict concavityof u1 and x ∈ (yx0 , y1x) (by b3(x) = 0) implies u1(x, x) > u1(x, y1x). But thiscontradicts our presumption that u1(a, a) < u1(a, y1a) for each a ∈ [0, 1].Thus, there exists no pure PBNE with two actions. But CS proved that inthe context of direct communication games if there exists a PBNE with Nactions, there also exists a PBNE with n actions for 1 ≤ n ≤ N . Proposition1 then implies that there cannot exist pure PBNE with 2 or more actions.Now we show that there exists a mixed PBNE with 2 actions. Consider

the following strategy profile with parameter > 0: Player 1 sends messagem1 ∈ M1 if θ ∈ [0, ) and m0

1 ∈ M1 if θ ∈ [ , 1]. If player 2 receives m1, hesends message m2, and if he receives m0

1, he sends m2 with probability p andm02 with probability 1− p. Conditional on off-path messages M1 \ m1,m

01,

he sends a message from m2,m02. Player 3 plays action y∗ if he receives m2

and y1 if he receives m02. Conditional on off-path messages M2 \ m2,m

02,

he plays an action from y∗, y1.It is easy to see that there exists an intermediator indifferent between

inducing y∗ or inducing y1, conditional on m01. This would hold for example

for an intermediator with quadratic loss function and state-independent biasy∗+y1

2−E(θ|m0

1). Hence, below we only need to check whether the strategiesof the sender and the receiver are compatible with PBNE.We show that there exists ¯ such that if < , there exists p ∈ (0, 1) and

y∗ such that this is indeed a PBNE.To see this, we need to establish the followings: If < , (i) u1( , y∗) =

u1( , y1), (ii) y0 < y∗ < y10, and (iii) y∗ is a best response conditional on

players 1 and 2’s strategies. (i) ensures that player 1 takes a best responseto the opponents’ strategies. (ii) ensures that p ∈ (0, 1). Player 3 takes abest response to the opponents’ strategies conditional on m0

2, by definition ofy1. Since given any on-path messages players put probability 0 on off-pathevents, and conditional on any off-path messages, players can have arbitrarybeliefs that make their choices optimal, (i), (ii), and (iii) are sufficient toestablish that the strategy profile constitutes a PBNE.Note that if we have u1( , y0) < u1( , y1) < u1( , y10), then we have (i) and

(ii), ignoring (iii). To see this, notice that this inequality ensures that thereexists y0 ∈ (y0, y1) such that u1( , y0) = u1( , y1). To see that y0 6∈ [y10, y1),recall that we have assumed that b1(0) < y10, which implies that

∂u1(0,y10)

∂y< 0.

By continuity, there exists 0 such that for all < 0, ∂u1( ,y10)

∂y< 0. Hence y0

40

cannot be contained in [y10, y1) if < 0.

Now we show that u1( , y0) < u1( , y1) < u1( , y10). Since u1(a, a) <u1(a, y1a) for each a ∈ [0, 1], in particular we have u1(0, 0) < u1(0, y10). Bycontinuity, there exists 00 such that for all < 00, u1( , y0) < u1( , y1). Also,we have shown that for all < 0, ∂u1( ,y10)

∂y< 0. Thus we have u1( , y1) <

u1( , y10) if < 0.Finally we prove (iii). Fix < min 0, 00 := . Then y∗ is uniquely

determined by condition (i). We prove that there exists p such that player 3takes a best response at y∗. Let y(p) be the best response when the mixingprobability is p. Notice that y(p) is continuous in p. Because of strict concav-ity, the best response is uniquely determined conditional on any probabilitydistribution on the state, hence y is a function. Note that y(0) = y0 andy(1) = y10. This implies that there exists p ∈ (0, 1) such that y(p) = y∗, sincewe know that y0 < y∗ < y10. Thus we have proved (iii).Overall, we have found that there exists ¯ such that if < , there exists

p ∈ (0, 1) and y∗ such that this is indeed a PBNE.Notice that player 3 has an option to play y10, conditioning on any mes-

sages. Since strict concavity implies the uniqueness of the best responseconditional on any probability distribution on the state, this implies that,given the conditions in this proposition, the PBNE we constructed gives astrictly higher ex ante payoff to player 3 than in any pure strategy PBNE.This completes the proof. ¥

41

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