Communication via a strategic mediator

Journal of Economic Theory 145 (2010) 869–884

www.elsevier.com/locate/jet

Note

Communication via a strategic mediator ✩

Maxim Ivanov ∗

Department of Economics, McMaster University, 1280 Main Street Hamilton, ON, Canada L8S 4M4

Received 3 December 2008; final version received 9 August 2009; accepted 16 August 2009

Available online 2 September 2009

Abstract

This paper investigates communication between an informed expert and an uninformed principal via astrategic mediator. We demonstrate that, for any bias in the parties’ preferences, there exists a strategicmediator that provides the highest expected payoff to the principal, as if the players had communicatedthrough an optimal non-strategic mediator.© 2009 Elsevier Inc. All rights reserved.

JEL classification: C72; D82; D83

Keywords: Communication; Information; Cheap talk; Mediation

1. Introduction

It is well known that introducing a mediator in communication between parties with conflict-ing interests can improve the efficiency of the interaction. The main role of the mediator is toprivately collect information from players and use it to give private recommendations to decisionmakers. Though the mediator does not have private information and cannot enforce her recom-mendations, the possibility of manipulations of received information (adding noise, randomizingover recommendations, collapsing information, etc.) often provides the players the incentive toreveal more information relative to direct talk. The fundamental assumption behind this result,

✩ I am grateful to Andreas Blume, Kalyan Chatterjee, Navin Kartik, Vijay Krishna, Hao Li, Timofiy Mylovanov,Gregory Pavlov, and seminar participants at the University of Toronto for helpful comments. I also owe special thanks totwo anonymous referees and the editor for their suggestions. Misty Ann Stone provided invaluable help with copy editingthe manuscript. All mistakes are my own. The paper has been previously circulated under the title “Communication viaa Biased Mediator”.

* Fax: +1 905 521 8232.E-mail address: [email protected].

0022-0531/$ – see front matter © 2009 Elsevier Inc. All rights reserved.doi:10.1016/j.jet.2009.08.001

870 M. Ivanov / Journal of Economic Theory 145 (2010) 869–884

which is widely used in the literature, is that the mediator is non-strategic. That is, she has nointerest in the outcome of communication. We refer to this property as the mediator’s neutrality.1

In practice, however, mediators are likely to have their own agenda. A natural question arises:how to implement the optimal outcomes in real life? That is, how to take advantage of the ben-efits of mediation if all potential candidates for the mediator have their own goals and, hence,tend to manipulate information in their favor? The literature has suggested some answers to thisimplementation problem, such as adding technological noise in messages (Blume et al. [2]),multiple rounds of communication (Krishna and Morgan [11]), or direct communication if thereare sufficiently many players and the players’ state and action spaces are finite (Forges [5]). Inthis paper, we offer a new alternative: the mediator is a player without commitment power andwith conflicting objectives. As the main finding of the paper, we establish the possibility of im-plementing the optimal outcome of the non-strategic, or neutral, mediator in the game with astrategic intermediary.

Information typically passes through several parties on the way from the source to decisionmakers. Legislators get information from special committees, who obtain it from lobbyists. ThePresident seeks advice from cabinet members, who receive information from their departments.Boards of Directors rely on information from managers, who obtain it from the associated di-visions. However, if the intermediary party has her own interest in the final decision, little canprevent the mediator from placing her desired “spin” on information. As a result, a natural ques-tion about the efficiency of mediation comes to attention. On one hand, involving a strategicmediator could be detrimental for effective communication since she wants to distort obtainedinformation in her favor. This may decrease the quality of the information conveyed to the de-cision makers. On the other hand, this manipulation of information affects the incentives of theinformed parties and may potentially force them to reveal more information to the mediator. Inthis paper, we investigate the trade-off between the benefits of non-strategic mediation and thepotential losses incurred due to the intermediary’s strategic behavior.

Before involving the strategic mediator in the communication process, the decision makerhas to consider the following questions: can the strategic mediator bring any benefits over directtalk? What is the performance of the strategic mediator compared with the optimal non-strategicone? Does the best mediator have an opposing or alike viewpoint relative to those of the otherplayers? What is the optimal magnitude of the mediator’s bias? To address these issues, wemodify a simple model of communication, à la Crawford and Sobel [4], hereafter CS, between aprincipal and a perfectly informed expert, in which the expert attempts to influence the principal’sdecision via sending costless and non-verifiable messages. In our model, the players communi-cate through a mediator, who collects information from the expert and gives recommendationsto the principal in a way that serves her own objectives.2 The preferences of the expert and themediator are parameterized by their inherent biases relative to that of the principal. The principalhas a choice over potential mediators, who differ by the values of their biases.3

1 Neutrality of the mediator is assumed by Myerson [16] and Goltsman et al. [8]. Krishna and Morgan [11] and Blumeet al. [2] point out that the equilibria in their environments can be replicated as mediated ones, given that there are norestrictions on the set of mediator’s strategies.

2 In this paper, we do not consider the possibility of interaction between the expert and the principal without themediator. That is, if the principal designs the communication protocol, commitment not to “spy” is credible.

3 For example, according to the U.S. Senate Committee System, each party assigns its own members to the Senate’scommittees, in which the chair of each committee and a majority of its members represent the majority party. Similarly,in forming the Cabinet, the President has a choice over advisors with different viewpoints.

M. Ivanov / Journal of Economic Theory 145 (2010) 869–884 871

The main result of this paper is that, for any value of the expert’s bias, there exists a strategicmediator who leads to the highest expected payoff to the principal as if they communicated viathe optimal non-strategic mediator.4 We characterize the bias of the optimal mediator and showthat it is opposing to the expert’s one. In contrast, if the mediator’s bias is between those ofthe other players, then mediation cannot improve upon direct talk between the expert and theprincipal. Finally, we show that our construction of the optimal equilibria is robust to slightchanges in the mediator’s bias.

The closely related paper by Goltsman et al. [8] characterizes the highest expected payoffthat can be obtained by the principal in non-strategic mediated communication. We depart fromthat model by allowing the mediator to behave strategically. A recent work by Ambrus et al. [1]investigates the set and the properties of equilibria in a game of communication through a chainof strategic intermediaries with given preferences. We focus on a different issue—investigatingthe potential welfare efficiency of the strategic mediator, who can be selected by the principal.Kydd [13] analyzes the benefits of an imperfectly informed strategic mediator in bargaining. Hedemonstrates that the mediator can be more effective if her interests are close to those of theuninformed party compared with the mediator who is interested in minimizing the probability ofdisagreement. We depart from Kydd [13] in two dimensions. First, the mediator’s informationin our setup is strategically determined by the expert rather than exogenously. Second, insteadof two participants in the bargaining process, the players’ payoffs in our model depend on thedecision of a single player (the principal). Li [15] investigates the effect of the expert’s andthe intermediary’s reputational concerns on the structure of communication and establishes thestrategic complementarity between the biased expert’s and the biased intermediary’s incentivesto communicate truthfully.

Krishna and Morgan [11] investigate a multi-stage version of the CS model, in which theexpert might be allowed to update his report depending on the outcome of interaction in theprevious stage. They demonstrate that active participation by the principal in the communica-tion process along with multiple stages of communication lead to Pareto improvement over asingle-stage case and that any equilibrium can be induced via a non-strategic mediator. Blumeet al. [2] consider direct talk with a noise in the communication channel and show that addinga proper amount of noise can be welfare improving. Similarly, the equilibria in their model areoutcome equivalent to those in mediated communication. However, the equilibria identified byKrishna and Morgan [11] and Blume et al. [2] cannot be sustained in the case of the strategicmediator. Forges [5] introduces a scheme of direct communication among at least three players,which implements every outcome of any communication mechanism if the players’ informationstructures and the action sets are finite. Neither of these assumptions holds in our model.

The rest of the paper is structured as follows. Section 2 presents the formal model and outlinesthe properties of mediated equilibria. In Section 3, we characterize the optimal strategic mediator.Section 4 establishes the inefficiency of a moderately biased mediator. Section 5 concludes thepaper.

2. The model

In this section we outline a simple model of communication between an informed expert (E)and an uninformed principal (P) through a strategic mediator (M). Before communication starts,

4 In this respect, the paper can be connected to Che and Kartik [3], Gerardi and Yariv [7], and Hori [9], all of whichconsider a different rationale for choosing a biased agent.


the principal chooses the mediator from a pool of potential mediators with different preferences.For our analysis, we use the well-known uniform-quadratic setting of the CS model. We chosethis setting because, to the best of our knowledge, this is the only setup with the characterizedoptimal non-strategic mediator (Goltsman et al. [8]). At the same time, the efficiency of theoptimal non-strategic mediator is the benchmark for estimating the potential losses due to themediator’s strategic motives.

The payoffs of all players depend on the state of nature θ , which is distributed uniformly onΘ = [0,1], and on the principal’s action (or decision) a ∈ R. The utility functions of the playersare of the quadratic form

Ui(a, θ) = −(a − θ − bi)2, i ∈ {E,M,P },

where parameter bi reflects a bias in interests of party i. The bias of the principal is normalizedto be 0. The bias of the expert is set to be positive: bE = b > 0. Finally, the principal chooses themediator with the bias bM = β ∈ R.

The timing of the game is as follows. First, the expert privately observes θ and sends a signals to the mediator. Then, the mediator conveys a message m to the principal. Finally, the principalmakes a decision, which is potentially based on the message.

2.1. Equilibrium

An expert’s strategy specifies a conditional probability density ρ(s|θ) over the signal space Sfor each type θ ∈ Θ . Similarly, a mediator’s strategy specifies a conditional probability densityσ(m|s) over the space of messages M for each signal s ∈ S . Finally, the principal’s action rulea(m) maps mediator’s messages into actions.5

For a fixed bias of the mediator, a perfect Bayesian equilibrium (hereafter, an equilibrium)(ρ,σ, a,F,G) consists of (i) a signaling strategy of the expert ρ(s|θ); (ii) a messaging strategyof the mediator σ(m|s); (iii) an action rule of the principal a(m); (iv) a belief function of themediator F(θ |s), which assigns a probability distribution over states θ to every s; and (v) a be-lief function of the principal G(θ |m), which assigns a probability distribution over states θ toevery m. It is required that for any message m, the principal maximizes his expected utility givenhis beliefs G(θ |m). Also, for any signal s, the mediator maximizes her expected utility givenF(θ |s) and a(m). The expert maximizes her expected utility given σ(m|s) and a(m). Finally,F(θ |s) and G(θ |m) are derived from ρ(s|θ) and σ(m|s) using Bayes’ rule wherever possible.6

Note that the only decision relevant information included in the signal s for the mediator is itsposterior value vs = E[θ |s], since the mediator’s preferences can be decomposed as

UM(a|s) = E[(a − β − θ)2

∣∣s] = −(a − β − vs)2 − Ds(vs) = UM(a, vs) − Ds(vs),

where Ds(vs) = E[(θ − vs)2|s]. Thus, a strategy ρ generates the mediator’s type space Ω = {ω |

ω = vs, s ∈ S}, where ω ∈ Ω is the mediator’s type.Similarly, given a message m, the principal’s expected utility is

UP (a|m) = UP (a, vm) − Dm(vm),

where vm = E[θ |m] and Dm(vm) = E[(θ − vm)2|m].

5 Due to the strict concavity of the principal’s preferences over actions, he never mixes between actions.6 For all messages m /∈ M, we define the principal’s beliefs in such a way that he interprets them as some m0 ∈ M.

The mediator’s beliefs about signals s /∈ S are defined in the same fashion.


The players’ equilibrium strategies are determined by the principal’s best response

a(m) = vm = E[E[θ |s]∣∣m] = E[vs |m], (1)

and the incentive-compatibility constraints for the expert and the mediator. Given the set of in-duced actions A, the mediator of type vs prefers an action a to a′ > a if it is closer to her optimalaction vs + β . The symmetry of UM(a|s) with respect to a implies

UM(a|s) � UM(a′|s) ⇐⇒ a + a′

2� vs + β. (2)

The incentive compatibility constraints of the expert require

s(θ) ∈ arg maxs∈S

∫M

UE

(a(m), θ

)σ(m|s) dm. (3)

Using (1), the players’ expected payoffs can be decomposed as

Vi = −∫

Θ,S,M

(a(m) − θ

)2σ(m|s)ρ(s|θ) dmds dθ − b2

i , i ∈ {E,M,P },

which gives Vi = VP − b2i , i ∈ {E,M}. That is, all equilibria can be ex-ante Pareto ranked.

2.2. Equilibrium characterization

In this subsection, we focus on the properties of the players’ equilibrium strategies. Our firstresult describes the principal’s behavior. All proofs are collected in Appendix A.

Lemma 1. In any equilibrium, the set of induced actions A is finite.

The main implication of the lemma above is that there is no separation for any subset of Θ

with a positive measure. That is, any principal’s decision comes with informational losses (upto a subset of zero measure). This is due to the mediator’s incentive compatibility constraints,which imply that the distance between any two actions must be at least 2|β|. If β = 0, then themediator’s and the principal’s preferences coincide. That is, the strategic mediation is outcomeequivalent to CS communication, which involves a finite number of actions.

Now, consider the expert. The signaling strategy is finite monotone partitional if the expertpartitions Θ into intervals Θk = [θk−1, θk], k = 1, . . . ,N < ∞, where each θ ∈ Θk generatesa single posterior value ωk = E[θ |θ ∈ Θk]. The following lemma states that without loss ofgenerality we can consider only this type of signaling strategies.

Lemma 2. Any equilibrium is outcome equivalent to an equilibrium, in which the expert’s strat-egy is finite monotone partitional.

Thus, the structure of mediated equilibria is similar to that in CS communication. The expertpartitions Θ into a collection of intervals and reveals the interval that contains the state. Eachinterval generates some mediator’s type, which maps the expert’s signals into recommendationsover a finite number of actions.


Fig. 1. An equilibrium with a strategic mediator.

Together, the two lemmas highlight the contrast between communication with exogenousnoise in the expert-principal channel and endogenous noise introduced by the strategic inter-mediary. As shown by Blume et al. [2], exogenous noise can make the mediator’s informationstructure and the action set quite sophisticated. For example, it may result in equilibria with anuncountable number of partition’s elements and actions, which are not sustainable in the caseof the strategic mediator. Intuitively, such a mediator cannot introduce noise in an arbitrary waybecause any distortion of information must serve her goals. In particular, the concavity of themediator’s objective function excludes randomizing over more than two actions. Also, the medi-ator of a higher type never induces a lower action. Therefore, the single crossing property—anincrease in the difference in the payoffs of two actions as θ grows—can be extended to the mix-tures over induced actions. This determines the partitional form of the expert’s strategy.

3. Optimal strategic mediator

In a recent paper, Goltsman et al. [8] demonstrate that a non-strategic mediator cannot providean expected payoff to the principal above VP (b) = − 1

3b(1 − b), b � 1/2. (For b > 1/2, even anon-strategic mediator cannot improve on the uninformative outcome. Therefore, it is only ofinterest to focus on the case b � 1/2.) The upper bound V (b) is tight, since it can be reached inthe equilibria identified by Blume et al. [2] and, if b � 1/8, by Krishna and Morgan [11]. In thispaper, we construct a new class of equilibria related to the aforementioned ones. The examplebelow illustrates the equilibrium construction and the algorithm of finding the optimal strategicmediator.

Example 1. Let the expert’s bias be b = 1/3. An optimal equilibrium with the non-strategicmediator involves only two actions, a1 = 1/3 and a2 = 5/9 (see Fig. 1). The expert informs themediator if the state is below or above the cut-off θ1 = 1/9. In the first case, the mediator inducesthe lower action a1. Otherwise, the mediator mixes between a1 and a2 with the probabilities5/32 and 27/32, respectively. In order to find the mediator’s bias, we need a condition, whichguarantees that the mediator of a higher type (1+θ1)/2 = 5/9 is indifferent between two actions.That is, her optimal decision 5/9 + β has to be equidistant from these actions. This is possibleonly if the mediator’s bias is β = −1/9. In addition, the negative value of the bias guarantees thatthe mediator of the lower type θ1/2 = 1/18 strictly prefers the lower action. Finally, a routinecheck shows that the principal’s actions and the expert’s strategy are the equilibrium ones.

The following result generalizes the above logic.


Theorem 1. For any b � 1/2, there exist a mediator with bias β0(b) ∈ (−2b,0] and an equilib-rium in the game with this mediator that provides VP = VP (b).

The optimal mediated equilibrium is similar to the most informative CS equilibrium. It differs,however, by possibly involving an extra action and randomizing between the two lowest actions.7

The construction of the equilibrium is as follows. (Hereafter, we refer to this type of the optimalequilibria as S-type.) First, the expert partitions the state space into three groups of intervals. Thefirst and the second groups consist of single intervals. The last group contains N(b) intervals,such that

N(b) = ⌊(2b)−1/2⌋ − 1, (4)

where �x is the highest integer less or equal x.8 Next, the mediator’s types that are generatedby the first interval and the third group purely induce the associated actions. However, the typegenerated by the second interval mixes over two lowest actions. That is, the mediator’s types inthe third group separate themselves, whereas the lowest type is partially pooled with the secondlowest one.

While the details on the equilibrium construction are left to Appendix A, we provide here abrief summary. Given the action rule and the mediator’s messaging strategy, the interval cutoffs{θk}N(b)+1

k=1 are given by the expert’s “no arbitrage” conditions. Also, the principal plays the bestresponse to the other players’ strategies. Now, consider the mediator. The value of β is chosento make the 2nd mediator’s type indifferent between a1 and a2. Also, the fact that the intervallengths θk = θk − θk−1, and, thus, the distances between the actions increase in k, guaranteesthat no mediator’s type above the 2nd one wants to deviate from purely inducing the associatedaction. Finally, we use the single degree of freedom—the probability of mixing between a1 anda2—which affects all endogenous variables: the expert’s cutoff types, the mediator’s types, andthe induced actions. It is chosen to satisfy a1 = b. Also, the lowest mediator’s type purely inducesa1 because of β < 0. Together, these two conditions imply that the expected payoff of the lowestexpert’s type θ = 0 is 0, which is the necessary and sufficient condition for the optimality ofmediated equilibria (Goltsman et al. [8]).

Intuitively, the efficiency gains come from the conflict between the expert’s and the mediator’sobjectives. The expert’s incentives to overstate information are offset by the mediator’s incentivesto understate it. In the equilibrium, the mediator punishes the expert for sending the 2nd lowestsignal by claiming, with some probability, that it is the lowest one. In response, the principal shiftsthe lowest action up toward the expert’s first-best decision, which increases its attractiveness fora bigger range of the expert’s low types. As a result, the expert reveals more information aboutlow states. In order to balance this conflict between the expert and the mediator, their objectivesshould not be too far apart, which is reflected in the fact that the mediator’s bias never exceedsthe expert’s one by more than twice in the absolute value.

Fig. 2 demonstrates the equilibrium outcomes in the S-type equilibria, direct talk via a noisychannel (Blume et al. [2]), and multi-stage communication (Krishna and Morgan [11]) for the

7 Depending on the expert’s bias, the number of actions in the mediated equilibrium may be the same as that in themost informative CS equilibrium or exceeds it by 1. Thus, the best mediated payoff can be reached using no morethan N + 1 messages from the expert and the mediator, where N is the size of the most informative CS equilibrium.This complements Ganguly and Ray [6], who argue that neutral mediation cannot improve upon CS communication ifb < 1/2N2 and the size of both the expert’s and the mediator’s message spaces are restricted to be N .

8 For b � 1/8, we have N(b) = 0. That is, this component of the partition is an empty set.


Fig. 2. Equilibrium outcomes in different modes of communication.

3-interval partitional strategy of the expert (if, say, b = 1/10). There is a clear similarity betweenthe equilibria in our model and those in Blume et al. [2]. In both cases, the low expert’s typessometimes end up being confounded with higher types, which relaxes the incentive compatibilityconstraint for the critical type that separates the first from the second interval. In Krishna andMorgan’s [11] type of equilibria, the principal benefits from the ability to react more sensitively tothe low informative expert’s messages about high states by inducing a bigger number of actions.It is interesting to note that all three types of equilibria can be replicated by using the “lostpigeon” construction of correlated noise by Myerson [16].9 That is, the expert’s message to theprincipal is lost with a message-contingent probability (in the examples above, the “lost message”is m0), and reaches the addressee with the complement probability. Also, Fig. 2 illustrates whythe outcomes of noisy direct communication and multi-stage interaction cannot be sustained withthe strategic mediator. In the first case, if the mediator with concave preferences over actions isindifferent between inducing a1 and a3, she can strictly benefit by inducing a2.10 In the secondcase, mixing between a2 and a3 requires the mediator have a positive bias, whereas mixingbetween a3 and a4 implies that the mediator’s bias must be negative.

Fig. 3 shows the behavior of the optimal mediator’s bias β0(b) in the S-type equilibrium. Themediator’s bias is neither monotone in b nor continuous.11 It is interesting to note that efficientcommunication with a less biased expert may require a more biased mediator. This is due to achange in the principal’s decisions in response to a smaller expert’s bias. Given a fixed numberof intervals, a decrease in b results in a more uniform allocation of the induced actions over theunit interval. In particular, this increases the distance between the lowest two actions. Since themediator has to be indifferent between these actions, the absolute value of β0(b) increases.

Since the construction above pins down the bias of the optimal mediator, it is natural to askhow robust the equilibrium structure is to slight changes in the mediator’s bias. The lemmabelow shows that for any equilibrium of the S-type, there exist a continuum of equilibria aroundit, which provide the expected payoff near VP (b) as long as the mediator’s biases are in theneighborhood of β(b).

9 I thank the anonymous referee for pointing out this common feature.10 This logic does not work if the action space consists of two points as in Example 1. In this case, the equilibria instrategic mediation are equivalent to those in direct talk via a noisy channel.11 If b falls, but the number of induced actions does not change, |β0(b)| monotonically increases from 0 to 2b. Asb reaches 1/2N2, so that N(b) grows by 1, the mediator’s bias falls from 2b to 0, and the monotone behavior of β

continues until b = 1/2(N + 1)2.


Fig. 3. The bias of the optimal mediator.

Lemma 3. For any equilibrium of the S-type and ε > 0, there exists δ > 0, such that if themediator’s bias β ∈ (β0(b) − δ,β0(b) + δ), then there is an equilibrium in the game with thismediator that results in VP > VP (b) − ε.

Intuitively, any mixed-strategy equilibrium with two-interval partitions and the two-point ac-tion set contains a continuum of equilibria around it which are obtained by small perturbationsin the probabilities of inducing different actions. Since the principal adjusts his strategies to themodified probabilities, then for any distance between two actions in a mixed-strategy equilib-rium, there exists a continuum of equilibria with the distances around the original one. Applyingthis logic to any equilibrium of the S-type implies that a perturbation in the mediator’s bias mustchange the distance between the lowest two actions. This also changes slightly the expert’s sig-naling strategy, the principal’s best response, and the probabilities of mixing between the lowesttwo actions, but does not affect the mediator’s preferences over induced actions.

4. Inefficiency of a moderately biased mediator

As demonstrated in the literature on mediated bargaining, effective mediation may requireinvolving a moderately biased intermediary, whose interests are between those of the communi-cating parties (Kydd [12]). In the context of this paper, this means that the mediator’s bias belongsto the interval (0, b). Given the positive results above about the efficiency of the biased medi-ator, it seems that a mediator with a positive moderate bias can bring benefits over direct talk,since she has more incentive to distort the expert’s information toward the principal’s objectivescompared with the expert. This is, however, never the case in our model.

Lemma 4. In any equilibrium with the mediator’s bias β ∈ [0, b), the mediator never plays amixed strategy. Thus, mediated communication cannot improve on direct talk.

We prove this result by contradiction. If ω′ is the highest mediator’s type that mixes betweentwo actions, then there must be a type that purely induces the higher action. This implies thatthe highest expert’s type θ ′ that generates ω′ has to be indifferent between these actions as well.Since ω′ � θ ′, the indifference conditions θ ′ + b = a+a′ = ω′ + β result in the contradiction
2


β � b. At the same time, the mediator cannot benefit from distorting the expert’s signals, so thatmediated communication can still replicate the most informative CS equilibrium.12

5. Conclusion

Typically, standard models that demonstrate the advantage of communication via a mediator,hinge on the assumption of the mediator’s neutrality. Without such a mediator, the benefits ofmediation can be impaired by the mediator’s strategic behavior. This paper shows that there areno efficiency losses if the strategic mediator is chosen properly.

A few comments are in order. First, note that the assumption about the principal’s authorityover the choice of the mediator is not crucial. Since the players’ expected payoffs are perfectlyaligned, this choice can be delegated to the expert, given that she chooses the mediator beforelearning the state. Second, the idea that the presence of a strategic mediator improves on di-rect communication does not appear to be restricted to the uniform-quadratic even though theachievement of the efficiency bound depends on this setting. The main argument is that the prin-cipal’s gains stem from the conflict between the expert’s and the principal’s interests rather thanfrom the distribution of states and the shape of the utility functions. In particular, stochastic un-derstating of information by the negatively biased mediator shifts the principal’s actions to theright. This, in turn, increases the incentives of the positively biased expert to reveal more in-formation for low states and improves the ex-ante payoff to the principal.13 A simple examplesupports this argument for the settings different from the uniform-quadratic ones.14 Also, Am-brus et al. [1] provide a sufficient condition for the existence of an intermediary that can sustaininformative communication for more general settings given that the only equilibrium in directtalk is non-informative.

Finally, the class of all equilibria in strategic mediation is quite rich and is not restricted toonly those, in which only one mediator’s type mixes over actions. There exist equilibria, in whichmultiple mediator’s types mix over actions or the single mixing type is not the second lowest one.However, the efficiency of these equilibria remains an open question.

Appendix A

Proof of Lemma 1. If β = 0, then Lemma 1 in Ivanov [10] says that for any Ω , we have|a − a′| � 2|β|, a, a′ ∈ A, a = a. Because (1) implies that A is bounded by [0,1], it is finite.Also, the mediator never plays a mixed strategy if β = 0 (for completeness, we prove it forβ ∈ [0, b) in Lemma 4 below). Since each expert’s signal induces a single action, any equilibrium

12 More generally, the most informative N -interval CS equilibrium of the communication game between the principaland an expert can be sustained in communication via the biased mediator as long as |β| � b + θ1/2, where θ1 is the firstexpert’s cutoff type in the CS equilibrium (Li [14]).13 If the players’ preferences differ from the quadratic ones, the ex-ante Pareto ranking of the equilibria does not neces-sarily hold, so that even if the principal’s ex-ante payoff can be improved by a strategic mediator, the ex-ante payoffs ofthe other players can potentially go down.14 Consider the triangle distribution with the density f (θ) = 2(1 − θ), θ ∈ [0,1], the utility functions Ui(a, θ) =−|a − θ − bi |3/2, and b = 1/7. In this case, the only informative CS equilibrium is characterized by the 2-intervalpartition with the cutoff θCS

1 = 0.037 and the actions {aCS1 , aCS

2 } = {0.018,0.341}, which result in V CSP

= −1/11.23

and V CSE

= −1/8.53. It is dominated by the equilibrium in strategic mediation with β = −0.143, θ1 = 0.1, {a1, a2} ={0.102,0.385}, and randomizing by the mediator of the higher type between a1 and a2 with the probabilities 0.102 and0.898, respectively, which provides VP = −1/12.28 and VE = −1/8.76.


is outcome equivalent to that in CS communication so that A is finite by Lemma 1 in Crawfordand Sobel [4]. �

Denote by σωa the probability of inducing an action a by the mediator of type ω. Since

∂2UM(a|s)/∂a2 < 0, the mediator of any type can mix between two adjacent actions only. Thefollowing lemma characterizes other properties of {σω

a }ω∈Ω,a∈A.

Lemma 5. In any equilibrium, {σωa }ω∈Ω,a∈A satisfies the following conditions:

(A) if β � 0, ω = max{ω | ω ∈ Ω}, and a = max{a | a ∈ A}, then σ ωa = 1; if β � 0, ω = min{ω |

ω ∈ Ω}, and a = min{a | a ∈ A}, then σωa = 1,

(B) σω′a > 0 implies σω

a′ = 0 if a′ < a, ω > ω′ or a′ > a, ω′ < ω,

(C) if σω′a > 0, σω′

a′ > 0, where a′ > a, then for β � 0, there exists ω > ω′ such that σωa′ > 0, and

for β � 0, there exists ω < ω′ such that σωa > 0, and

(D) σω′a > 0 and σω′′

a > 0 imply σωa = 1, ω ∈ (ω′,ω′′) ∩ Ω .

Proof. (A) Suppose β � 0. From (1), we have a � ω � ω + β . Since ∂UM(a|ω)/∂a > 0 ifa < ω + β , then UM(a|ω) < UM(a|ω), a < a. If β � 0, then the argument for σ

ωa is the same.

(B) By contradiction, let σω′a > 0 and σω

a′ > 0 for some a′ < a,ω > ω′. Then, σω′a > 0 im-

plies UM(a,ω′) � UM(a′,ω′) and σωa′ > 0 implies UM(a′,ω) � UM(a,ω). Combining these

inequalities results in UM(a,ω) − UM(a′,ω) � 0 � UM(a,ω′) − UM(a′,ω′), which contradictsthe single-crossing property UM(a,ω) − UM(a′,ω) > UM(a,ω′) − UM(a′,ω′).

(C) Suppose β � 0. By contradiction, let σω′a > 0, σω′

a′ > 0, a′ > a, and σωa′ = 0, ω > ω′.

Since UM(a|ω′) = UM(a′|ω′), then ω′ + β < a′. Condition (B) for σω′a implies σω

a′ = 0, ω < ω′.That is, σω

a′ = 0, ω = ω′, which results in the contradiction a′ = ω′ > ω′ + β . If β � 0, applying

condition (B) to σω′a′ leads to the contradiction a = ω′ < ω′ + β . Finally, applying (C) to σω′

a and

σω′′a results in σω

a′ = 0, a′ = a, ω ∈ (ω′,ω′′), which gives (D). �Proof of Lemma 2. First, we show that any equilibrium is outcome equivalent to that witha finite type space. Suppose that mediator’s type ω induces actions aω

1 and aω2 � aω

1 with theprobabilities σω and 1 − σω, respectively. Then, a signal s generates the mediator’s type ω = vs ,which induces the distribution of actions (hereafter, a lottery) L(ω) = (aω

1 , aω2 , σω) in the lottery

space Λ = {L(ω) | ω ∈ Ω}. Similarly, the expert’s strategy ρ(s|θ) induces the distribution Γ (θ)

over lotteries in Λ for the expert’s type θ .In any equilibrium, consider the set ΩM of mediator’s types that induce two actions. By (2), it

is a subset of the finite set {(ai +ai+1)/2−β}#A−1i=1 . Then, split Ω\ΩM into the subsets {Ωa}a∈A

of types, which purely induce actions a ∈ A. For each a ∈ A, modify ρ(s|θ) by collapsing alltypes ω ∈ Ωa (e.g., by the uniform randomization over the signals that generate ω ∈ Ωa) intoωa = E[ω|ω ∈ Ωa]. By property (D) of Lemma 5, each Ωa is an interval. Since ωa is boundedby Ωa , this leads to ΩM ∩ {ωa}a∈A = ∅.

Given the initial a(m), property (D) of Lemma 5 implies σωaa = 1. Given this messaging

strategy for the modified types, a(m), and the initial strategy {σωa }ω∈ΩM,a∈A, it follows that Λ

does not change, and each s induces the same lottery as that in the initial equilibrium. Thus,no expert’s type θ can beneficially deviate from inducing Γ (θ). This means that a(m) is notaffected. Since Ω ′ = ΩM ∪ {ωa}a∈A is finite, this completes the first part of the lemma.


Now, we prove that each ω ∈ Ω ′ is generated by a single interval. If Ω ′ is a singleton, it isgenerated by Θ . Otherwise, consider two different lotteries L(ω) and L(ω′), ω′ > ω. By property(B) of Lemma 5, it follows that aω′

1 � aω2 . Also, aω′

2 > aω1 and (1 − σω′

) + σω + (aω′1 − aω

2 ) > 0

(otherwise, both lotteries degenerate into the single action aω′1 = aω

2 ). Then,

EUE

(θ,L(ω′),L(ω)

) = E[UE(a, θ)

∣∣L(ω′)] − E

[UE(a, θ)

∣∣L(ω)]

= (1 − σω′)(

UE

(aω′

2 , θ) − UE

(aω′

1 , θ))

+ σω(UE

(aω

2 , θ) − UE

(aω

1 , θ))

+ (UE

(aω′

1 , θ) − UE

(aω

2 , θ))

. (5)

Since

aω′2 � aω′

1 � aω2 � aω

1 , (6)

and aω′2 > aω

1 , at least one of the inequalities in (6) is strict. This, along with the single-crossingproperty of UE(a, θ), implies that all three components in (5) are (weakly) increasing, andat least one of them is strictly increasing. Thus, ∂

∂θEUE(θ,L(ω′),L(ω)) ≷ 0 if and only if

ω′ ≷ ω. Hence, if θ ′ and θ ′′ prefer L(ω1), then all θ ∈ (θ ′, θ ′′) prefer L(ω1). Otherwise, if thereis θ ∈ (θ ′, θ ′′) that prefers L(ω2), where, say, ω2 < ω1, then EUE(θ,L(ω2),L(ω1)) � 0. Since∂∂θ

EUE(θ,L(ω2),L(ω1)) < 0, it follows that EUE(θ,L(ω2),L(ω1)) > 0, θ < θ . This con-tradicts the fact that θ ′ prefers L(ω1). If ω2 > ω1, then ∂

∂θEUE(θ,L(ω2),L(ω1)) > 0, and the

same argument leads to the contradiction for θ ′′. �Proof of Theorem 1. In any equilibrium, all θ ∈ Θk = [θk−1, θk] generate the posterior

ωk = E[θ |θ ∈ Θk] = θk−1 + θk

2, ∀k.

Fix N = N(b), where N(b) is determined by (4). Equivalently, N(b) is such that 12(N(b)+2)2 <

b � 12(N(b)+1)2 . Consider the sequences {θk}N+2

k=0 , {ωk, ak}N+2k=1 , and the matrix {σωk

aj}k=1,...,N+2j=1,...,N+2,

which describe Ω and the players’ strategies (up to M and S ):

θ0 = 0, θ1 = 1

3

(2b

(N2 − 1

) − 23/2b1/2N + 1), θ2 = 3θ1 + 2b, (7)

θk = θ2 + (k − 2)(2bk + ξ − 6b), 3 � k � N + 2, ξ = 1 − θ2 − 2bN(N − 1)

N, (8)

ωk = θk−1 + θk

2, k = 1, . . . ,N + 2;

a1 = b, ak = ωk, k = 2, . . . ,N + 2, and (9)

σω2a2

= 3

8

3θ1 + 2b

θ1 + b, σω2

a1= 1 − σω2

a2, σωk

ak= 1, k = 2, and

σωkaj

= 0 otherwise. (10)

To show that (7)–(10) constitute an equilibrium, we start with the expert’s incentives. BecauseN mediator’s types ωk = θk−1+θk

2 , k = 3, . . . ,N + 2 purely induce ak = ωk , the expert’s “noarbitrage” conditions for θ , which generate ωk , k � 3, are

θk+1 − θk = θk − θk−1 + 4b,


or θk = θk−1 + 4b, where θk = θk − θk−1. Expression (8) is the solution to this second-order difference equation with the boundary conditions θ2 and θN+2 = 1. Also, we impose the“forward solution” condition ξ > 0, which is shown to hold below.

The expert’s type θ1 must be indifferent between a1 and L(ω2) = (a1, a2, σω2a1 ), which means

θ1 + b = a1 + a2

2. (11)

Finally, if N � 1, then type θ2 must be indifferent between L(ω2) and a3:

−(1 − σω2

a2

)(a1 − b − θ2)

2 − σω2a2

(a2 − b − θ2)2 = −(a3 − b − θ2)

2. (12)

Now, consider the action set. It is characterized by ak = θk−1+θk

2 , k = 2, . . . ,N + 2, and

a1 = 1

2

θ21 + (1 − σ

ω2a2 )(θ2

2 − θ21 )

θ1 + (1 − σω2a2 )(θ2 − θ1)

. (13)

Goltsman et al. [8] show that VP = VP (b) if and only if the expected payoff of the lowestexpert’s type θ = 0 is 0. This implies a1 = b and σ

ω1a1 = 1. Plugging a1 = b and a2 = θ1+θ2

2 into(11) results in θ2 = 3θ1 + 2b and

a2 = 2θ1 + b. (14)

Also, solving (13) for σω2a2 gives

σω2a2

= 3

8

3θ1 + 2b

θ1 + b. (15)

From (15), θ1 � 0 implies σω2a2 > 0. Also,

∂σω2a2

∂θ1= 3

8b

(θ1+b)2 > 0. That is, σω2a2 < σ

ω2a2 (θ1 = 2b) = 1

if and only if θ1 < 2b.If b ∈ (1/8,1/2], so that N = 0, the condition θ2 = 3θ1 + 2b = 1 leads to θ1 = (1 − 2b)/3,

and by (14) and (15), determines a2 and σω2a2 . Note that b � 1/2 implies θ1 = (1 − 2b)/3 � 0.

Also, b > 1/8 implies θ1 = (1 − 2b)/3 < 2b or σω2a2 < 1.

If b � 1/8, so that N � 1, then (8) gives θ3 = θ2 + ξ and a3 = θ2 + ξ2 = θ2 + b − bN − θ2−1

2N.

Plugging θ2, σω2a2 , and a2 as functions of θ1 into the left-hand side (LHS) of (12) gives

−(1 − σω2

a2

)(a1 − b − θ2)

2 − σω2a2

(a2 − b − θ2)2

= −(1 − σω2

a2

)θ2

2 − σω2a2

(2θ1 − θ2)2

= −1

8

2b − θ1

θ1 + b1(3θ1 + 2b)2 − 3

8

3θ1 + 2b

θ1 + b(2θ1 − 3θ1 − 2b)2

= −1

8

3θ1 + 2b

θ1 + b

((2b − θ1)(3θ1 + 2b) + 3(θ1 + 2b)

) = −2b(3θ1 + 2b) = −2bθ2,

and the right-hand side (RHS) of (12) is −(a3 − b − θ2)2 = −(bN + θ2−1

2N)2. Thus, we obtain the

quadratic equation with respect to θ2:

(θ2 + M − 1)2 − 4Mθ2 = θ22 − 2(M + 1)θ2 + (M − 1)2 = 0, (16)

where M = 2bN2. This equation has two solutions, θ1,22 = M + 1 ± 2M1/2 = 2bN2 + 1 ±

23/2b1/2N , which are equivalent to θ1,21 = 1

3 (2bN2 + 1 ± 23/2b1/2N − 2b). To prove that θ1 =1 (2b(N2 −1)+1−23/2b1/2N) is in [0,2b), consider θ1 as a function of N . It is convex in N and
3


has two roots, N1 = (2b)−1/2 − 1 and N2 = (2b)−1/2 + 1 > N1. By convexity, θ1 � 0 if N � N1or b � 1

2(N+1)2 . Also, θ1 < 2b if (2b)−1/2 − 2 < N < (2b)−1/2 + 2 or 12(N+2)2 < b < 1

2(N−2)2 .Thus, the unique integer N , such that θ1 ∈ [0,2b), is given by (4).

Consider the mediator with the bias β0 = − a2−a12 = − 2θ1+b−b

2 = −θ1. In this case, the medi-ator’s type ω2 is indifferent between a1 and a2, since by (2),

UM(a2,ω2) − UM(a1,ω2) = a2 + a1

2− ω2 − β0 = a2 − ω2 = 0.

We show now that no mediator’s type ωk , k = 2, can beneficially deviate from purely in-ducing ak . Property (A) of Lemma 5 guarantees that ω1 purely induces a1. For ωk , k � 3,note that ∂UM(a,ωk)/∂a < 0, a > ωk + β0. Since ak = ωk � ωk + β0, it follows that UM(a,ωk) < UM(ak,ωk), a > ak . Also, (2) implies that UM(ak,ωk)−UM(ak−1,ωk) � 0 if and only ifak+ak−1

2 � ωk + β0, or

ak + ak−1

2− ωk − β0 = ωk−1 + ωk

2− ωk − β0 = ωk−1 − ωk − 2β0

2� 0,

which gives ωk − ωk−1 � −2β0 = 2θ1, k � 3. Because

ωk − ωk−1 = θk + θk−1

2− θk−1 + θk−2

2= θk + θk−1

2

= θk−1 + θk−2

2+ 4b = ωk−1 − ωk−2 + 4b > ωk−1 − ωk−2, k � 3,

it is sufficient to show that ω3 − ω2 = θ3−θ12 � 2θ1, or θ3 � 5θ1. Since θ3 = 3θ1 + 2b + ξ >

4θ1 + ξ , it follows that θ3 > 5θ1 if ξ � θ1. From (8), we have

N(ξ − θ1) = 1 − θ2 − 2bN(N − 1) − Nθ1 = 1 − 3θ1 − 2b − 2bN(N − 1) − Nθ1

> 1 − 2b(N + 3) − 2b(1 + N(N − 1)

)� 1 − 4 + N2

(N + 1)2= 2N − 3

(N + 1)2,

where the first inequality follows from θ1 < 2b, and the second one follows from b � 12(N+1)2 .

Thus, ξ > θ1 � 0 for N � 2. For N = 1, we have b ∈ (1/18,1/8]. Then, (7) leads to θ1 =13 (1 − 23/2b1/2) < 1

3 (1 − 23/2

181/2 ) = 19 and ξ − θ1 = 1 − 4θ1 − 2b � 1 − 4

9 − 2 18 = 11

36 > 0. �Proof of Lemma 3. If b = 1

2N2 for some integer N � 1, then β0 = 0 and VP (b) is reachedin the N -interval CS equilibrium. To replicate this equilibrium in strategic mediation, the suffi-cient condition is |β| � b + θ1/2 = 2b, where θ1 = 2b is the first expert’s cutoff type in the CSequilibrium (Li [14]).

If 12(N+2)2 < b < 1

2(N+1)2 , then N(b) = N . Consider the optimal equilibrium (7)–(10) withthe mediator’s bias β0 = −θ1. We will construct a continuum of similar equilibria if β is close

to β0. In particular, we parametrize them by δ, such that aδ1 = b + δ, and show that ∂βδ

∂δ|δ=0 > 0.

In this case, since both βδ and aδ1 are strictly increasing in δ and differentiable at δ = 0 (and,

hence, continuous in some neighborhood of δ = 0), there is a one-to-one mapping between δ andβ in the neighborhood of δ = 0.

From (11), (12), and aδ2 = (θδ

1 + θδ2 )/2, we obtain

θδ2 = 3θδ

1 + 2b − 2δ, aδ2 = 2θδ

1 + b − δ, and

σ δ,ω2a2

= (3θδ1 − 4δ)(3θδ

1 + 2b − 2δ)

8(θδ − δ)(θδ + b − δ). (17)

1 1


Hence, the condition aδ1 = b + δ is equivalent to βδ = (aδ

1 − aδ2)/2 = δ − θδ

1 .If b ∈ (1/8,1/2), so that N(b) = 0, the condition θδ

2 = 3θδ1 + 2b − 2δ = 1 leads to θδ

1 =(1 + 2δ − 2b)/3 that determines aδ

2 and σδ,ω2a2 by (17). Note that b < 1/2 implies θ1 > 0, and,

by the continuity of θδ1 in δ at δ = 0, it follows that θδ

1 > 0 as δ → 0. Also, σω2a2 > 0 and the

continuity of σδ,ω2a2 in (δ, θδ

1 ) at (0, θ1) imply σδ,ω2a2 > 0 as δ → 0. Similarly, b > 1/8 implies

σω2a2 < 1, and, by the continuity of σ

δ,ω2a2 in (δ, θδ

1 ) at (0, θ1), we have σδ,ω2a2 < 1 as δ → 0.

Finally, ∂βδ

∂δ= ∂

∂δ(δ − θδ

1 ) = 1/3.If b < 1/8, then following the same lines as those in Theorem 1 determines the LHS of (12):

−(1 − σ δ,ω2

a2

)(aδ

1 − b − θδ2

)2 − σ δ,ω2a2

(aδ

2 − b − θδ2

)2 = −2bθδ2 − δ2.

Then, θδ3 = θδ

2 + ξδ = θδ2 + 1−θδ

2 −2bN(N−1)

Nresults in a3 = θδ

2 + ξδ

2 = θδ2 +b −bN + 1−θδ

22N

, whichdetermines the RHS of (12):

−(aδ

3 − b − θδ2

)2 =(

bN + θδ2 − 1

2N

)2

.

Hence, (12) becomes the quadratic equation with respect to θδ2 :

Φ(δ, θδ

2

) = (θδ

2

)2 − 2(M + 1)θδ2 + (M − 1)2 − 4δ2N2 = 0, (18)

where M = 2bN2. Indeed, (18) is Eq. (16) with a variation in the constant term. By the continuityof Φ(δ, θδ

2 ) with respect (δ, θδ2 ), the solutions θδ

2 → θ2 and, hence, θδ1 → θ1 as δ → 0. Also,

12(N+2)2 < b < 1

2(N+1)2 implies θ1 ∈ (0,2b) and σω2a2 ∈ (0,1). This gives θδ

1 ∈ (0,2b) as δ → 0,

and, by the continuity of σδ,ω2a2 in (δ, θδ

1 ) at (0, θ1), implies σδ,ω2a2 ∈ (0,1). Then,

∂θδ2

∂δ

∣∣∣∣δ=0

= − Φ ′δ(δ, θ

δ2 )

Φ ′θδ

2(δ, θδ

2 )

∣∣∣∣δ=0

= 8δN2

Φ ′θδ

2(δ, θδ

2 )

∣∣∣∣δ=0

= 0,

where Φ ′θδ

2(0, θδ

2 )|δ=0 < 0, since θ2 = θ02 is the smaller root of the quadratic equation (16). That

is, βδ = δ − θδ1 = δ − θδ

2 +2δ−2b

3 and ∂βδ

∂δ|δ=0 = 1

3 .Property (A) of Lemma 5 guarantees that ωδ

1 purely induces aδ1. Since ω3 − ω2 > 2θ1, the

continuity of ωδk, k = 1, . . . ,N + 2 in δ at δ = 0 implies ωδ

3 − ωδ2 > 2|βδ| = 2(θδ

1 − δ) as δ → 0,so that the incentive-compatibility constraints for types ωk , k = 3, . . . ,N + 2, are satisfied.

Finally, given the expected payoff VE(0) of the expert’s type θ = 0, VP − VP (b) = VE(0)/3in any equilibrium (Goltsman et al. [8]). By construction above, we have V δ

E(0) = −(aδ1 − b)2 =

−δ2. Thus, V δP is continuous in δ at δ = 0 and V δ

P > VP (b) − ε if βδ is such that δ2/3 < ε. �Proof of Lemma 4. By contradiction, suppose ΩM = ∅. Consider the highest mediator’s typeω′ = max{ω | ω ∈ ΩM}, which induces a and a′ > a, and let Θ ′ be the interval of the expert’stypes, which generate ω′. By property (C) of Lemma 5, if β � 0, there exists the mediator’s typeω′′ > ω′ that purely induces a′. Thus, θ ′ = sup{θ | θ ∈ Θ ′} must be indifferent between a′ andthe lottery over a and a′, or equivalently, between a and a′. The indifference conditions for ω′


and θ ′ imply

θ ′ + b = a + a′

2= ω′ + β.

Since ω′ � θ ′, we have θ ′ − ω′ = β − b � 0, which is violated if β < b. �References

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Communication via a strategic mediator