Simple Mediation in a Cheap-Talk Game ^†

Abstract

In the Crawford–Sobel (uniform, quadratic utility) cheap-talk model, we consider a simple mediation scheme (a communication device) in which the informed agent reports one of the N possible elements of a partition to the mediator and then the mediator suggests one of the N actions to the uninformed decision-maker according to the probability distribution of the device. We show that no such simple mediated equilibrium can improve upon the unmediated N-partition Crawford–Sobel equilibrium when the preference divergence parameter (bias) is small.

1. Introduction

We consider the (uniform, quadratic utility) cheap-talk model of [1] (hereafter referred to as the CS model) to study the effect of mediation in comparison to unmediated cheap-talk communication. In this game, the strategic interaction and information transmission between an uninformed decision-maker (called the receiver) and an informed agent (called the sender) has been studied, and what can be achieved by unmediated cheap-talk communication is well established. CS have proved that any (Bayesian–Nash) equilibrium in this cheap-talk game is equivalent to a partition equilibrium where the informed agent reveals one of the finitely many elements of the partition in which the true state of nature lies. The number of elements, N, in the most informative partition equilibrium in the CS model depends on the value of the preference divergence parameter, b.

Ref. [2] introduced an unmediated communication protocol which improved the welfare of two players relative to the CS equilibrium. Krishna and Morgan also constructed an example of mediated communication in the CS model, demonstrating the possibility of Pareto improvement.

Ref. [3] consider the effect of adding noise to the sender’s message in the Crawford–Sobel model. The noise can be interpreted as communication error. In addition, one can think of their communication scheme as a special kind of a mediator who passes on messages from the sender to the receiver with some exogenous noise added on. Specifically, with some probability, the mediator passes on the sender’s message to the receiver unchanged; otherwise, independent of the message sent by the sender, the mediator passes on a random message from some fixed error distribution. They show that for a sufficiently small amount of noise, it is possible to improve upon the N-partition CS equilibrium as long as $b\ne \frac{1}{2N^2}$ for any integer N and $b<\frac{1}{2}$.

Ref. [4] subsequently analysed the general form of mediation in the cheap-talk framework of CS. They derive the optimal unconstrained mediated mechanism and an upper bound on the receiver’s payoff that can be achieved by any mediated equilibrium. This upper bound is achieved by the construction of [3] when the level of noise is chosen appropriately.1 This bound is also the same as the value of the expected payoff to the receiver that is achieved by the equilibrium of the modified CS game constructed by Krishna and Morgan when $b<\frac{1}{8}$.

Here, we should point out that in all the above three models, welfare-improving mediation or noise involves a larger number of messages compared to the minimal number of messages required in the most informative CS equilibrium. None of the papers mentioned above addresses this issue of whether welfare-improving (non-strategic) mediation must necessarily involve additional messages and more actions induced in equilibrium.

We, in this paper, consider mediation schemes2 in which the informed agent reports one possible element of a partition to a mediator (a communication device) and then the mediator suggests an action to the uninformed decision-maker according to the probability distribution of the device. We ask the question whether it is sufficient to use simple schemes (involving the same number of messages, N, as in the most informative CS equilibrium) for mediation to be Pareto superior to the CS equilibrium.3

In particular, we concentrate on a specific form of mediated equilibria, that we call N-simple mediated equilibria, in which the mediator is restricted to use the same number (N) of inputs and outputs as the number of elements of the N-partition CS equilibrium. This clearly imposes a restriction on the mediated mechanisms that one may consider; however, this restriction is made in order to enable us to answer the above question. The mediator, associated with a specific probability distribution, can be interpreted as a communication scheme that the players mutually agree to use. In this scheme, the possible number of elements the sender is allowed to use and the receiver is expected to receive via the mediator is restricted to N, as in the most informative CS equilibrium. Note that b, and hence N, is commonly known to the players.

The terminology “simple mechanism” also arises elsewhere in the context of mechanism design. An interesting strand of this literature studies “robust mechanisms” which relax some of the common-knowledge assumptions about beliefs, priors or functional forms of preferences (see [10]). The objective is to identify “simple” mechanisms which are reasonably detail-free compared to traditional Bayesian mechanisms and hence robust to structural uncertainty about the environment. Simple mechanisms and rules that give simple predictions are also appealing as they are more realistic and easily understandable. Although the motivation of this literature is somewhat different from our paper, it helps to justify our focus on the “simplicity” of the language or message space used by a mediator in our model.4

Our setup of cheap talk and mediation can be used to understand real-world scenarios, for example, one involving a politician (an uninformed decision-maker), a civil servant (a mediator), and an expert scientist (an informed agent). We should point out that we are not formally modelling such a situation here, in particular, the role of a civil servant. However, one can certainly consider a situation in which the expert meets the civil servant and reports (stochastically) the true state of the nature, and the civil servant in turn suggests an action (again, probabilistically) to the politician. In such a setup, it is natural to presume that the civil servant would use a message space that is not richer or more complex than what would be used in direct communication between the expert scientist and the politician. The main question of this paper can be interpreted as asking whether such a “simple” civil servant is worth having as a mediating channel in a conversation between an uninformed politician and an informed expert.

The main result of our paper (Theorem 1) is that the N-partition CS equilibrium cannot be improved upon by the corresponding N-simple mediated equilibrium when the preference divergence parameter b is small (less than $\frac{1}{2N^2}$). In other words, when b is small, mediation needs to use more messages (relative to the minimal number of messages that can be used in the best CS equilibrium) in order to improve upon the N-partition CS equilibrium. If mediation or noise is simply a randomisation over the messages that the sender would have reported in an original unmediated CS equilibrium, then it cannot improve information transmission when the degree of preference misalignment is small.

Mediation can usually improve on the cheap talk outcome because the mediator introduces noise in the communication between the informed agent and the decision-maker (see [11]). By randomising over the actions recommended to the receiver, mediation makes it easier to satisfy the incentive compatibility constraints. However, in our setup where the message space is restricted, there is no such additional benefit from mediation when the interests of the sender and the receiver are sufficiently closely aligned.

Although [4] provided a necessary and sufficient condition for an incentive compatible mediation rule to be optimal and showed that two specific mediation rules proposed in the literature (that of [3] and by [2]) are indeed optimal for certain values of b, we do not know what the structure of other optimal mediation rules might be.

One might ask if there exists an optimal mediation rule which is also an N-simple mediated equilibrium. If the answer is yes, this could imply that a suitable randomisation over the messages used in the original unmediated CS equilibrium could improve information transmission and there would be no need to use additional messages. However, we cannot deduce the minimal number of messages required in all such optimal mediation rules from [4].

Theorem 1, in this paper, thus advances our understanding of the effect of mediation in the CS model. It partially answers the question of whether the minimal size of the message space needs to be larger for mediation to Pareto dominate the CS equilibrium. Although we do not answer the question of whether or not an N-simple mediated equilibrium can always improve on the CS equilibrium for large b, we provide a suggestive example.

2. The Model

2.1. Crawford–Sobel Game

Our setup is identical to the uniform-quadratic utility CS model, as presented in the literature. Informed readers may wish to skip this subsection.

There are $two$ agents. The informed agent, called the sender (S), precisely knows the state of the world, $\theta$, where $\theta \sim U\left[0,1\right]$, and they can send a message at no cost, based on his private information, to the other agent, called the receiver (R). The receiver, however, does not know $\theta$ but must choose some decision y based on the information contained in the signal. The receiver’s payoff is $U^R(y,\theta )=-{(y-\theta )}^2$, and the sender’s payoff is $U^S(y,\theta ,b)=-{(y-(\theta +b))}^2$, where $b>0$ is a parameter that measures the ‘bias’ in their preferences.

CS have shown that any equilibrium of this game is essentially equivalent to a partition equilibrium where only a finite number of actions are chosen in equilibrium and each action corresponds to an element of the partition. For b $<\frac{1}{2N(N-1)}$, where $N\ge 2$ is an integer, there is a partition equilibrium5 in which the state space is partitioned into N elements, characterised by $0=a_0<a_1<a_2<\dots \dots <a_{N-1}<a_N=1$, where $a_k=\frac{k}{N}+2bk(k-N)$; in this equilibrium, S sends a message for each element $\left[a_{k-1},a_k\right)$, and given this message, R takes the optimal action $y_k=\frac{a_{k-1}+a_k}{2}$. We call this the N-partition CS equilibrium. For $\frac{1}{2N(N+1)}\le b<\frac{1}{2N(N-1)}$, the “best” equilibrium (the one that maximises the receiver’s expected payoff, $E{U}^R$) is the N-partition CS equilibrium6. For such an equilibrium, the receiver’s expected payoff is $E{U}^R=-\frac{1}{12N^2}-\frac{b^2(N^2-1)}{3}$, while the sender’s expected payoff is $E{U}^S=$$E{U}^R-b^2$.

2.2. Mediated Equilibrium

Within the CS framework, we now consider mediation, a possible structure of which could be as follows: S sends a message based on his private information to the mediator; the mediator then chooses an action according to a commonly known probability distribution and recommends it to R. We here consider a specific form of direct mediation (in the spirit of canonical mechanisms, as initiated and analysed extensively by [13,14,15]) and formally define such a mediated talk below.

Definition 1. 

An $N\times M$ mediated talk is $[{\left\{x_k\right\}}_{k=0}^N,{\left\{y_j\right\}}_{j=1}^M,{\left\{p_{kj}\right\}}_{k=1,\dots N;j=1,..M}]$ where $0=x_0<x_1<x_2<\dots \dots <x_{N-1}<x_N=1$, each $y_j\in [0,1]$ for $j\in \{1,2,\dots M\}$, each $p_{kj}\in [0,1]$ for $k\in \{1,2,\dots N\}$, $j\in \{1,2,\dots M\}$ with ${\sum }_{j=1}^M{p}_{kj}=1$.

In an $N\times M$ mediated talk, S reports one of the N possible elements, $[x_{k-1},x_k)$, in which the true state $\theta$ may lie, to the mediator and given the report $\theta \in [x_{k-1},x_k)$, the mediator then recommends to R one action, $y_j$, out of the M possible actions with probability $p_{kj}$.

Our mechanism7 (an $N\times M$ mediated talk) is said to be in equilibrium if it is incentive compatible for both players, that is, if (i) S has the incentive to be truthful to the mediator given the probabilities $p_{kj}$, and (ii) R has the incentive to obey the mediator’s recommendation $y_j$, given the posterior probabilities on the state of nature.

In what follows, we will focus only on $N\times N$ direct8 mediated equilibria. We will call such an equilibrium an N-simple mediated equilibrium.

Let $g_k\left(\theta \right)$ denote the difference in expected utility from inducing the distributions ${\left\{p_{k+1j}\right\}}_{j=1}^N$ and ${\left\{p_{kj}\right\}}_{j=1}^N$ that a type $\theta$ sender would obtain. Formally, $g_k\left(\theta \right)={\sum }_{j=1}^N\left(p_{kj}-p_{k+1j}\right){\left[y_j-\left(\theta +b\right)\right]}^2$.

Definition 2. 

For any specific value of b, an $N\times N$ (or, N-simple) mediated equilibrium is an $N\times M$ (with $M=N$) mediated talk that satisfies

(i) incentive compatibility for S: for all $k\in \{1,2,\dots ,N-1\}$, $g_k\left(x_k\right)=0$ and $g_k^{\prime }\left(\theta \right)\ge 0$$\forall \theta$.

(ii) incentive compatibility for R: $y_j=\underset{y}{\mathit{argmax}}-{\sum }_{k=1}^N{q}_{kj}\frac{1}{(x_k-x_{k-1})}{\int }_{x_{k-1}}^{x_k}{(y-\theta )}^{2}d\theta$ for all $j\in \{1,2,\dots ,N\}$, where $q_{kj}$, the posterior probability that $\theta \in [x_{k-1},x_k)$, is given by $q_{kj}=\frac{(x_k-x_{k-1})p_{kj}}{{\sum }_{k=1}^N(x_k-x_{k-1})p_{kj}}$.

In $\left(i\right)$, $g_k\left(x_k\right)=0$ essentially corresponds to the “arbitrage condition” in the CS equilibrium. The incentive compatibility for the sender requires that the $x_k$ type sender is indifferent between inducing ${\left\{p_{k+1j}\right\}}_{j=1}^N$ and ${\left\{p_{kj}\right\}}_{j=1}^N$. In addition, a sender with type $\theta \in \left(x_k,x_{k+1}\right)$ should weakly prefer the distribution ${\left\{p_{k+1j}\right\}}_{j=1}^N$ over ${\left\{p_{kj}\right\}}_{j=1}^N$ and a sender with type $\theta \in \left(x_{k-1},x_k\right)$ should weakly prefer the distribution ${\left\{p_{kj}\right\}}_{j=1}^N$ over ${\left\{p_{k+1j}\right\}}_{j=1}^N$, implying that $g_k^{\prime }\left(\theta \right)\ge 0$. Note that $g_k^{\prime }\left(\theta \right)\ge 0$ at $\theta =x_k$ and since $g_k^{\prime }\left(\theta \right)$ is constant, it has to be non-negative everywhere.

In $\left(ii\right)$, the incentive compatibility for R requires that when $y_j$ has been recommended, R indeed chooses the action $y_j$ because it maximises his expected utility given his posterior beliefs.

2.3. Characterisation

An N-simple mediated equilibrium can be characterised easily. Incentive compatibility for R, as in Definition $2\left(ii\right)$, requires for all $j=1,\dots ,N$, ${\sum }_{k=1}^N{p}_{kj}\left[{(y_j-x_{k-1})}^2-{(y_j-x_k)}^2\right]=0$. This implies:

$$y_j=\frac{1}{2}\left[\frac{(1-{\sum }_{k\ne j}{p}_{jk})(x_j^2-x_{j-1}^2)+{\sum }_{k\ne j}{p}_{kj}(x_k^2-x_{k-1}^2)}{(1-{\sum }_{k\ne j}{p}_{jk})(x_j-x_{j-1})+{\sum }_{k\ne j}{p}_{kj}(x_k-x_{k-1})}\right]$$

(1)

The incentive compatibility for S, as in Definition $2\left(i\right)$, requires ${\sum }_{j=1}^{N-1}(p_{k+1j}-p_{kj})\left(y_j-y_N\right)\ge 0$ and ${\sum }_{j=1}^N(p_{kj}-p_{k+1j}){\left[y_j-(x_k+b)\right]}^2=0$ for all $k=1,\dots ,N-1$.

The latter implies for all $k=1,\dots ,N-1$,

$$\begin{array}{c}2\left(x_k+b\right)=\frac{{\sum }_{j=1}^{N-1}\left(p_{kj}-p_{k+1j}\right)(y_j^2-y_N^2)}{{\sum }_{j=1}^{N-1}\left(p_{kj}-p_{k+1j}\right)\left(y_j-y_N\right)}\hfill \\ =\frac{(1-{\sum }_{j\ne k}{p}_{kj}-p_{k+1k})\left(y_k^2-y_N^2\right)+(p_{kk+1}-1+{\sum }_{j\ne k+1}{p}_{k+1j})(y_{k+1}^2-y_N^2)+{\sum }_{j\ne k,k+1}\left(p_{kj}-p_{k+1j}\right)(y_j^2-y_N^2)}{(1-{\sum }_{j\ne k}{p}_{kj}-p_{k+1k})\left(y_k-y_N\right)+(p_{kk+1}-1+{\sum }_{j\ne k+1}{p}_{k+1j})(y_{k+1}-y_N)+{\sum }_{j\ne k,k+1}\left(p_{kj}-p_{k+1j}\right)\left(y_j-y_N\right)}\hfill \end{array}$$

(2)

Thus, an N-simple mediated equilibrium is characterised by $[{\left\{x_k\right\}}_{k=0}^N,{\left\{y_j\right\}}_{j=1}^N$, ${\left\{p_{kj}\right\}}_{k=1,\dots ,N;j=1,..,N}]$ satisfying Equations (1) and (2) with the constraints that ${\sum }_{j=1}^{N-1}(p_{k+1j}-p_{kj})\left(y_j-y_N\right)\ge 0$ for all $k=1,\dots ,N-1$.

3. Results

To state and prove our results, we take $b<\frac{1}{2N(N-1)}$, for which the N-partition CS equilibrium exists. We first observe that the N-partition CS equilibrium is actually equivalent to a particular N-simple mediated equilibrium, namely, one with $x_k=a_k=\frac{k}{N}+2bk(k-N)$ for all $k\in \{1,\dots ,N\}$; $y_j=\frac{a_{j-1}+a_j}{2}$ for all $j\in \{1,\dots ,N\}$ and $p_{kj}=0$ for all k, $j\in \{1,\dots ,N\}$, $k\ne j$.

Note that in the class of simple mediated equilibria, $E{U}^S=E{U}^R-b^2$, that is, ex ante, the sender’s and receiver’s welfare ranking agree. For any simple mediated equilibrium, it is also clear that $E{U}^R=-{\sum }_{k=1}^N[(1-{\sum }_{j\ne k}{p}_{kj}){\int }_{x_{k-1}}^{x_k}{(y_k-\theta )}^{2}d\theta +{\sum }_{\begin{array}{c}j=1\\ j\ne k\end{array}}^N{p}_{kj}{\int }_{x_{k-1}}^{x_k}{(y_j-\theta )}^{2}d\theta ]$, which implies

$$E{U}^R=-\frac{1}{3}{\sum }_{k=1}^N[(1-{\sum }_{j\ne k}{p}_{kj})\{{(y_k-x_{k-1})}^3-{(y_k-x_k)}^3\}+{\sum }_{\begin{array}{c}j=1\\ j\ne k\end{array}}^N{p}_{kj}\{{(y_j-x_{k-1})}^3-{(y_j-x_k)}^3\}].$$

We are interested in the question of when can the N-partition CS equilibrium not be improved upon by an N-simple mediated equilibrium, that is, when is the N-partition CS equilibrium is indeed the best among the set of N-simple mediated equilibria. Our main result answers the above question by solving the following constrained maximisation problem that we call the final problem.

Final problem: Maximise $E{U}^R$ among the set of N-simple mediated equilibria (as characterised in the previous section).

Before we state and prove our main result, as a first step, we consider the following constrained maximisation problem that we call the initial problem:

Initial problem: $\underset{{\left\{x_k\right\}}_{k=1}^{N-1},{\left\{p_{kj}\right\}}_{k\ne j}}{\mathrm{Maximise}}$$E{U}^R$ subject to Equations (1) and (2).

The difference between the final problem and the initial problem is simply the set of restrictions that ${\sum }_{j=1}^{N-1}(p_{k+1j}-p_{kj})\left(y_j-y_N\right)\ge 0$, for all $k=1,\dots ,N-1$.

Our first lemma below proves that for $b<\frac{1}{2N^2}$, a “corner” solution satisfies the necessary first-order conditions of the initial problem.

Lemma 1. 

For $b<\frac{1}{2N^2}$, the necessary conditions for a solution of the initial problem are satisfied at $p_{kj}=0$, for all $k\ne j$; k, $j\in \{1,\dots ,N\}$.

Lemma 1 also characterises the values of ${\left\{x_k\right\}}_{k=1}^{N-1}$ at this solution of the initial problem. They are indeed the CS values given by $x_k=x_k^{CS}=\frac{k}{N}+2bk(k-N)$, for all $k\in \{1,\dots ,N-1\}$. The formal proof of Lemma 1 has been relegated to the Appendix A.

The following result shows that the N-partition CS equilibrium satisfies the first-order conditions for a (local) optimiser of $E{U}^R$ among the set of N-simple mediated equilibria. The result is an immediate consequence of Lemma 1, and thus, the formal proof is postponed to the Appendix A.

Corollary 1. 

For $b<\frac{1}{2N^2}$, the necessary conditions for a local maximum of the final problem are satisfied at $p_{kj}=0$ for all $k\ne j$; k, $j\in \{1,\dots ,N\}$ and $x_k=x_k^{CS}=\frac{k}{N}+2bk(k-N)$ for all $k\in \{1,\dots ,N-1\}$.

This corollary shows that the CS equilibrium values constitute a candidate solution to the final problem.

We are now ready to state our main result.

Theorem 1. 

Consider $b<\frac{1}{2N(N-1)}$, for which the N-partition CS equilibrium exists. If $b<\frac{1}{2N^2}$ ($<\frac{1}{2N(N-1)}$), then no N-simple mediated equilibrium can improve upon the N-partition CS equilibrium.

Our theorem above proves that for $b<\frac{1}{2N^2}$, the N-partition CS equilibrium is actually a global maximum among the set of N-simple mediated equilibria. Furthermore, it suggests that there may exist N-simple mediated equilibria that can improve upon the N-partition CS equilibrium when $b\in [\frac{1}{2N^2},\frac{1}{2N(N-1)})$, as we demonstrate using an example in Section 4.1 below.

We first note that the global maximum exists by appealing to the Weierstrass Theorem, since the objective function is continuous and is defined over a compact set.

To prove the theorem, we first reconsider $E{U}^R$ of any N-simple mediated talk and write the following expression:

$$-3E{U}^R={\sum }_{k=1}^N[(1-{\sum }_{j\ne k}{p}_{kj})\{{(y_k-x_{k-1})}^3-{(y_k-x_k)}^3\}+{\sum }_{\begin{array}{c}j=1\\ j\ne k\end{array}}^N{p}_{kj}\{{(y_j-x_{k-1})}^3-{(y_j-x_k)}^3\}].$$

We work with the above expression, involving the variables $[{\left\{x_k\right\}}_{k=0}^N,{\left\{y_j\right\}}_{j=1}^N$, ${\left\{p_{kj}\right\}}_{k=1,\dots N;j=1,..N}]$, satisfying Equation (1) from the previous section, $y_j=\frac{1}{2}\left[\frac{(1-{\sum }_{k\ne j}{p}_{jk})(x_j^2-x_{j-1}^2)+{\sum }_{k\ne j}{p}_{kj}(x_k^2-x_{k-1}^2)}{(1-{\sum }_{k\ne j}{p}_{jk})(x_j-x_{j-1})+{\sum }_{k\ne j}{p}_{kj}(x_k-x_{k-1})}\right]$. We redefine the above expression as a function, f, explicitly as a function of the variables ${\left\{x_k\right\}}_{k=1}^{N-1}$ and ${\left\{p_{kj}\right\}}_{k\ne j}$ such that the domain satisfies Equation (1). Let us thus write

$$\begin{array}{c}f\left(x_1,\dots ,x_{N-1};p_{12},p_{13},\dots ,p_{1N};p_{21},p_{23}\dots ,p_{2N};\dots \dots \dots ;p_{N1},\dots ,p_{NN-1}\right)\hfill \\ =\sum _{k=1}^N[(1-\sum _{j\ne k}{p}_{kj})\{{(y_k-x_{k-1})}^3-{(y_k-x_k)}^3\}+\sum _{\begin{array}{c}j=1\\ j\ne k\end{array}}^N{p}_{kj}\{{(y_j-x_{k-1})}^3-{(y_j-x_k)}^3\}].\hfill \end{array}$$

The function, f, is therefore the negative of the original objective function with only the constraint (1) incorporated but without taking into account the incentive compatibility constraint for S.

Abusing notation, we treat f as a function of ${\left\{p_{kj}\right\}}_{j\ne k}$, for any fixed level of $\left(x_1,\dots ,x_{N-1}\right)$, and also, as a function of ${\left\{x_k\right\}}_{k=1}^{N-1}$, for fixed values of $p_{kj}$, for all $k\ne j$. We first observe the following result which characterises a property of the function, f.

Lemma 2. 

For any $\left(x_1,\dots ,x_{N-1}\right)$, $\frac{\partial f}{\partial p_{kj}}>0$, at $p_{kj}=0$, for all $k\ne j$; k, $j\in \{1,\dots ,N\}$.

This means that at such a corner value of the probabilities where $p_{kj}=0$, for all $k\ne j$, k, $j\in \{1,\dots ,N\}$, there is locally an open neighbourhood around this corner value where the value of the function f is larger than at the corner for any $\left(x_1,\dots ,x_{N-1}\right)$. So, Lemma 2 shows that f, as a function of ${\left\{p_{kj}\right\}}_{j\ne k}$, attains a local minimum at $p_{kj}=0$, for all $k\ne j$; k, $j\in \{1,\dots ,N\}$.

We then consider the global minimisation of f with respect to $p_{kj}$ for all $k\ne j$, and we prove the following lemma.

Lemma 3. 

For any $\left(x_1,\dots ,x_{N-1}\right)$, f, as a function of ${\left\{p_{kj}\right\}}_{j\ne k}$, attains a (global) minimum at $p_{kj}=0$ for all $k\ne j$; k, $j\in \{1,\dots ,N\}$.

The next lemma indicates an important property of f as a function of $\left(x_1,\dots ,x_{N-1}\right)$, for a particular value of ${\left\{p_{kj}\right\}}_{j\ne k}$.

Lemma 4. 

At $p_{kj}=0$, for all $k\ne j$, f, as a function of ${\left\{x_k\right\}}_{k=1}^{N-1}$, is strictly convex.

The proof of Theorem 1 is now a direct consequence of the above lemmata. We provide the formal argument in the Appendix. Theorem 1 proves that for $b<\frac{1}{2N^2}$, a global maximum among the set of N-simple mediated equilibria must coincide with the N-partition CS equilibrium.

4. Remarks

4.1. Example

We provide here an example for $N=2$ where simple mediation improves on the corresponding CS equilibrium when the bias b is appropriately larger than the bound mentioned in Theorem 1. Recall that a 2-partition CS equilibrium exists for $b<\frac{1}{4}$ and for $\frac{1}{12}\le b<\frac{1}{4}$, the best CS partition equilibrium involves two elements. Theorem 1 confirms that the 2-partition CS equilibrium cannot be improved upon by any 2-simple mediated equilibrium when b$<\frac{1}{8}$. Our result also suggests that a 2-simple mediated equilibrium may improve upon the 2-partition CS equilibrium when b is large enough, that is, for $\frac{1}{8}\le b<\frac{1}{4}$. We illustrate this comment for $b=\frac{1}{6}$. Here, the 2-partition CS equilibrium is characterised by $a=\frac{1}{6}$, $y_1=\frac{1}{12}$, and $y_2=\frac{7}{12}$ with utilities $E{U}^R=-\frac{7}{144}\simeq -0.0486$ and $E{U}^S=-\frac{11}{144}\simeq -0.0764$.

From the characterisation presented in Section 2.3, a 2-simple mediated equilibrium is given by9$(x,y_1,y_2,p_{11},p_{12},p_{21},p_{22})$, where x, $y_1$, $y_2\in (0,1)$, $p_{11},p_{12},p_{21},p_{22}\in [0,1]$ and $p_{11}+p_{12}=1$, $p_{21}+p_{22}=1$. The incentive compatibility constraints for S and for R can all be combined into one equation given by $\frac{(1-p_{12})x^2+p_{21}(1-x^2)}{4[(1-p_{12})x+p_{21}(1-x)]}+\frac{p_{12}{x}^2+(1-p_{21})(1-x^2)}{4[p_{12}x+(1-p_{21})(1-x)]}-x=b$.

Thus, a 2-simple mediated equilibrium in this setup can be characterised by three variables ($p_{12},p_{21},x$), where $x\in (0,1)$ and $p_{12},p_{21}\in [0,1]$, satisfying the above equation.

It is now easy to check that for $b=\frac{1}{6}$, $x=0.2245201023$, $y_1=0.1745967377$, $y_2=0.6077768002$, $p_{12}=0.03$, and $p_{21}=0.04$ constitute a 2-simple mediated equilibrium with utilities $E{U}^R\simeq -0.0483$ and $E{U}^S\simeq -0.0760$ and can improve upon the corresponding 2-partition CS equilibrium.

The interpretation of the above example is as follows. One can see that the partitioning point x and the two decisions $y_1$ and $y_2$ of the 2-simple mediated equilibrium are all larger than the corresponding values of the 2-partition CS equilibrium. The fact that the lower interval is bigger and the higher element of the partition is smaller in size means that more information is being transmitted. This is possible because the mediator is allowed to randomise between $y_1$ and $y_2$.

4.2. Relationship with Goltsman et al. (2009)

One might be interested in knowing how our theorem compares with the corresponding results in [4] and in particular, if there is a connection between our Theorems 1 and 2 of [4]. In Theorem 2 of [4], an optimal mediation rule is provided. One interesting corollary of this theorem is that for $b=\frac{1}{2N^2}$, this particular optimal mediation rule cannot improve upon the N-partition CS equilibrium. This implies that the N-partition CS equilibrium is optimal, irrespective of the number of messages that the players are allowed to use, when $b=\frac{1}{2N^2}$. However, since Theorem 2 of [4] is about a specific optimal mediation rule which uses more messages than our N-simple mediated equilibrium and in general, there might be a continuum of optimal mediation rules. We thus feel that this theorem is not useful in answering the question posed in our paper.

A more meaningful approach might be to ask if the technique used by [4] to prove their Theorem 2 can provide any insight or an alternative way of proving our result. Ref. [4] introduced a lemma (Lemma 2 in their paper) to derive a necessary and sufficient condition for an incentive compatible mediation rule to be optimal and to provide an upper bound on the objective function using an incentive compatible mediation rule. One might try to identify such a condition and an upper bound in the more constrained setting of N-simple mediation rules. If the characterisation of optimal mediation rules in Lemma 2 of [4] could be appropriately modified to derive a characterisation of optimal N-simple mediated equilibria, then this would provide another method of proof of our Theorem 1. We would like to point out that we do not derive such a characterisation in this paper and that such an alternative proof is not straightforward, either.

It is also worth mentioning two recent papers on the connection between communication equilibrium and correlated equilibrium [5,18] that are relevant to our work. Ref. [18] proved that (essentially) every communication equilibrium of any finite Bayesian game with two players can be implemented as a strategic form correlated equilibrium of a game extended by a cheap-talk phase before the original Bayesian game. On the other hand, specific to the CS model, [5] constructed a strategy-correlated equilibrium, that sends messages to both players before the sender sends any message to the receiver, to achieve the best possible payoff from the mediated equilibrium of the CS model. Importantly, in his construction, unlike our work, neither player needs to send messages to the device. Following these new results, one may be interested to know whether our N-simple mediated equilibrium can be obtained as a correlated equilibrium in the sense of [18], or as a strategy-correlated equilibria, as in [5]. In particular, one may further ask whether or not these new constructible correlated or strategy-correlated equilibria will involve only a few (N many) messages. Clearly, these are important questions for future research.