Facilitating trust and trustworthiness: The
impact of mediation∗
Saori Chiba†

Michiko Ogaku‡

30 June 2024

Abstract
This paper examines how mediation can foster trust and trustworthinessbetween socially distant parties, focusing on investor-receiver interactions.
We analyze three mediation scenarios: an omniscient mediation, a communication device mediation, and an influential mediation affecting game
dynamics. Using communication game models, we find that while omniscient mediation enhances transaction likelihood, a communication device
mediation provides no advantage over unmediated interactions. An influential mediation, however, can motivate trustworthy behavior when receivers
are incentivized to act contrary to recommendations. Unexpectedly, wedemonstrate a case where the communication revelation principle (Sugaya
and Wolitzky 2021) doesn’t hold under Nash equilibrium when the mediator influences the game. This study contributes to the understanding oftrust building mechanisms in socially distant transactions and has implications for game theory involving communication and practical mediation
strategies.
∗
Financial support from the Trust Forum Foundation and the Kyoto University Foundation
is gratefully acknowledged.
†
Faculty of Economics, Kyoto Sangyo University, Motoyama Kamigano, Kita-ku, Kyoto,603-8555, Japan‡
Faculty of Economics, Nagasaki University, 4-2-1 Katafuchi, Nagasaki, 850-8506 Japan

1

Key words: Trust, Trustworthiness, Bayes correlated equilibrium, Communication revelation principle, Sequential communication equilibrium
JEL: D82, D83.

1

Introduction

Large social distance1 is, in itself, a significant obstacle to building and maintaining trust and trustworthiness.2 Conversely, spatial and non-spatial proximity havebeen shown to enhance the reciprocal establishment of such trust and trustworthiness.3 Trust and trustworthiness are key to initiating transactions and subsequently building relationships. How can we facilitate the initiation of successfultransactions when the social distance between people cannot be reduced?
This paper examines how mediation can foster trust and trustworthiness between
an investor and a receiver who are not socially close, thereby facilitating theinitiation of transactions. We consider three scenarios with differing mediation
characteristics. First, we consider an ideal case where the mediator possesses
perfect knowledge of each party’s type. We test this using a mechanism whose
solution concept is a Bayes correlated equilibrium (Bergemann and Morris 2016).
As intuition suggests and our results confirm, such omniscient mediation could
significantly increase the likelihood of successful transactions.
However, mediators often lack complete information about the parties involved.
Therefore, we analyze a case where the mediator does not know the parties’ types
1

This social distance could be interpreted as any socio-economic difference.
Alesina and La Ferrara (2002) found that interpersonal trust was lower in more racially heterogeneous communities using data from the General Social Survey. Several authors supported the
hypothesis put forward by Alesina and La Ferrara (2002). For example, Hoffman et al. (1996)
provided the experimental evidence that smaller social distance lead to more other-regarding
actions in dictator games. Binzel and Fehr (2013) conducted a lab-in-the-field experiment withresidents of an informal settlement of Cairo. They showed that trust is higher among friends(members of a given social network) than among strangers, and that higher trust among friends
is related to the trustor’s belief in the other’s trustworthiness.
3
Fisman et al. (2017) provided empirical evidence that cultural proximity (shared codes, belief
and ethnicity) between lenders and borrowers raises the volume of and reduces default rate using
data from a state-owned bank in India.
22and only acts as a communication device, as defined by Forges (1986). In this
scenario, the mediator receives input (self-reported types) from the players andpasses output (recommendations) to them, without direct involvement in their original game. We test this using a mechanism whose solution concept is a sequential
communication equilibrium (Myerson 1986). Unfortunately, we find that the mediator’s role in this case becomes insignificant, with the use of a communication
device providing no advantage over the parties transacting without a mediator.
This result stems from the adverse selection incentive of the bad type receiver,suggesting that if a mediator is just a communication device, the mediator shouldat least know the types of receivers.
Given these contrasting outcomes, we finally examine an alternative form of
mediation where the mediator can directly influence the original game by justifying or discounting parties’ actions. This influential mediator can be modeledby adapting a mediator in Sugaya and Wolitzky (2021). We modify it so thatthe receiver is motivated to disobey recommendations. In the presence of a largesocial distance where the decision to betray is rational for the receiver, behaving
trustworthily after being recommended to betray provides a positive surprise forthe investor and psychological benefits for the receiver. Under this assumption, weshow that the mediator could motivate trustworthy behavior, albeit because goodactions become more meaningful when not simply following recommendations.
Our analysis of the last case yields an unexpected result. We find that ourmodel could produce an example where the communication revelation principle,
defined by Sugaya and Wolitzky (2021), does not hold under the solution concept
of Nash equilibrium. The communication revelation principle is an extension of
the revelation principle for multistage games with communications. This violation
occurs due to the mediator’s recommendation inducing the receiver to act contrary
to expectations. Indeed, the principle requires that players canonically follow therecommendations. If the mediator is just a communication device, this violation3

does not occur in our example.
Using a mediator who could potentially influence the game dynamics, Sugaya
and Wolitzky (2021) showed that the communication revelation principle does
not hold under the solution concept of sequential communication equilibrium, although it does hold under the solution concept of Nash equilibrium. According toour finding, if the mediator acts solely as a communication device, “does the communication revelation principle still hold under the solution concept of sequential
communication equilibrium?” is still an open question.
This paper is structured as follows. Section 2 describes our basic game. Section 3 presents results of the basic game, where the players choose their actions
without any communication. Section 4 provides results of the mediated game, examining three cases: (1) where the mediator knows the types perfectly, (2) where
the mediator is only a communication device, and (3) where the mediator could
potentially influence the game directly. Section 5 concludes. All the proofs are in
the appendix.

2

Model

We use a game with spacial matching adapted from Tabellini (2008) and Okada(2020). The game involves two roles of players: an investor (or player 1) anda receiver (or player 2). Players of each type are continuously distributed along
a line segment of size y. Each player is randomly matched with a player of the
opposite role. Their distance, denoted by the random variable Y on a probability
space (Ω, Σ, P ) with the values in (0, y], follows a distribution with probability
density function g(y).
The matched players may initiate a transaction. Each player has two possible
actions: player 1 chooses to invest (T R) or not invest (N ), while player 2 chooses
to return some rewards (T W ) or not (BE). Let Ai be the set of actions of player
i and A := A1 × A2 .
4

Players could obtain both material payoffs and psychological benefits fromtheir transaction. Table 1 presents the material payoffs, where λ > 0 is player 1’s
initial endowment, Λλ is player 2’s gross earning upon receiving λ, and α ∈ (0, 1)
is the allocation rule for the gross earnings.
TW

BE

TR

αΛλ, (1 − α)Λλ

0, Λλ

N

λ, 0

λ, 0

Table 1: Material payoffs
If players were to consider only material payoffs, player 2 would have an incentive to betray (choose BE), which would lead player 1 to choose N . Consequently,(N, BE) would be the equilibrium.
Psychological benefits represent concepts of trust and trustworthiness. We
make the following assumption:
Assumption 1: d1 > λ, d2 > αλΛ > λ.
This assumption gives the possibility of successful transactions depending on thesocial distance between players, where player 1 chooses T R and player 2 chooses
T W . The value of psychological benefits for each player i is di e−θi y , where di is thebenefit value in a match with zero distance, and θi > 0 is the decaying rate. Thesebenefits are earned regardless of the other player’s action and make (T R, T W ) thesocially optimal outcome.
Let ui : Ai × A−i × Θ → R be the utility function of player i, where Θ :={g, b} × {g, b} × (0, ȳ] is the set of all states. Each element θ ∈ Θ comprises ofplayer 1’s type, player 2’s type (defined later), and their distance y, respectively.

5For (ai , a−i , θ) ∈ Ai × A−i × Θ, we define:ui (ai , a−i , θ) = ζi (ai , a−i ) + di e−θi y δi ,where ζi (ai , a−i ) is the material payoff for player i when (ai , a−i ) ∈ Ai × A−i ischosen, and δi is 1 if ai is T R or T W , and 0 if otherwise. Under Assumption 1, if
the social distance y is sufficiently small, T R and T W become dominant strategies
for the players.
We further assume that each player has a type: “good” (g) or “bad” (b).
Hereafter we use “good (bad) type” or “type (j = good, bad)” to describe player
type. The proportion of good players among players i is ni . Good players are
more tolerant of social distance than bad players. Let θij be the decay rate forplayer i of type j, with 0 < θig < θib . Consequently, good players choose T R and
T W over a wider interval of y.
The basic game G consists of: (i) the set of actions Ai and utility function ui
for each player i and (ii) a common prior ψ ∈ ∆++ (Θ).
We also define signals on types. Let T := T1 × T2 be a set of random vectors on
the probability space (Ω, Σ, P ). Each element t ∈ T and Y are independent. Let
π be the distribution of t ∈ T . For each i, Ti := Ti1 × Ti2 denotes the set of signalsthat player i receives at the game’s start. A typical element is ti = (t1i , t2i ) ∈ Ti ,where t1i is a perfect signal about player i’s type, and t2i is a signal about the otherplayer −i’s type. Although t11 and t12 are independent of each other, t21 and t22 arepotentially correlated.
Let S 1 be an information structure composed of (T 1 , π 1 ), given at the game’s
start. This is the information structure studied in Okada (2020). We also consider
an information structure composed of T = (T, π), which we’ll use when discussing
a mediator-provided information structure. The two tuple (G, S 1 ) is a standard
incomplete information game played by the players in our model.

6Let βi : Ti × (0, y] → ∆(Ai ) be a strategy for player i in a match with distancey ∈ (0, ȳ] when given signals ti ∈ Ti , where ∆(Ai ) is the set of all Borel probability
measures on Ai . Let Bi (C|t−i , y) be the average probability that an individual in
the population of player i chooses either T R or T W in a match with distance y.
Player 1 of type j is indifferent between choosing T R and N if and only if:jαΛλB2 (C|t1 , y) + d1 e−θ1 y = λ.

(1)

The distance y satisfying (1) is given by:

y = I1j (B2 (C|t1 , y)) =

1
d1
.
j log
λ(1 − αΛB2 (C|t1 , y))θ1

Similarly, for player 2, given B1 (C|t2 , y), the distance y that makes player 2 indifferent between choosing T W and BE is:y = I2j (B1 (C|t2 , y)) =

1
d2
.
j log
λ(αΛB1 (C|t2 , y))θ2

We consider that these Iij (·) as mappings from [0, 1] to R+ . Let Iij (p) = +∞if Iij (p) cannot be defined for a given p ∈ [0, 1]. We call Iij (·) a trust threshold
function. Let νij (y) be the inverse function of Iij , νij (y) = (Iij )−1 (y) for y ∈ (0, y].
We define ν1j (y) = 0 for y < I1j (0) and ν2j (y) = 0 for y < I2j (1).
Note that I1j is increasing in the probability B2 (C|t11 , t21 , y) and I2k is decreasingin the probability B1 (C|t12 , t22 , y). Furthermore, since good players have lower decay
rates, the inequality Iig > Iib holds for i = 1, 2.

3

Unmediated gameTo understand the results without mediation, we can refer to Okada (2020).
Okada (2020) demonstrates the Bayes Nash equilibrium of the incomplete information game (G, S 1 ). A partial excerpt of his findings is as follows. Let
7

(a, b)+ := max(a, b) and (a, b)− := min(a, b).
Proposition 1. (Okada 2020) Suppose Assumption 1 holds. For (I1b (n2 ), I2b (1))+ <y < I2b (n1 ):• Good players 1 and 2 take T R and T W respectively with probability 1• Bad players employ mixed strategies:– Bad player 1 chooses T R with probability (ν2b (y) − n1 )/(1 − n1 )– Bad player 2 chooses T W with probability (ν1b (y) − n2 )/(1 − n2 )As we will demonstrate in the following section, introducing a mediator withperfect knowledge of each player’s type improves upon this outcome.

4

Mediated game

We now consider the case where (G, S 1 ) is played with mediation. Before thegame, the mediator privately recommends actions to each player. Each player
can either follow or dismiss this recommendation. The optimal condition occurs
when the mediator perfectly knows each player’s type and aims to maximize social
welfare, which in this model achieved when the action pair (T R, T W ) is played.
We first consider this ideal case. The equilibrium concept used is Bayes correlated
equilibrium, as defined by Bergemann and Morris (2016).

4.1

Case: Mediator with perfect knowledge of types

The mediator’s recommendations are defined as follows:
Let σ : Θ → ∆(A1 × A2 ) be the mediator’s decision rule in the game (G, S 1 ).
Recommendations follow this the decision rule σ. The mediator’s goal is to increase
the probability of the action pair (T R, T W ) being played. We assume the mediator
only recommends actions that players will follow, aligns with our equilibrium
concept.
8

Definition 1. (Bayes correlated equilibrium) A decision rule σ is a Bayes correlated equilibrium of (G, S 1 ) if it is obedient for (G, S 1 ).
In Definition 1, obedience satisfies the following constraints.
Definition 2. (Obedience) A decision rule σ is obedient for (G, S 1 ) if, for i ∈{1, 2}, t1i ∈ {g, b} and ai ∈ Ai , we have:X

π 1 (t1i , t1−i )σ(ai , a−i |θ)ui (ai , a−i , θ)a−i ,t1−iX≥

π 1 (t1i , t1−i )σ(ai , a−i |θ)ui (a′i , a−i , θ)a−i ,t1−i

for all a′i ∈ Ai .
The optimal recommendation is given as a solution to the following mediator’s
problem:
Problem A: For each y for t11 , t12 ∈ {g, b},

max1 1X

σ(T R,T W |t1 ,t2 ,y) 1 1
t1 ,t2 ∈{g,b}π 1 (t11 , t12 )σ(T R, T W |t11 , t12 , y)subject to the obedience constraints.
We focus on the case where the mediator’s recommendation is potentially effective, in particular when the players’ social distance is in the interval (I2b (1), I2b (n1 )],with the assumption of I1g (0) < I2b (1).
Assumption 2: I1g (0) < I2b (1).
Since social distance y is publicly observable, it often determines action choices.
For very small y affecting both players (y ≤ I2b (1)), (T R, T W ) is achieved withprobability one without mediation. For y that is very small only from the perspective of player 1 (y ∈ (I2g (1), I1b (0)]), (T R, BE) is inevitable even with mediation. We focus on y > max(I1g (0), I2b (1)), where mediation can significantly
9impact outcomes. In this range, good players can potentially choose (T R, T W )with probability one, while bad players could choose (N, BE).
Proposition 2.

Suppose Assumptions 1 and 2 hold. For players with social dis-

tance y ∈ (I2b (1), I2b (n1 )] playing (G, S 1 ), the mediator’s optimal recommendations
are as follows:
If the relative weights of the indifference probabilities satisfy the inequalityν1b (y) ≥n2 (1 − ν2g (y))(1 − ν1b (y)),(1 − n2 )(1 − ν2b (y))

(2)

then

σ(T R, T W |θ1 ) = 1, σ(T R, BE|θ1 ) = 0,n2 (1 − ν2g (y))(1 − ν1b (y)),(1 − n2 )(1 − ν2b (y))
n2 (1 − ν2g (y))(1 − ν1b (y))
b,σ(T R, BE|θ2 ) = 1 − ν1 (y) +(1 − n2 )(1 − ν2b (y))ν2g (y) − n1
, σ(T R, BE|θ3 ) = 0,σ(T R, T W |θ3 ) =1 − n11 − ν2g (y),σ(N, BE|θ3 ) = 0, σ(N, T W |θ3 ) =1 − n1"#ν2b (y) − n1 b
n2 (1 − ν2g (y))(1 − ν1b (y))σ(T R, T W |θ4 ) =ν1 (y) −,1 − n1(1 − n2 )(1 − ν2b (y))"#
1 − ν2b (y)n2 (1 − ν2g (y))(1 − ν1b (y))
b· ν1 (y) −σ(N, T W |θ4 ) =,1 − n1(1 − n2 )(1 − ν2b (y))"#ν2b (y) − n1n2 (1 − ν2g (y))(1 − ν1b (y))
b1 − ν1 (y) +,σ(T R, BE|θ4 ) =1 − n1(1 − n2 )(1 − ν2b (y))"#
1 − ν2b (y)n2 (1 − ν2g (y))(1 − ν1b (y))
b1 − ν1 (y) +,σ(N, BE|θ4 ) =1 − n1(1 − n2 )(1 − ν2b (y))σ(T R, T W |θ2 ) = ν1b (y) −where θ1 = (g, g, y), θ2 = (g, b, y), θ3 = (b, g, y), θ4 = (b, b, y), respectively.
The optimal recommendation in Proposition 2 advises the bad player 2 tochoose T W more often than in his mixed strategy. Additionally, since ν1b (y) ≥n2 implies (2), good players 1 and 2 are recommended to choose T R and T W
10respectively with probability one for a larger interval of social distance than in theunmediated game. This leads to the following result:Corollary 1. The optimal recommendation in the mediated game given in Proposition 2 provides a higher probability of a successful transaction than the unmediatedgame shown in Proposition 1.
However, as we will see in the following subsections, if the mediator does notknow each player’s type, the mediation effect will be limited.

4.2

Case: Mediator as a communication device

We now consider the case where the mediator does not know the players’ types
and is not directly involved in their transactions. This scenario aligns with theconcept of a communication device as described by Forges (1986), where the mediator receives inputs from players and passes output to players without directly
participating in the original game. We demonstrate that in this case, a canonical communication equilibrium (as defined in Forges (1986)) is achieved, with themediator only recommending the actions that players would have taken without
mediation.
To address potential ambiguity in player identification when the mediator functions solely as a communication device, we assume a fixed communication order:
Player 1 privately communicates with the mediator first, followed by player 2.
As in the previous case, we focus on the scenarios where the players’ social distance falls within the interval (I2b (1), I2b (n1 )], maintaining the assumption that
I1g (0) < I2b (1).
The mediator’s problem, requiring inputs (type reporting) to provide outputs(recommendations), is formulated as follows.

11Problem B: For each y for t11 , t12 ∈ {g, b},

max1 1X

σ(T R,T W |t1 ,t2 ,y) 1 1
t1 ,t2 ∈{g,b}π 1 (t11 , t12 )σ(T R, T W |t11 , t12 , y)subject to truth-telling and obedience constraints.
The truth-telling constraints are defined as:
Definition 3. (Truth-telling constrants) A decision rule σ for (G, S 1 ) induces
players’ truth telling if, for i ∈ {1, 2}, t1i ∈ {g, b}, we have:X

π 1 (t1i , t1−i )σ(ai , a−i |t1i , t1−i , y)ui (ai , a−i , t1i , t1−i , y)ai ,a−i ,t1−i

≥

X

′

π 1 (t1i , t1−i )σ(ai , a−i |t1i , t1−i , y)ui (a′i , a−i , t1i , t1−i , y)ai ,a−i ,t1−i′for all a′i ∈ Ai and t1i ∈ {g, b}.
The solution to Problem B can be viewed as a one-period version of the sequential communication equilibrium described in Myerson (1986). While not explicitly
stated, it is understood that the mediator’s recommendations are restricted to{T R} for good player 1 and {T W } for good player 2, aligning with the rule ofrecommendations in Myerson (1986) that excludes codominated actions.
Problem B does not explicitly describe the conditional probability systems introduced in Myerson (1986). However, it is known that any sequential equilibriumin the sense of Kreps and Wilson (1982) is a sequential communication equilibrium
as defined by Myerson (1986). Therefore, it suffices to show that the solution to
Problem B can be viewed as a one-period version of sequential equilibrium. Thiscan be demonstrated as follows: Proposition 4 shows that the equilibrium of Problem B is equivalent to the Bayes Nash equilibrium of a one-period game without amediator, where two players can have two possible types independently. Furthermore, the assessment at the terminal nodes can be considered reasonable in the12sense of Fudenberg and Tirole (1991), and in this case, the concept of reasonableassessments is equivalent to the concept of consistent assessment as defined by
Kreps and Wilson (1982). As shown by Fudenberg and Tirole (1991), this implies
the equivalence of perfect Bayesian equilibrium and sequential equilibrium, where
the assessments are considered only at the terminal nodes.
If the mediator ignores the truth-telling constraints and maintains the optimal
recommendation from Proposition 2, the bad player 2 has an incentive to misreporttheir type as good and then choose N with probablity one. The truth-telling andthe obedience constraints ensure that it is always rational for player i of type j
to be honest and obedient at both the reporting and action stages, provided they
have not previously lied.
However, the addition of truth-telling constraints renders the mediator ineffective, as shown in the following propositions.
Proposition 3. The optimal decision rule σ reduces to the strategy profile in theunmediated game shown in Proposition 1.
This result is intuitive: the rationality of honesty for bad player 2 requires that
player 1 chooses T R with the same probability against both good and bad player
2. This can be implemented when player 1 chooses the strategy without knowing
player 2’s type. For a successful transactions, it is better not to inform bad player
2 of player 1’s type. If uninformed, bad player 2 chooses the mixed stragegy forboth types of player 1, which is sufficient for good players 1 and 2 to choose T R
and T W respectively with probability one.
More generally, when a mediator functions solely as a communication device,
the optimal mediation rule is equivalent to a Bayes Nash equilibrium of (G, S ∗ ),where S ∗ is either an expansion of S 1 or S 1 itself. This can be easily proved byextending the theorem by Bergemann and Morris (2016), which states that if a
mediator has knowledge of the players’ types, a decision rule is a Bayes correlated13equilibrium if and only if there exists a Bayes Nash equilibrium of (G, S ∗ ) thatinduces σ. The ”if” part of this theorem still holds true even when the mediator
serves as a communication device, provided that the decision rule satisfies truthtelling constraints and is a Bayes correlated equilibrium.
Proposition 4. Suppose a mediator is simply a communication device. If a decision rule σ of the mediator for (G, S 1 ) satisfies truth-telling constraints and is a
Bayes correlated equilibrium, then there exists a Bayes Nash equilibrium of (G, S ∗ )that induces σ, where S ∗ is either an expansion of S 1 or S 1 itself.
Truth-telling constraints typically limit the informativeness of the expansion
S ∗ of S 1 as seen in Proposition 4. If there are no conflicts of interest between
players, these constraints may not be binding, allowing the mediator to function
as if it had perfect knowledge. Otherwise, in extreme cases, players may dismiss
their communication device entirely. This case corresponds to Proposition 3.

4.3

Case: Influential mediator

Given the contrasting outcomes discussed above, we now examine a case where
the mediator can directly influence the original game. The mediator’s recommendation justifies or discounts players’ actions, potentially altering their payoffs
based on whether they follow the recommended action. Under certain assumptions, we demonstrate that even if it lacks type knowledge, if it is influential a
mediator could motivate trustworthy behavior, thereby facilitate successful transactions. Furthermore, we unexpectedly find that this case exemplifies a situation
where the communication revelation principle, as defined by Sugaya and Wolitzky(2021), does not hold under either sequential communication equilibrium or Nashequilibrium solution concepts.
The case is as follows: The mediator’s recommendation discounts the players’
actions. Player 1 may greatly appreciate if player 2 chooses T W , even when14the mediator must recommend BE. In this scenario, player 2 might gain extra
psychological benefits by giving player 1 such a pleasant surprise. This situation
does not arise with a mediator serving only as a communication device, but canbe considered by adapting a mediator model proposed by Sugaya and Wolitzky(2021). In their model, the mediator is allowed to take an action in each periodthat directly affects the players’ utilities. While it may not be natural to considera case where the mediator invests or returns a part of profits, this action could beinterpreted as a record of the mediator’s recommendation. If the mediator records
a recommended action for a player, this could influence whether the players follow
the recommendation.
We consider a scenario where the social distance falls within the range I2g (1) <y ≤ I1b (0). In this range Okada (2020) demonstrated that in the absence ofa mediator, players adopt pure strategies: player 1 takes T R and player 2 takes
BE. Since the players adopt pure strategies, a communication device alone cannot
alter the equilibrium presented by Okada (2020).
We assume a mediator who records the action recommended for player 1. Whilemaintaining both the monetary and psychological benefits for player 1 as before,
we modify the psychological benefits of player 2 as follows: Suppose that if themediator recommends BE, if player 2 disobeys, and if this provides player 1 witha pleasant surprise, then player 2 gains φ as an extra psychological benefit suchthatb(1 − α)Λλ + d2 e−θ2 y + φ > Λλ.
The monetary benefits of player 2 remain unchanged. Under these conditions,both types of player 2 will be motivated to disobey the BE recommendation.
First, let’s consider the outcome of this change under the solution concept ofsequential communication equilibrium. We restrict the range of the mediator’s recommendation for both types of player 1 to {T R} because N is the codominated15action for them. Furthermore, we must restrict the ranges of the mediator’s recommendations for both types of player 2 to empty sets. This is because both T W
and BE are codominated actions for both types of player 2. If T W is recommended, they would disobey because both types of player 1 always choose T R withprobability 1, given that the social distance in I2g (1) < y ≤ I1b (0) is too large forthem to take T W . They also disobey BE because they are better off by disobeying the BE recommendation. If the range of the mediator’s recommended actions
is the empty set, both types of player 2 play BE. Consequently, the equilibrium
is equivalent to the Bayes Nash equilibrium in the absence of the mediator. In
this situation, the communication revelation principle does not hold. Truth-telling
constraints are satisfied because the mediator’s recommendations are the same foreach type of player. Obedience constraints hold in the sense that players do notdisobey the mediator’s recommendation. However, the communication revelation
principle requires players to canonically follow the recommended actions.
Next, let’s examine the outcome of this change under the Nash equilibrium
solution concept. In this analysis, we assume that all players are good types,
i.e., n1 = n2 = 1. We do not restrict the range of the mediator’s recommendations. If the mediator were only a communication device, it would be classified as
autonomous, as defined in Forges (1986), where the mediator does not receive any
inputs.
Under these assumptions, we observe that player 2 deviates from both T W
and BE recommendations. Although the mediator may recommend BE because
it guarantees a transaction success with probability one, this recommendation fails
to meet the criteria of the communication revelation principle, which limits theactions to those that canonically follow the mediator’s recommendations.
Proposition 5. The communication revelation principle does not hold under the
Nash equilibrium solution concept if the mediator is allowed to take an action ineach period.
16

If we interpret the revelation principle as a principle stating that a social
choice function defined on the set of types is implementable with an induced
direct mechanism, such a principle holds for this example. Knowing that player 2 iscontrarian, the mediator can induce the most preferable outcome by recommending
the opposite action to player 2 than the one the mediator wishes them to take,when recommending to player 1 the action that the mediator wishes them to take.

5

Conclusion

This paper applies various communication games with a mediator to an investment game where trust and trustworthiness are crucial. Our findings provide
multiple important implications. First, when the mediator knows the types of theinvestor and receiver, the mediation increases the probability of successful transactions, even without mandatory compliance. Second, however, if the mediatorlacks knowledge of the parties’ types, the effect is limited. When acting solely
as a communication device without influencing the original game, the mediation
effect disappears and the recommendations fail to convey useful information forimproving transactions. This suggests that, in order to facilitate transactions
where trust is paramount, a communication device mediator should be selectedfrom among those who, at minimum, know the receivers’ types. Third, when the
mediator can influence the parties’ original game, a positive mediation effect canbe expected. We specifically examine cases where a mediator’s recommendation
discounts the players’ actions, and where the trustworthy behavior of disobeying
a betrayal recommendation provides psychological benefits to the receiver. Successful mediation occurs when the mediator recommends to the receiver the opposite action than desired, while recommending the desired action to the investor.
Finally, and unexpectedly, our model demonstrates that the communication revelation principle may not hold under the Nash equilibrium solution concept whenthe mediator is influential, which could change the payoffs of the parties’ original
17

game. This study contributes to the understanding of mechanisms that foster
trust and trustworthiness in socially distant transactions, with implications forboth communication game theory and practical mediation strategies.

References
Alberto Alesina and Eliana La Ferrara. Who trusts others? Journal of Public
Economics, 85(2):207–234, 2002.
Dirk Bergemann and Stephen Morris. Bayes correlated equilibrium and the comparison of information structures in games. Theoretical Economics, 11(2):487–
522, 2016.
Christine Binzel and Dietmar Fehr. Social distance and trust: Experimental evidence from a slum in cairo. Journal of Development Economics, 103:99–106,
2013.
Raymond Fisman, Daniel Paravisini, and Vikrant Vig. Cultural proximity andloan outcomes. American Economic Review, 107(2):457–492, 2017.
Francoise Forges.

An approach to communication equilibria.

Econometrica:

Journal of the Econometric Society, pages 1375–1385, 1986.
Drew Fudenberg and Jean Tirole. Perfect bayesian equilibrium and sequential
equilibrium. journal of Economic Theory, 53(2):236–260, 1991.
Elizabeth Hoffman, Kevin McCabe, and Vernon L. Smith. Social distance andother-regarding behavior in dictator games. American Economic Review, 86(3):653–660, 1996.
David M Kreps and Robert Wilson. Sequential equilibria. Econometrica: Journalof the Econometric Society, pages 863–894, 1982.

18

Roger B Myerson. Multistage games with communication. Econometrica: Journalof the Econometric Society, pages 323–358, 1986.
Akira Okada. The cultural transmission of trust and trustworthiness. Journal of
Economic Behavior & Organization, 169:53–69, 2020.
Takuo Sugaya and Alexander Wolitzky. The revelation principle in multistage
games. The Review of Economic Studies, 88(3):1503–1540, 2021.
Guido Tabellini. The scope of cooperation: Values and incentives. Quarterly
Journal of Economics, 123(3):905–950, 2008.

Appendix A

Proofs

Proof of Proposition 2
To reduce the notation for fixed y, let θ1 = (g, g, y), θ2 = (g, b, y), θ3 = (b, g, y),θ4 = (b, b, y), respectively. Writeqi = σ(T R, T W |θi ), ri = σ(T R, BE|θi ), si = σ(N, BE|θi ), ti = σ(N, T W |θi ).

Write the average probabilities of action pairs (T R, T W ), (T R, BE), (N, BE),(N, T W ) from the perspective of player 2 as follows: For the pairs (i, j) =(1, 3), (2, 4)

q ij : = n1 qi + (1 − n1 )qj , rij := n1 ri + (1 − n1 )rj
sij : = n1 si + (1 − n1 )sj , tij := n1 ti + (1 − n1 )tj .

Similarly, write the average probabilities of (T R, T W ), (T R, BE), (N, BE),(N, T W ) from the perspective of player 1 as follows: For the pairs (i, j) =19

(1, 2), (3, 4)
q 1ij : = n2 qi + (1 − n2 )qj , r1ij := n2 ri + (1 − n2 )rj
1

s1ij : = n2 si + (1 − n2 )sj , tij := n2 ti + (1 − n2 )tj .

The mediator’s problem is rewritten as for any fixed ymax

σ(T R,T W |t1 ,y)n2 q 13 + (1 − n2 )q 24

subject to1s112 ≥(1 − ν1g (y))(s112 + t12 )(A.3)

q 112 ≥ν1g (y)(q 112 + r112 )

(A.4)
1

s134 ≥(1 − ν1b (y))(s134 + t34 )(A.5)

q 134 ≥ν1b (y)(q 134 + r134 )(A.6)

q 13 ≤ν2g (y)(q 13 + t13 )(A.7)

s13 ≤(1 − ν2g (y))(r13 + s13 )(A.8)

q 24 ≤ν2b (y)(q 24 + t24 )(A.9)

s24 ≤(1 − ν2b (y))(r24 + s24 )(A.10)

qi , ri ,si , ti ∈ [0, 1]
1 =qi + ri + si + ti for i = 1, 2, 3, 4.

The constraints from (A.3) to (A.10) are obedience constraints.
Proof of Proposition 2. Consider the case of I2b (1) < y ≤ I2b (n1 ). Since larger
values of q 13 and q 24 increase the value of objective function, the constraints (A.7)
and (A.9) are binding. Writeh = q 13 + t13 and h′ = q 24 + t24 .

20

Then, from (A.7) and (A.9)

q 13 = ν2g (y)h, t13 = (1 − ν2g (y))h,q 24 = ν2b (y)h′ , t24 = (1 − ν2b (y))h′ .
Since r13 + s13 = 1 − h and r24 + s24 = 1 − h′ , using parameters Y ≥ ν2g (y) andY ′ ≥ ν2b (y), s13 , r13 , s24 , r24 could be written as

s13 = (1 − Y )(1 − h), r13 = Y (1 − h),s24 = (1 − Y ′ )(1 − h′ ), r24 = Y ′ (1 − h′ ).
1

Similarly, let l = s134 + t34 , and using parameter X ≥ ν1b (y) and X ′ ≤ ν1b (y), s134 ,1

t34 , q 134 , r134 are written as1

s134 = (1 − X ′ )l, t34 = X ′ lq 134 = X(1 − l), r134 = (1 − X)(1 − l).

Show (A.3) is binding: Using the relations n1 s112 = n2 s13 + (1 − n2 )s24 − (1 − n1 )s134
11

and n1 t12 = n2 t13 + (1 − n2 )t24 − (1 − n1 )t34 , (A.3) is rewritten ashi(1 − n1 )l(X ′ − ν1g (y)) + n2 (1 − Y ) + (1 − n2 )(1 − Y ′ ) ν1g (y)hi≥ ν1g (y)n2 (1 − Y ) + (1 − ν1g (y))n2 (1 − ν2g (y)) hhi+ ν1g (y)(1 − n2 )(1 − Y ′ ) + (1 − ν1g (y))(1 − n2 )(1 − ν2b (y)) h′(∗)

To increase the value of the objective function, which is increasing in h and h′ , for
given l, X ′ , Y and Y ′ , the pairs of h and h′ rise until the above equation holds.
Thus, (A.3) is binding.

21

Show X ′ = ν1b (y), Y = ν2g (y), Y ′ = ν2b (y) and l = n2 (1 − ν2g (y)) + (1 − n2 )(1 −ν2b (y))/(1 − n1 ): From binding (∗), both h and h′ are increasing in X ′ . So X ′ =ν1b (y).
From binding (∗), the partial derivative of h with respect to (1 − Y ) is positivefor h′ ≥ ν1g (y), Y ′ ≥ ν2b (y), and that with respective to (1 − Y ′ ) is also positive. So
the smaller the values of Y and Y ′ , the larger the value of h. Similar arguments
hold for h′ as well. Thus, Y = ν2g (y) and Y ′ = ν2b (y).
Substituting X ′ = ν1b (y), Y = ν2g (y) and Y ′ = ν2b (y) into (∗), we obtain(1−n1 )(ν1b (y)−ν1g (y))l = n2 (1−ν2g (y))(h−ν1g (y))+(1−n2 )(1−ν2b (y))(h′ −ν1g (y)). (∗∗)
1

On the other hand, from n1 s112 + n1 t12 ≥ 0,n2 (1 − ν2g (y)) + (1 − n2 )(1 − ν2b (y))
l≤. (∗ ∗ ∗)1 − n1Since from (∗∗), l should be large as much as possible, the above inequality is
1binding at the maximum. This would be the case that n1 s112 + n1 t12 = 0.
Show binding (∗ ∗ ∗) implies q1 + r1 = q2 + r2 = 1, s1 = s2 = t1 = t = 2 = 0: From1

n1 s112 + n1 t12 = 0, we have q1 + r1 = q2 + r2 = 1 and s1 = s2 = t1 = t2 = 0.
Show the objective function n2 ν2g (y)h + (1 − n2 )ν2b h′ satisfying (∗∗) and binding(∗ ∗ ∗) is increasing in h: From (∗∗) and binding (∗ ∗ ∗), we have the equationhin2 (1−ν2g (y))h+(1−n2 )(1−ν2b (y))h′ = ν1b (y) n2 (1−ν2g (y))+(1−n2 )(1−ν2b (y)) . (†)

Using (†), the objective function could be expressed as a linear function of h such

22

that

n2 ν2g (y)h + (1 − n2 )ν2b (y)h′ =n2 (ν2g (y) − ν2b (y))
n2 (1 − ν1g (y))
b
b
h
+
(1
−
n
)ν
(y)ν
(y)
1
+
.
2 1
2
1 − ν2b (y)(1 − n2 )(1 − ν2b (y))"#

Since ν2g (y) > ν2b (y), this function is increasing in h.
Show the optimal h, h′ and corresponding X: From the equation n2 q 13 + (1 −n2 )q 24 = n1 q 112 + (1 − n1 )q 134 , we havehin1 q 112 = n2 ν2g (y)h + (1 − n2 )ν2b (y)h′ − X n2 ν2g (y) + (1 − n2 )ν2b (y) − n1 .
Set q1 = h and q2 = h′ . It does not restrict the value of objective function. (Since
s1 = t1 = 0 and h = 1 make r1 = 0, h = 1 is only allowed if q1 = h = 1. q2 = h′can be set because q2 = h′ implies q4 = (ν2b (y) − n1 )h′ /(1 − n1 ) and y ≤ I2b (n1 ).)Then, using (†) we obtainn2 (ν2g (y) − ν2b (y))(1 − n1 )gbhX[n2 ν2 (y) + (1 − n2 )ν2 (y) − n1 ] =1 − ν2b (y)ν b (y)(ν2b (y) − n1 )+ [n2 (1 − ν2g (y)) + (1 − n2 )(1 − ν2b (y))] 1
1 − ν2b (y)(a) For ν1b (y) ≥(††)

n2 (1−ν2g (y))(1−ν1b (y)): Since the value objective function is increasing(1−n2 )(1−ν2b (y))

in h, set h = 1. Then, from (††) X is determined such thatX=

n2 (ν2g (y) − ν2b (y))(1 − n1 ) + [n2 (1 − ν2g (y)) + (1 − n2 )(1 − ν2b (y))]ν1b (y)(ν2b (y) − n1 )< 1.
[n2 ν2g (y) + (1 − n2 )ν2b (y) − n1 ](1 − ν2b (y))(A.11)The last inequality follows from the fact: On the (h, X) plane, the line (††) andthe line X = h intersects at (h, X) = (ν1b (y), ν1b (y)). Since the slope of (††) is lessthan 1, when h = 1, X < h = 1.

23

From (†)′

hn2 (1 − ν2g (y))(1 − ν1b (y))
b= ν1 (y) −.
(1 − n2 )(1 − ν2b (y))(A.12)Show remained probabilities: Since l, h, h′ , X ′ and X are determined, q1 , r1 , q2 ,r2 , qi , ri , si and ti , i = 3, 4 are given by

q1 = 1, r1 = 0,n2 (1 − ν2g (y))(1 − ν1b (y))
n2 (1 − ν2g (y))(1 − ν1b (y))
b
(y)
+
,
r
=
1
−
ν
2
1
(1 − n2 )(1 − ν2b (y))(1 − n2 )(1 − ν2b (y))ν g (y) − n1
q3 = 2, r3 = 0,
1 − n11 − ν2g (y),s3 = 0, t3 =1 − n1"#ν2b (y) − n1 b
n2 (1 − ν2g (y))(1 − ν1b (y))
q4 =
ν1 (y) −,1 − n1(1 − n2 )(1 − ν2b (y))"#
1 − ν2b (y)n2 (1 − ν2g (y))(1 − ν1b (y))
bt4 =· ν1 (y) −,1 − n1(1 − n2 )(1 − ν2b (y))"#
gbν2b (y) − n1(y))(y))(1
−
ν
n
(1
−
ν
2
2
1
r4 =
1 − ν1b (y) +,1 − n1(1 − n2 )(1 − ν2b (y))"#
n2 (1 − ν2g (y))(1 − ν1b (y))
1 − ν2b (y)b
1 − ν1 (y) +.
s4 =1 − n1(1 − n2 )(1 − ν2b (y))
q2 = ν1b (y) −Proof of Proposition 3
The mediator’s problem is rewritten as for any fixed ymax

n2 q 13 + (1 − n2 )q 24

σ(T R,T W |t1 ,y)subject to the truth telling constraints for each type of player1 and 2, obedience
constraints from (A.3) to (A.10), qi , ri , si , ti ∈ [0, 1], 1 = qi + ri + si + ti for
i = 1, 2, 3, 4.

24

It is easy to see that only the truth-telling constraints for the bad player 2
need to change the optimal recommendation in proposition 2. The truth-telling
constraints for bad player 2 that change the optimal recommendation are written
as follows.

q 24 (1 − α + αν2b (y)) + r24 + t24 αν2b (y)≥ q 13 (1 − α + αν2b (y)) + r13 + t13 αν2b (y).

(A.13)(1 − α)q 24 + r24 + αν2b (y)(q 24 + t24 )≥ q 13 + r13(A.14)(1 − α)q 24 + r24 + αν2b (y)(q 24 + t24 )≥ (1 − α + αν2b (y))(q 13 + r13 ) + αν2b (y)(t13 + s13 )(A.15)(1 − α)q 24 + r24 + αν2b (y)(q 24 + t24 )≥ q 13 + (1 − α)r13 ) + αν2b (y)(r13 + s13 )(A.16)

These constraints induce bad player2 to report his type honestly. Each constraint
differs in the bad player’s strategy after his misreporting. (A.13) requires honest
reporting and obedience is preferable to the case where he misreports and then has
the strategy to obey. (A.13) requires honest reporting and obedience is preferredto the case where he misreports and has the strategy of taking N with probability
1 afterwards. (A.15) requires that this is preferred to the case where he misreports
and has the strategy of taking C with probability 1 afterwards. (A.15) requiresthat this is preferable to the case where he misreports and has the strategy ofdisobeying afterwards.
Proof of Proposition 3. The fact that only the truth-telling constraints for the badplayer 2 need to change the optimal recommendation in proposition 2 implies that25

(A.9) is binding. Writeh = q 13 + t13 and h′ = q 24 + t24 .

Then, from (A.9)
q 24 = ν2b (y)h′ , t24 = (1 − ν2b (y))h′ .
From (A.7), using the parameter X ≤ ν2g (y), q 13 and t13 are written as

q 13 = Xh, t13 = (1 − X)h.
Since r13 + s13 = 1 − h and r24 + s24 = 1 − h′ , using the parameters Y ≥ ν2g (y)and Y ′ ≥ ν2b (y), s13 , r13 , s24 and r24 are written as

s13 = (1 − Y )(1 − h), r13 = Y (1 − h)s24 = (1 − Y ′ )(1 − h′ ), r24 = Y ′ (1 − h′ ).
11Similarly, let l = s134 + t34 , and using the parameter X ′ ≤ νb1 (y), s134 and t34 are
written as1

s134 = (1 − X ′ )l, t34 = X ′ l.
If X ≥ ν2b (y), then (A.16) is binding and we have
ν2b (y)h′ + Y ′ (1 − h′ ) = Xh + Y (1 − h) + (1 − h)αν2b (y).
If X ≤ ν2b (y), then (A.15) or (A.16) binds and we have
ν2b (y)h′ + Y ′ (1 − h′ ) = (1 − α)[Xh + Y (1 − h)] + αν2b (y) orν2b (y)h′ + Y ′ (1 − h′ ) = Xh + Y (1 − h) + (1 − h)αν2b (y).

26In both cases, if h′ > ν2b (y), then h is increasing in h′ , which makes the mediator’s objective function n2 Xh + (1 − n2 )ν2b (y)h′ increasing in h. If h′ = ν2b (y),then the objective function is again increasing in h. So h = 1 could be a necessary
condition for the optimal recommendation.
Suppose h = 1. If X ≥ ν2b (y), then since (A.16) and (A.14) bind, we have
X = ν2b (y)h′ + Y ′ (1 − h′ ).

(A.17)

If X < ν2b (y), then (A.15), (A.14) or (A.16) bind. If (A.16) is binding, then wehave(1 − α)X + αν2b (y) = ν2b (y)h′ + Y ′ (1 − h′ ).
However, this implies that Y ′ < ν2b (y). A contradiction. Thus, we need to havethe equation (A.17).
Show that (A.3) is binding. Using the relation n1 s112 = n2 s13 + (1 − n2 )s24 − (1 −11n1 )s134 and n1 t12 = n2 t13 + (1 − n2 )t24 − (1 − n1 )t34 , and substituting s13 = 0, h = 1
and (A.17) into (A.3), (A.3) is rewritten as(X ′ − ν1g (y))(1 − n1 )l + (ν1g (y) − n2 )(1 − Y ′ )≥ [(ν1g (y) − n2 )(1 − Y ′ ) + (1 − ν1g (y))(1 − ν2b (y))]h′(A.18)

To increase the value of the objective function that is increasing in h′ , increase h′until the equality in (A.18) holds. So (A.3) is binding.
Show X ′ = ν1b (y), Y ′ = ν2b (y) and X = ν2b (y). Since (A.18) is binding, h′ isincreasing in X ′ . So X ′ = ν1b (y). Also, since the partial derivative of h′ withrespect to (1 − Y ′ ) is positive, Y ′ should be the smallest, and we have Y ′ = ν2b (y).
From (A.17), we have X = ν2b (y).

27

1

1

Show s112 = t12 = 0 and l = (1 − ν2b (y))/(1 − n1 ). From n1 s112 + n1 t12 ≥ 0, l isbounded from above as follows.

l≤1 − ν2b (y)n2 (1 − X) + (1 − n2 )(1 − ν2b (y))=.
1 − n1
1 − n1(A.19)

Since the binding (A.18) implies that l should be as large as possible for larger h′ ,1

(A.19) is binding and this is when n1 s112 + n1 t12 = 0.
Show h′ = (ν1 (y) − n2 )/(1 − n2 ) for y ≥ I1b (n2 ). Since (A.18) and (A.19) arebinding, we haveh′ =

ν1b (y) − n2.
1 − n2This h′ is zero or positive for y ≥ I1b (n2 ).
Show qi , ri , si , ti for i = 1, . . . , 4. The optimal values of parameters h = 1, h′ =(ν1b (y) − n2 )/(1 − n2 ) and X = ν2b (y) are valid for (I2b (1), I1b (n1 ))+ < yleqI2b (n1 ) if

q1 = 1, r1 = s1 = t1 = 0,
ν1b (y) − n2
1 − ν1b (y), r2 =, s2 = 0, t2 = 0
1 − n2
1 − n2ν b (y) − n1
1 − ν2b (y)q3 = 2, r3 = 0, s3 = 0, t3 =1 − n1
1 − n1bb(ν (y) − n1 )(ν1 (y) − n2 )q4 = 2(1 − n1 )(1 − n2 )b(ν (y) − n1 )(1 − ν1b (y))
r4 = 2(1 − n1 )(1 − n2 )(1 − ν2b (y))(1 − ν1b (y))
s4 =(1 − n1 )(1 − n2 )(1 − ν2b (y))(ν1b (y) − n2 )t4 =.
(1 − n1 )(1 − n2 )q2 =This result is equivalent to the result in the silent game shown in proposition
1.
28

Proof of Corollary 1
Proof. The optimal recommendation in proposition 2 is obtained under the condition that the obedience constraint inducing to take T R is not binding for bad
player 1. This is true for good player 1 for y ∈ (I2b (1), I2g (1)). It implies that bad
player 2 is recommended to take T W more than his mixed strategy in the silent
game. In addition, since ν1b (y) ≥ n2 implies (2), good players 1 and 2 are recommended to take T R and T W respectively with probability 1 for a larger intervalof social distance. As a result, the recommendation provides a higher probabilityof successful transactions than in the silent game shown in proposition 1.

Proof of Proposition 4
It is proved by a minor change of the proof of Theorem 1 of Bergemann and Morris(2016).
Proof of Proposition 3. Suppose that σ is a Bayes correlated equilibrium satisfying
truth-telling constraints. Then we have for each y, i, t1i ∈ T 1 , ai ∈ A and a′i ∈ AiX

π 1 (t1i , t1−i )σ((ai , a−i )|t1i , t1−i , y)ui ((ai , a−i ), t1i , t1−i , y)a−i ,t1−i≥X

π 1 (t1i , t1−i )σ((ai , a−i )|t1i , t1−i , y)ui ((a′i , a−i ), t1i , t1−i , y).

a−i ,t1−i

Note that π(t1i , t1−i |y) = π(t1i , t1−i ) from the independence of types t1 and social
distances y, and social distance y is observable for all players and the mediator.
Let S ∗ = ((Ti∗ )2i=1 , π ∗ ) be an expansion of S 1 , where Ti∗ = Ti1 × Ti2 , Ti2 = Ai for
i = 1, 2, and π ∗ is defined such thatπ ∗ ((t1i , ai )2i=1 |y) = π(t1 |y)σ(a|t1 , y) = π(t1 )σ(a|t1 , y)for t1 ∈ T 1 , a ∈ A and y ∈ (0, y].

29Consider the strategy βj∗ for player j in the game (G, S ∗ ) such that




1

if a′j = aj ,0

if a′j ̸= aj ,βj∗ (a′j |(t1j , aj )) =


for all t1j ∈ Tj1 and aj ∈ Aj . Then for a social distance y ∈ (0, y], the expectedutility of player i who is observing signal (t1i , ai ) and chosing action a′i ∈ Ai in thegame (G, S ∗ ) is propotional toXπ ∗ ((t1i , t1−i ), (ai , a−i )|y) a′−i ,a−i ,t1−iYβj∗ (a′j |t1j , aj ) ui ((a′i , a′−i ), t1i , t1−i , y)

j̸=i

=

X

π ∗ ((t1i , t1−i ), (ai , a−i )|y)ui ((a′i , a−i ), t1i , t1−i , y)a−i ,t1−i=Xπ(t1 )σ(a|t1 , y)ui ((a′i , a−i ), t1i , t1−i , y).

a−i ,t1−i

So the Bayes correlated equilibrium σ satisfying truth-telling constraints implies
Bayes Nash equilibrium conditions under the strategy profile β ∗ . By its constraction, the stragegy profile β ∗ induces σ.

30