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Kim: Endogenous Choice of a Mediator
consider a mediator to be represented as a mechanism. The definitions of what follows are done in
full generality, although I later come to a specific problem in the context of international conflict
in Section 2.2, in which I will be interested in the case of two players, each with two types, and two
outcomes.
2.1 The Base Setup
The two-person bargaining problem is characterized by the following structures:
Γ = (D, d 0 , T 1 , T 2 , u 1 , u 2 , p 1 , p 2 ),
whose components are interpreted as follows. D is the set of feasible bargaining outcomes available
to the two players from which they can choose. d 0 ∈ D denotes the disagreement outcome, which
the two players get if they fail to coordinate on choosing a mediator. For each i ∈ {1, 2}, T i is
the set of possible types t i for player i. 6 The players are uncertain about each other’s type, and
the players’ types are unverifiable. For each i ∈ {1, 2}, u i is player i’s utility payoff function from
D × T 1 × T 2 into R, such that u i (d, t) denotes the payoff to player i if d ∈ D is the outcome and
t ∈ T is the true vector of the players’ types. The payoffs are in von Neumann-Morgenstern utility
scale. Each p i is a conditional probability distribution function that represents player i’s beliefs
about the other player’s type as a function of his own type. That is, p i (t −i |t i ), i ∈ {1, 2}, denotes
the conditional probability that player i of type t i would believe about the other player’s type being
t −i .
Let T = T 1 × T 2 denote the set of all possible type combinations t = (t 1 , t 2 ). For mathematical
simplicity, I assume D and T are finite sets. Without loss of generality, I assume the utility payoff
scales are normalized so that u i (d 0 , t) = 0 for all i and all t. That is, each player could guarantee
himself a payoff of zero by refusing to involve a mediator.
For simplicity, I assume that the players’ beliefs are consistent with some prior probability
distribution ¯p in ∆(T ), which is common knowledge, under which the players’ types are independent
random variables. That is, I assume that for every i, there exists a probability distribution ¯p i in
∆(T i ) such that ¯p i (t i ) is the prior marginal probability that player i’s type will be t i and
p i (t −i |t i ) = ¯p −i (t −i ), ∀i ∈ N, ∀t −i ∈ T −i , ∀t i ∈ T i .
6 Each t i ∈ T i represents player i’s characteristics, such as preferences, strengths, capabilities, or endowments.
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