Job Market Paper - Personal Web Pages - University of Chicago

Kim: Endogenous Choice of a Mediator
denotes the Lagrange multiplier for the participation constraints. Moreover, (λ, α, β) satisfy the
interim incentive efficiency conditions for the mechanism µ.
I explain more about the virtual utility payoffs later in this subsection, but for now, note that the
virtual utility differs qualitatively from the actual utility. The above theorem asserts that a neutral
bargaining solution can be characterized as an incentive feasible mediator who is not only efficient
in terms of maximizing the sum of the players’ virtual utility payoffs, but is also equitable in terms
of balancing out intertype compromise. Any allocation vector ϖ that satisfies the conditions in the
theorem for some positive λ and some nonnegative α and β can be called virtually equitable for Γ.
The first equation in the theorem defines the intertype-equity conditions such that the weighted
sums of players’ possible conditionally expected utilities should be equal. The third condition says
that a neutral bargaining solution should generate type-contingent expected utilities that are equal
to or interim Pareto superior to a limit of virtually equitable utility allocations. The theorem
implies that a player makes equity comparisons in virtual utility terms rather than in actual utility
terms. 16
The theorem gives a set of conditions that must hold for every mediator in the set of neutral
bargaining solutions. Moreover, because the set of all interim incentive efficient mechanisms would
satisfy the two axioms, I can think about picking a neutral bargaining solution over the set of
all symmetric interim incentive efficient mediators S(Γ). Let NS(Γ) denote the set of neutral
bargaining solutions identified for the class of two-person Bayesian bargaining game Γ that satisfies
assumptions (A1) through (A3). Then, for the bargaining problem considered in this paper, only
one mediator exists among S(Γ) that satisfies the conditions in the above theorem.
Theorem 1. For any two-person Bayesian bargaining problem Γ that satisfies (A1) – (A3) with
independent and symmetric priors, the neutral bargaining solution is unique.
Proposition 3. The neutral bargaining solution in S(Γ) is
If ¯p(s) ≤ p ∗ : NS(Γ) = {µ 1,0 }.
If ¯p(s) ∈ (p ∗ , p ∗∗ ): NS(Γ) = {µ 1,¯z(¯p(s)) }.
If ¯p(s) ≥ p ∗∗ : NS(Γ) = {µ 0,0 }.
16 The players can negotiate over the set of virtual utility maximizing allocations without revealing information
about their types.
18