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Kim: Endogenous Choice of a Mediator
Let v i (d, t, λ, α, β) = [(λ(t i ) + ∑ s i ∈T i
α(s i |t i ) + β(t i ))u i (d, t) − ∑ s i ∈T i
α(t i |s i )u i (d, (t −i , s i ))]/¯p(t i ).
This v i (d, t, λ, α, β) is called the virtual utility payoff to player i from outcome d, when the type
profile is t, with respect to the utility weights λ and the Lagrange multipliers α and β. With
this setup, I arrive at the most tractable conditions for computing the interim incentive efficient
mediators, which are used to prove Proposition 1 and Corollary 1. Without loss of generality, I can
focus on i = 1.
Theorem (Theorem 10.1, Myerson (1991)). An incentive feasible mediator who mediates according
to an incentive compatible and individually rational mediation mechanism µ is incentive efficient
if and only if there exist vectors λ = (λ(t i )) ti ∈T i
, α = (α(s i |t i )) si ∈T i ,t i ∈T i
, and β = (β(t i )) ti ∈T i
such that
λ(t i ) > 0, α(s i |t i ) ≥ 0, β(t i ) ≥ 0, ∀t i ∈ T i , ∀s i ∈ T i
∑
d∈D
µ(d|t) ∑ i∈N
α(s i |t i )(U i (µ|t i ) − U ∗ i (µ, s i |t i )) = 0, ∀t i ∈ T i , ∀s i ∈ T i ,
β(t i )U i (µ|t i ) = 0, ∀t i ∈ T i ,
∑
v i (d, t, λ, α, β) = max v i (d, t, λ, α, β), ∀t ∈ T.
d∈D
i∈N
(7.1)
Lemma (Thresholds). There exists nonnegative p ′ , p ∗ , and p ∗∗ such that
p ′ ≡
p ∗∗ > p ∗ , and p ∗∗ > p ′ .
−u 1 (d 1 , sw)
u 1 (d 1 , ss) − u 1 (d 1 , sw) =
−u 2 (d 1 , ws)
u 2 (d 1 , ss) − u 2 (d 1 , ws) ,
p ∗ u 1 (d 1 , ww)
≡
u 1 (d 1 , ww) + u 1 (d 1 , ws) = u 2 (d 1 , ww)
u 2 (d 1 , ww) + u 2 (d 1 , sw) ,
p ∗∗ u 1 (d 1 , ss)u 1 (d 1 , ww) − u 1 (d 1 , sw)u 1 (d 1 , ws)
≡
u 1 (d 1 , ss)u 1 (d 1 , ww) − u 1 (d 1 , sw)u 1 (d 1 , ws) + u 1 (d 1 , ss)u 1 (d 1 , ws)
u 2 (d 1 , ss)u 2 (d 1 , ww) − u 2 (d 1 , ws)u 2 (d 1 , sw)
=
u 2 (d 1 , ss)u 2 (d 1 , ww) − u 2 (d 1 , ws)u 2 (d 1 , sw) + u 2 (d 1 , ss)u 2 (d 1 , ws) ,
Proof of Lemma 1. p ′
is computed such that the participation constraints bind for the strong type
given a mediator µ 0,0 , where µ 0,0 (d 1 |t) = 1 for all t, that is: U 1 (µ 0,0 |s)|¯p(s)=p
′ = 0
←→p ′ µ 0,0 (d 1 |ss)u 1 (d 1 , ss) + (1 − p ′ )µ 0,0 (d 1 |sw)u 1 (d 1 , sw) = 0,
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