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Kim: Endogenous Choice of a Mediator
of G δ with respect to the posterior beliefs (¯q 2,i (t 2 ), ¯q 1,i (t 1 )) if and only if:
U j (G δ |t j , Σ(¯q·,i )) = ∑ t −i
¯q −j,i (t −j ){ψ j,i (t j )u j (d 0 , t) + (1 − ψ j,i (t j )) · (ψ −j,i (t −j )u j (d 0 , t)
+ (1 − ψ −j,i (t −j )) ∑ r∈R
σ i (r|t) ∑ d∈D
δ(d|r)u j (d, t))}
≥ U j (G δ , ˆr j , ψ ′ j,i|t j , Σ(¯q·,i ))
= ∑ t −i
¯q −j,i (t −j ){ψ ′ j,i(t j )u j (d 0 , t) + (1 − ψ ′ j,i(t j )) · (ψ −j,i (t −j )u j (d 0 , t)
+ (1 − ψ −j,i (t −j )) ∑ r∈R
σ i (r|t) ∑ d∈D
δ(d|r −j , ˆr j )u j (d, t))},
∀j, ∀t j ∈ T j , ∀ˆr j ∈ R j , ∀ψ ′ j,i ∈ [0, 1].
(T2)
Condition (T2) asserts that player j with type t j should not expect any other report ˆr j or any
other war strategy ψ ′ j,i to be better for him in Gδ than the strategies selected by his σ j,i and ψ j,i
when i vetoes and G δ is played.
Let ¯q = {¯q·,1 , ¯q·,2 } be a collection of vote beliefs vectors following any veto by a single player
when the other player votes for ratification. Let Σ(¯q) = {Σ(¯q·,1 ), Σ(¯q·,2 )} denote a corresponding
collection of equilibria in the resulting play of G δ with those vote beliefs.
Unanimous ratification of γ followed by truthful revelation in γ is a sequential equilibrium in
the two-stage ratification game if and only if γ is incentive compatible and γ is individually rational
relative to G δ , that is, there exists ¯q and Σ(¯q) such that for each i and each t i :
∑
¯p(t −i ) ∑ γ(d|t)u i (d, t)
t −i d∈D
≥ ∑ ¯p(t −i ) ∑ γ(d|t −i , ˆt i )u i (d, t);
t −i d∈D
and ∑ ¯p(t −i ) ∑ γ(d|t)u i (d, t)
t −i d∈D
≥ ∑ ¯p(t −i ){ψ i,i(t ′ i )u i (d 0 , t)
t −i
+ (1 − ψ ′ i,i(t i )) · (ψ −i,i (t −i )u i (d 0 , t)
(T3)
+ (1 − ψ −i,i (t −i )) ∑ r∈R
σ i (r|t) ∑ d∈D
δ(d|r −i , ˆr i )u i (d, t))},
∀ˆt i ∈ T i , ∀ψ ′ i,i ∈ [0, 1], ∀ˆr i ∈ R i .
That is, (T3) asserts that player i cannot gain by vetoing the alternative γ (and then going to war
with probability ψ ′ i,i and reporting ˆr i if G δ is played) when t i is his true type and the other player
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