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Kim: Endogenous Choice of a Mediator
payoffs are in the virtual utility scales v i (·, ·, λ, α, β) with respect to λ, α, and β, instead of u i (·, ·),
and these virtual utility payoffs are transferable among the players. If (λ, α, β) satisfy the interim
incentive efficiency conditions for µ and α i (s i |t i ) > 0, then we say that type t i jeopardizes type s i . A
positive Lagrange multiplier α i (s i |t i ) implies that t i can only jeopardize s i in µ if the informational
incentive constraint is binding for µ.
In the bargaining problem Γ considered in this paper, I obtain the following corollaries.
Corollary 3. When ¯p(s) ≤ p ∗ , for the neutral bargaining solution µ 1,0 , the conditions in Myerson
(1984b)’s Theorem are satisfied for all ε ≥ 0 by any λ i (s) > λ ∗ for some λ ∗ > ¯p(s), α i (s|w) =
α i (w|s) = 0, and β i (s) = β i (w) = 0 for all i.
When ¯p(s) ≤ p ∗ , in the (λ, α, β) virtual bargaining problem with transferable virtual utility
payoffs, no incentive constraints are binding. In other words, the constraints do not matter. The
fact that there are no binding constraints is exactly what makes λ i (t i ) weights different from the
probability weights. In particular, we have λ i (s) > ¯p(s) and λ i (w) < ¯p(w) for all i. For the neutral
bargaining solution µ 1,0 when ¯p(s) ≤ p ∗ , the type-dependent weights λ i (t i ) are necessarily distorted
in a way that they scale up the actual utility of the strong type and scale down the actual utility
of the weak type, as if the strong type were more important. Then, the neutral bargaining solution
can be interpreted as if some principal maximizes the ex ante social welfare putting extra welfare
weights on the strong types in the virtual utility characterization. 17
The more interesting cases in the neutral bargaining solutions should be when the incentive
constraints bind; for the benchmark class in my framework, these cases would correspond to when
¯p(s) ∈ (p ∗ , p ∗∗ ).
Corollary 4. When ¯p(s) ∈ (p ∗ , p ∗∗ ), for the neutral bargaining solution µ 1,¯z(¯p(s)) , the conditions
in Myerson (1984b)’s Theorem are satisfied for all ε ≥ 0 by any λ i (s) > λ ∗∗ for some λ ∗∗ > ¯p(s),
some α i (s|w) > 0, α i (w|s) = 0, and β i (s) = β i (w) = 0 for all i.
When ¯p(s) ∈ (p ∗ , p ∗∗ ), the constraint that a player of a weak type should not be tempted to
pretend that her type is strong is a binding constraint for the neutral bargaining solution µ 1,¯z(¯p(s)) .
Because the weak type wants to imitate the strong type, the weak type jeopardizes the strong type.
17 Otherwise, the mediator would not have been an equilibrium in any communication game defined. This can be
seen as an analog to being renegotiation-proof. Putting more weight on the strong type’s utility allows the equilibrium
to be interim incentive efficient and inscrutable.
20