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Kim: Endogenous Choice of a Mediator
incentive constraint for the weak type is binding in a bargaining process, which implies that the
strong type of a player begins to act according to virtual preferences which exaggerate the difference
from the weak type that he needs to distinguish himself from.
For example, when ¯p(s) = 0.4, mediator µ 1,¯z(0.4) = µ 1,0.1429 is the unique neutral bargaining
solution that gives the interim incentive efficient allocations such that
U i (µ 1,0.1429 |s) = 1.3714, U i (µ 1,0.1429 |w) = 2.4, ∀i.
The conditions in Theorem (Myerson, 1984b) are satisfied for all ε by λ, α, and β such that
λ i (s) ≥ 0.8829, λ i (w) = 1 − λ i (s),
α i (w|s) = 0, α i (s|w) = 4 7 λ i(s), β i (s) = β i (w) = 0, ∀i.
That is, the weak type jeopardizes the strong type, because the only problematical incentive constraint
is that the weak type of any player should not have any incentive to claim that her type is
strong if it is actually weak. Therefore, the strong type’s virtual utility is constructed as a positive
multiple of his actual utility minus a positive multiple of the utility payoff for the other type that
jeopardizes the strong type; and the weak type’s virtual utility is a positive multiple of her actual
utility.
Using the parameters λ i (s) = 0.9, λ i (w) = 0.1, α i (s|w) = 0.5143, α i (w|s) = 0, and β i (s) =
β i (w) = 0, the virtual utility functions are v i (d 0 , t, λ, α, β) = 0 for all i and all t (because u i (d 0 , t) = 0
for all i and all t), and for all i:
v i (d 1 , (s, t −i ), λ, α, β) = (0.9 · u i (d 1 , (s, t −i )) − 0.5143 · u i (d, (w, t −i ))) /(0.4),
v i (d 1 , (w, t −i ), λ, α, β) =(0.1 + 0.5143)u i (d, (w, t −i ))/(0.6).
That is, the virtual utility functions are as follows:
Table 3: The Virtual Utilities when ¯p(s) = 0.4 and λ i (s) = 0.9
t (s, s) (s, w) (w, s) (w, w)
d = d 0 0, 0 0, 0 0, 0 0, 0
d = d 1 0, 0 -7.4, 7.17 7.17, -7.4 4.1, 4.1
If the players made interpersonal-equity comparisons in terms of virtual utility, then each type
t i could justly demand half of the expected virtual utility that t i contributes by cooperating.
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