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Kim: Endogenous Choice of a Mediator cooperative game. Likewise, I can argue that the observation of the players mimicking other players in the noncooperative game comes directly from a cooperative approach. Therefore, both approaches take into account, either directly or indirectly, the information leakage problem. By either approach, I can conclude that the selection of a mediator by privately informed players is endogenous and depends on the information that the players reveal by expressing their preferences for mediation. Consequently, with both approaches, I can define the smallest solution set for the given bargaining problem: When two privately informed players bargain over the choice of a mediator, the reasonable solution of a mediator in a cooperative sense is the neutral bargaining solution; and the equilibrium solution in a noncooperative sense is the threat-secure mediator. Remarkably, I obtain the same unique solution that is farthest away from ex ante incentive efficiency. Moreover, in the class of bargaining games considered in this paper, the intuition behind a noncooperative game can be founded on the very intuition behind a cooperative bargaining theory. 6 An Illustrative Example Let us consider an example to illustrate the core idea of the symmetric interim incentive efficient set, the neutral bargaining solution, and the threat-secure set. Online Appendix C provides a more extensive illustration with detailed computations. Suppose that there are two players in the economy, and each player is one of two possible types: s (strong) or w (weak). There are two possible decisions called d 0 (war) and d 1 (peace). By the revelation principle, players bargain over the selection of a mediator, and they have candidates for the job from the set of all incentive feasible mediators, each of which is associated with a mediation mechanism. Consider the normalized utility payoffs (u 1 , u 2 ) that depend on the decisions and types as shown in the following table. The payoffs satisfy (A1) through (A3): Table 1: An Example t (s, s) (s, w) (w, s) (w, w) d = d 0 0, 0 0, 0 0, 0 0, 0 d = d 1 4, 4 -1, 7 7, -1 4, 4 36
Kim: Endogenous Choice of a Mediator 6.1 The Interim Incentive Efficient Mediators The symmetric interim incentive efficient mechanisms have the structure of µ y,z such that µ y,z puts probability 0 on d 0 for t ∈ {ss, ww}, probability y ≥ 0 on d 0 for t ∈ {sw, ws}, and probability z ≥ 0 on d 0 for t ∈ {ss} with further restrictions on y and z depending on the different regions of ¯p(s): t (s, s) (s, w) (w, s) (w, w) d = d 0 0, 0 [z] 0, 0 [y] 0, 0 [y] 0, 0 [0] d = d 1 4, 4 [1-z] -1, 7 [1-y] 7, -1 [1-y] 4, 4 [1] With these payoffs, we have p ′ = 0.2, p ∗ = 0.36, and p ∗∗ = 0.45. Figure 6.1 plots y that a symmetric interim incentive efficient mechanism µ y,z can possibly put on d 0 when t = {sw, ws} for each case characterized in Proposition 1, except that the dotted line plots ¯z(¯p(s)). Figure 6.1: Interim Incentive Efficient Mediators S(Γ) = {µ y,z } When ¯p(s) < 0.2 (Case 1 ), then the mediator with µ y(¯p(s)),0 where y(¯p(s)) = 1−5¯p(s) 1−¯p(s) is the unique ex ante incentive efficient mediator. The dark orange line depicts y(¯p(s)) (decreasing and concave in ¯p(s)) which is the lowest probability on war that an interim incentive efficient mediator can put for t ∈ {sw, ws}. When ¯p(s) ≥ 0.2 (Cases 2 , 3 , & 4 ), then the mediator who promises to implement µ 0,0 such that µ 0,0 (d 1 |t) = 1 ∀t uniquely maximizes the two players’ ex ante expected 37
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