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Kim: Endogenous Choice of a Mediator 3 Cooperative Approach My goal is to develop a formal argument of endogenous choice of a mediator that determines the smallest possible set of solutions, so as to get the strongest possible predictions. At some level, there is potentially an endless variation to an extensive form communication game of the negotiation process over selecting mediators, which I am not explicitly modeling. Instead, I take payoffs and the existence of incentive feasible mediators as primitives. The revelation principle implies that if all incentive compatible direct mechanisms have some property, then every equilibrium of every game form has this property. With a cooperative approach, I am explicitly taking into account the information leakage problem by imposing relevant properties or axioms that must be satisfied for a reasonable solution. In this section, I characterize the interim incentive efficient set and the neutral bargaining solution using the solution concepts in Holmström and Myerson (1983) and Myerson (1984b); and I show the inefficiency in bargaining. All proofs are in Appendix A or Online Appendix B. 3.1 The Interim Incentive Efficient Set An incentive feasible mediator associated with a mechanism µ is interim incentive efficient if and only if there exists no other incentive feasible mediator with a mechanism that is interim Pareto superior to µ. If every player would surely prefer µ ′ over µ when he knows his own type, whatever his type might be, then µ ′ is interim Pareto superior to µ. Interim welfare is evaluated after each player learns his own type but before he learns the other player’s type. An incentive feasible mediator associated with a mechanism µ is ex ante incentive efficient if and only if there exists no other incentive feasible mediator with a mechanism that is ex ante Pareto superior to µ. Ex ante welfare is evaluated before any types are specified. If a mediator is incentive efficient, then another mediator who does not know any player’s actual type could not propose any other incentive feasible mediation mechanism that every player is sure to prefer according to an appropriate welfare criterion. 12 One might naively consider ex ante incentive efficiency to be a requirement for a bargaining solution. But if a player expresses a preference for the ex ante incentive efficient outcome, it would be giving information to the other side that might be detrimental. Therefore, we have no reason a priori to expect the ex ante incentive efficient outcome to be negotiated with privately informed 12 Note that for any given set of incentive feasible mechanisms, the set of ex ante incentive efficient mechanisms is a subset of the set of interim incentive efficient mechanisms. 10
Kim: Endogenous Choice of a Mediator parties. Therefore, as a first cut, I use the notion of interim incentive efficiency, which is a minimal requirement in a game in which each player knows his own type but does not know the other’s type at the initial decision making stage. Let S(Γ) denote the set of incentive feasible mediators for Γ, each of whom is ready to offer to mediate through some incentive compatible and individually rational mediation mechanism that is on the interim incentive efficient frontier. In fact, each interim incentive efficient mediator can be entirely defined by the probability weight he puts on the outcome of war and peace for each type realization. Here, I focus on the symmetric interim incentive efficient mediation mechanisms, but the result continues to hold when non-symmetric mediation mechanisms are allowed. Therefore, the following structure is useful to construct because I am going to use it routinely throughout the paper. Definition 1. A mediation mechanism µ y,z is defined to have the following structure: µ y,z (d 0 |t) = 0, µ y,z (d 1 |t) = 1 for t ∈ {ww} µ y,z (d 0 |t) = y, µ y,z (d 1 |t) = 1 − y for t ∈ {sw, ws} µ y,z (d 0 |t) = z, µ y,z (d 1 |t) = 1 − z for t ∈ {ss}, where y ≥ 0 and z ≥ 0. That is, µ y,z implements d 0 (war) with probability zero when t = {ww}, with the probability y ≥ 0 when t = {sw, ws}, and with the probability z ≥ 0 when t ∈ {ss}. I restrict attention to mechanisms that treat t = {sw} and t = {ws} cases symmetrically. Remark 1. Every symmetric interim incentive efficient mechanism has the structure of µ y,z defined in Definition 1. as Therefore, each symmetric interim incentive efficient mediator is indexed by y and z. following lemma gives three thresholds that separate four cases along the range of ¯p(s). The Lemma 1 (Thresholds). Let p ′ , p ∗ , and p ∗∗ be such that p ′ ≡ p ∗ ≡ p ∗∗ ≡ −u 1 (d 1 , sw) u 1 (d 1 , ss) − u 1 (d 1 , sw) , u 1 (d 1 , ww) u 1 (d 1 , ww) + u 1 (d 1 , ws) , and 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) . 11
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