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Kim: Endogenous Choice of a Mediator Myerson, Roger B. 1979. “Incentive Compatibility and the Bargaining Problem.” Econometrica 47(1):61–74. Myerson, Roger B. 1983. “Mechanism Design by an Informed Principal.” Econometrica 51(6):1767– 1797. Myerson, Roger B. 1984a. “Cooperative Games with Incomplete Information.”International Journal of Game Theory 13(2):69–96. Myerson, Roger B. 1984b. “Two-Person Bargaining Problems with Incomplete Information.” Econometrica 52(2):461–488. Myerson, Roger B. 1991. Game Theory: Analysis of Conflict. Cambridge, M.A.: Harvard University Press. Myerson, Roger B. and Mark A. Satterthwaite. 1983. “Efficient Mechanisms for Bilateral Trading.” Journal of Economic Theory 29(2):265–281. Nalebuff, Barry. 1987. “Credible Pretrial Negotiation.”The RAND Journal of Economics 18(2):198– 210. Nash, John F. 1950. “The Bargaining Problem.” Econometrica 18(2):155–162. Nash, John F. 1953. “Two-Person Cooperative Games.” Econometrica 21(1):128–140. Rosenthal, Robert W. 1978. “Arbitration of Two-Party Disputes under Uncertainty.” Review of Economic Studies 45(3):595–604. 48
Kim: Endogenous Choice of a Mediator Appendix A: Proofs Supplementary Explanations and Proofs of Section 3 An interim incentive efficient mechanism in the incentive feasible set F is any mechanism that is an optimal solution to an optimization problem of the form max µ j ∈F ∑ p(t) ∑ µ(d|t) ∑ λ i (t i )u i (d, t) t∈T d∈D i∈N where p(t) = ∏ 2 i=1 ¯p(t i) and the utility weight λ i (t i ) is a positive number for each player i and each type t i . The type sets and outcomes sets are all finite. Under the finiteness assumption, the set of incentive feasible mechanisms is a convex polyhedron. By the supporting hyperplane theorem, an incentive feasible mechanism µ is incentive efficient if and only if there exist some positive numbers λ i (t i ) for each type t i of each player i such that µ is an optimal solution to the optimization problem max µ:D×T →R i∈N ∑ ∑ t i ∈T i λ i (t i )U i (µ|t i ) s.t. U i (µ|t i ) ≥ U ∗ i (µ, s i |t i ), ∀i ∈ N, ∀t i ∈ T i , ∀s i ∈ T i , U i (µ|t i ) ≥ 0, ∀i ∈ N, ∀t i ∈ T i , ∑ µ(d|t) = 1 and µ(d|t) ≥ 0 ∀d ∈ D, ∀t ∈ T, d∈D where Ui ∗(µ, s i|t i ) = ∑ t −i ∈T −i ∑d∈D ¯p(t −i)µ(d|t −i , s i )u i (d, t). Because the objective and constraints are all linear in µ, this optimization problem is a linear programming problem. Because players are symmetric in type-dependent payoffs, I can set λ 1 (s) = λ 2 (s) ≡ λ(s) and λ 1 (w) = λ 2 (w) ≡ λ(w); and normalize such that λ(s) + λ(w) = 1 without loss of generality. For the above optimization problem, a Lagrangean function can be formed. Let α i (s i |t i ) denote the Lagrange multiplier for the incentive compatibility constraints and β i (t i ) denote the Lagrange multiplier for the individual rationality constraints. Also, I can set α 1 (w|s) = α 2 (w|s) ≡ α(w|s), α 1 (s|w) = α 2 (s|w) = α(s|w), β 1 (s) = β 2 (s) ≡ β(s), and β 1 (w) = β 2 (w) ≡ β(w). Let α = (α(w|s), α(s|w)) and β = (β(s), β(w)). Then the Lagrangean function can be written as ∑ ∑ λ(t i )U i (µ|t i ) + ∑ ∑ α(s i |t i ) (U i (µ|t i ) − Ui ∗ (µ, s i |t i )) + ∑ ∑ β(t i )U i (µ|t i ). i∈N t i ∈T i i∈N t i ∈T i i∈N t i ∈T i 49
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