Implementation theory by Maskin E., Sjostrom T.

By Maskin E., Sjostrom T.

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Formally, the frontier is linear in state θ if, for all b, b0 ∈ B that are both Pareto optimal in state θ, the lottery λb + (1 − λ)b0 is also Pareto optimal, where λ is the probability of b. 31 32 Under the hypothesis of Theorem 14, a mechanism in effect induces a two-person zero-sum game (renegotiation ensures that outcomes are Pareto efficient, and the linearity of the Pareto frontier means that payoffs sum to a constant). In zero-sum games, any reÞned Nash equilibrium must yield both players the same payoffs as all other Nash equilibria.

J. K. Sen and K. , Social Choice Re-examined, Vol. II (IEA Conference Volume No. 117), pp. 82-95. London: Macmillan. Harris, M. and R. Townsend (1981) “Resource allocation with asymmetric information”, Econometrica, 49: 33-64. Harsanyi, J. and R. Selten (1988), A General Theory of Equilibrium Selection in Games, MIT Press. Herrero, M. and S. Srivastava (1992) “Implementation via backward induction”, Journal of Economic Theory, 56:70-88. Hong, L. (1995), “Nash implementation in production economies”, Economic Theory, 5:401-418.

Then there is i ∈ N and θi , θ0i ∈ Θi such that E {ui (f(θ), θi ) | θ i } < E {ui (f (θ−i , θ 0i ), θ i ) | θi } (7) where θ = (θ−i , θi ). Let σ ∈ BN E Γ be such that h ◦ σ = f. If agent i’s type θi uses the equilibrium strategy σ i (θi ), his expected payoff is E {ui (h(σ(θ)), θi ) | θi } = E {ui (f(θ), θi ) | θ i } (8) If instead he were to send the message m0i = σ i (θ0i ), he would get E {ui (h(σ −i (θ−i ), σ i (θ0i ))) | θi )} = E {ui (f (θ−i , θ0i ), θi ) | θi } (9) But inequality (7) and equations (8) and (9) contradict the deÞnition of Bayesian Nash equilibrium.

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