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By Jackson M.O.

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Consider M c 2 fa; bg. Thus, Mci is found by eliminating strategies that can only ever lead gm to c or d. Since F is implemented, there must be some strategy pro le c m such that c = a, and so Mc is non-empty. Given the assumptions on preferences and the gm niteness of the mechanism, there exists a Nash equilibrium in mixed strategies to c g at R1; R2. Given the de nition of Mc and the fact that preferences on lotteries M; are monotone, it follows that this equilibrium must put positive probability on at least one of the outcomes a and b, as these are reachable by some strategy of 1 given any mixed strategy of 2, and are preferred by 1 to c and d over which 1 is indi erent.

149 160. Dutta, B. and A. Sen 1994, The Necessity of In nite Mechanisms ," Journal of Economic Theory, Vol. 64, pp. 130 141. , A. Sen and R. Vohra 1995, Nash Implementation Through Elementary Mechanisms in Economic Environments," Economic Design, Vol. 1, pp. 173 204. Eliaz, K. 1999, Fault Tolerant Implementation," mimeo: Tel Aviv University. Gevers, L. " In W. , Social Choice and Public Decision Making: Essays in Honor of Kenneth J. Arrow, Cambridge University Press: Cambridge. Gibbard, A.

Satterthwaite 1977, On the Equivalence of Strong Positive Association and Strategy Proofness," Journal of Economic Theory, Vol. 14, pp. 412 418. Myerson, R. and M. Satterthwaite 1983, E cient Mechanisms for Bilateral Trading," Journal of Economic Theory, Vol. 29, pp. 265 281. I. 1994, A Necessary and Su cient Condition for Walrasian Social Choice," Journal of Economic Theory, Vol. 62, pp. 186 208. Nakamura, S. 1988, Feasible Nash Implementation of Competitive Equilibria in an Economy with Externalities," mimeo: University of Minnesota.

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A crash course in implementation theory by Jackson M.O.

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