Repeated Games and the Central Limit Theorem

Bernard De Meyer

No 1993003, LIDAM Discussion Papers CORE from Université catholique de Louvain, Center for Operations Research and Econometrics (CORE)

Abstract: The value vn ( P ) of the n times repeated zero sum game with incomplete information on one side is a concave function on the simplex p(K) that decreases to cav(u)( p ) as n grows. The rate of convergence 1/[ square root n ] that was given in Aumann's demonstration (See [A-M 68]) using a rough bound on martingale variation was proved to be the true one by Mertens and Zamir (See [M-Z 76] and [M-Z 77]) who analyzed a particular game with two states of nature, for which '[ psi_n ]( P ) = [ squareroot_n ][ vn[vn( P ) - cav(u)( p )] was showed to converge to a limit [ psi]( P ) related to the normal density. In our previous paper [DM-89]' we generalized the Mertens and Zamir's reasoning to a class of games [ delta_sigma_0 ] : there we show how the recurrence formula for vn rewritten as one for [ psi_n] becomes a partial differential equation (the heuristic equation) for [ psi] and proved that any solution of this differential problem with some boundary conditions was necessarily the limit of the [ psi_n]. We next proved that for a subclass R[ sigma] of [ delta_sigma_0 ] the heuristic equation had, as in the Mertens and Zamir's game, a solution related to the normal density. In this paper we explain the occurrence of the normal density as a consequence of the Central Limit Theorem.

Date: 1993-03-01
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