Games with randomly disturbed payoffs: A new rationale for mixed-strategy equilibrium points

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Abstract

Equilibrium points in mixed strategies seem to be unstable, because any player can deviate without penalty from his equilibrium strategy even if he expects all other players to stick to theirs. This paper proposes a model under which most mixed-strategy equilibrium points have full stability. It is argued that for any gameΓ the players' uncertainty about the other players' exact payoffs can be modeled as a disturbed gameΓ *, i.e., as a game with small random fluctuations in the payoffs. Any equilibrium point inΓ, whether it is in pure or in mixed strategies, can “almost always” be obtained as a limit of a pure-strategy equilibrium point in the corresponding disturbed gameΓ * when all disturbances go to zero. Accordingly, mixed-strategy equilibrium points are stable — even though the players may make no deliberate effort to use their pure strategies with the probability weights prescribed by their mixed equilibrium strategies — because the random fluctuations in their payoffs willmake them use their pure strategies approximately with the prescribed probabilities.

The author is grateful to the National Science Foundation for supporting this research by Grant GS-3222 to the University of California, Berkeley, through the University's Center for Research in Management Science. Special thanks are due to ProfessorReinhard Selten, of the University of Bielefeld, West Germany, for suggesting Theorem 1 and for many valuable discussions. The author is indebted for helpful comments also to ProfessorRobert J.Aumann of the Hebrew University, Jerusalem; to ProfessorsGerard Debreu andRoy Radner of the University of California, Berkeley; and to Dr.Lloyd S.Shapley of RAND Corporation, Santa Monica.