Uncoupled Dynamics Do Not Lead to Nash Equilibrium

It is notoriously difŽ cult to formulate sensible adaptive dynamics that guarantee convergence to Nash equilibrium. In fact, short of variants of exhaustive search (deterministic or stochastic), there are no general results; of course, there are many important, interesting and well-studied particular cases. See the books of Jörgen W. Weibull (1995), Fernando VegaRedondo (1996), Larry Samuelson (1997), Drew Fudenberg and David K. Levine (1998), Josef Hofbauer and Karl Sigmund (1998), H. Peyton Young (1998), and the discussion in Section IV below. Here we provide a simple answer to the question: Why is that so? Our answer is that the lack of a general result is an intrinsic consequence of the natural requirement that dynamics of play be “uncoupled” among the players, that is, the adjustment of a player’s strategy does not depend on the payoff functions (or utility functions) of the other players (it may depend on the other players’ strategies, as well as on the payoff function of the player himself). This is a basic informational condition for dynamics of the “adaptive” or “behavioral” type. It is important to emphasize that, unlike the existing literature (see Section IV), we make no “rationality” assumptions: our dynamics are not best-reply dynamics, or better-reply, or payoffimproving, or monotonic, and so on. What we show is that the impossibility result is due only to an “informational” requirement—that the dynamics be uncoupled.

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