Playing-Against-The-Sum

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Better-Reply Dynamics and Global Convergence to Nash Equilibrium in Playing-Against-The-Sum Games
Martin Dindo Department of Mathematics University of North Carolina Chapel Hill, NC 27599-3250 email: dindos@math.unc.edu
Claudio Mezzetti Department of Economics University of North Carolina Chapel Hill, NC 27599-3305 email: mezzetti@email.unc.edu
May 24, 2001
Abstract We consider n-person, generic, quasi-concave games with continuous action spaces and in which the payo¤ of a player depends on her own action and the sum of the actions of opponents. We study a discrete-time, stochastic adjustment process (the better-reply dynamics) in which players move towards better replies. Our main result is a sucient condition for this process to converge globally to a Nash equilibrium of the game. This condition requires that actions be either locally strategic substitutes or locally strategic complements for all players at each Nash equilibrium that is locally asymptotically stable under an associated deterministic, adjusted best-reply dynamics. We provide an example of a 2-person game with a unique Nash equilibrium at which the derivatives of the best-reply functions have di¤erent signs and in which the better-reply dynamics does not converge to the equilibrium.
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