Convergence Issues in Competitive Games

Abstract

We study the speed of convergence to approximate solutions in iterative competitive games. We also investigate the value of Nash equilibria as a measure of the cost of the lack of coordination in such games. Our basic model uses the underlying best response graph induced by the selfish behavior of the players. In this model, we study the value of the social function after multiple rounds of best response behavior by the players. This work therefore deviates from other attempts to study the outcome of selfish behavior of players in non-cooperative games in that we dispense with the insistence upon only evaluating Nash equilibria. A detailed theoretical and practical justification for this approach is presented. We consider non-cooperative games with a submodular social utility function; in particular, we focus upon the class of valid-utility games introduced in [13]. Special cases include basic-utility games and market sharing games which we examine in depth. On the positive side we show that for basic-utility games we obtain extremely quick convergence. After just one round of iterative selfish behavior we are guaranteed to obtain a solution with social value at least $\frac13$ that of optimal. For n-player valid-utility games, in general, after one round we obtain a $\frac{1}{2n}$-approximate solution. For market sharing games we prove that one round of selfish response behavior of players gives $\Omega({1\over \ln n})$-approximate solutions and this bound is almost tight. On the negative side we present an example to show that even in games in which every Nash equilibrium has high social value (at least half of optimal), iterative selfish behavior may “converge” to a set of extremely poor solutions (each being at least a factor n from optimal). In such games Nash equilibria may severely underestimate the cost of the lack of coordination in a game, and we discuss the implications of this.