Reciprocal Upper Semicontinuity and Better Reply Secure Games: A Comment

Abstract

A convex, compact, and possibly discontinuous better reply secure game has a Nash equilibrium. We introduce a very weak notion of continuity that can be used to establish that a game is better reply secure and we show that this notion of continuity is satisfied by a large class of games. THE CLASS OF BETTER REPLY SECURE GAMES was introduced by Reny (1999) who showed the existence of a pure Nash equilibrium for games in this class. Furthermore, Reny provided two conditions that, when combined, become suf- ficient for a game to be better reply secure. The first condition is payoff secu- rity, which means that given a joint strategy x, every player can find a strategy that yields almost the same payoff at x, even when the other players slightly deviate from x. The second condition is reciprocal upper semicontinuity (rusc). This condition, roughly speaking, requires the payoff of one of the players to jump up whenever the payoff of another player jumps down. We introduce the notion of weak reciprocal upper semicontinuity (wrusc), a strict weakening of rusc, and prove that a game that is wrusc and payoff secure is better reply secure, and therefore has a pure strategy Nash equilibrium.