The Complexity of Decision Problems about Nash Equilibria in Win-Lose Games

Among the most fundamental problems in Al-gorithmic Game Theory are those concerning the Nash equilibria of a strategic game: states where no player could unilaterally deviate to improve her utility. Such algorithmic problems, including their decision, search and approximation variants , have been studied extensively in the last few years. The fundamental theorem of Nash [4,5] that Nash equilibria are guaranteed to exist makes the search problem for Nash equilibria total, which implies that the search problem is not N P-complete unless N P = coN P. Decision problems about Nash equilibria result naturally by twisting the search problem in one of several simple ways that deprive it from its existence guarantees. Here is a (non-exhaustive) list of decision problems about Nash equilibria: Given a strategic game, does it have: (i) A Nash equilibrium where each player has utility at least a given number? [3],(ii) A Nash equilibrium where each player has utility at most a given num-ber?, (iii) At least two Nash equilibria? [3], (iv) A Nash equilibrium whose support contains a set of strategies? [3], (v) A Nash equilibrium whose support is contained in a set of strategies? [3], (vi) A Nash equilibrium whose support has size greater than a given number? [3], (vii) A Nash equilibrium whose support has size smaller than a given number? [3], (viii) A Nash equilibrium in which the total utility of players is at least a given number? [2], (ix) A Nash equilibrium in which the total utility of players is at most a given number?, (x) A rational Nash equilibrium (i.e., one with all probabilities rational)? [1]. Some of these decision problems are N P-complete for symmetric two-player games; this was originally shown by Gilboa and Zemel [3] and later by Conitzer and Sandholm [2] via a unifying reduction from the satisfiability problem (which covered some additional decision problems over those considered in [3]). The last problem in the list is N P-complete even for three-player games [1] — recall that all Nash equilibria of a two-player game are rational, so that the problem is trivial for two-player games. In this work, we settle the complexity of the natural decision problems about Nash equilibria previously considered in [2,3] (or introduced here) for win-lose games, i.e., games in which all utilities are 0 or 1. Specifically, we show, as our main result, that these decision problems are N P-complete for two-player win-lose …