The influence of neighbourhood and choice on the complexity of finding pure Nash equilibria

Game theory is a mathematical framework for representing interactions of rational players trying to achieve possibly contradictory goals. Players’ strategies are said to be in Nash equilibrium if no player can do better by unilaterally changing her strategy. Nash’s famous theorem [1] guarantees the existence of at least one such equilibrium when players are allowed to play mixed strategies, i.e., probabilistic combinations of actions. If strategies are restricted to deterministic choice of actions, called pure strategies, the existence of an equilibrium is no longer guaranteed (see, e.g., [2]). Equilibria in the latter case are referred to as pure strategy Nash equilibria, or pure Nash equilibria in short. Complexity issues related to pure Nash equilibria have recently been investigated in [3]. It was shown that even for a very restricted class of games in graphical normal form, where each player is allowed to play at most 3 different actions and her payoff depends on at most 3 other players, determining whether a given game has a pure Nash equilibrium is NP-complete. Moreover, some tractable classes of strategic games have been identified.