Approximate Pure Nash Equilibria via Lovász Local Lemma

In many types of games, mixed Nash equilibria is not a satisfying solution concept, as mixed actions are hard to interpret. However, pure Nash equilibria, which are more natural, may not exist in many games. In this paper we explore a class of graphical games, where each player has a set of possible decisions to make, and the decisions have bounded interaction with one another. In our class of games, we show that while pure Nash equilibria may not exist, there is always a pure approximate Nash equilibrium. We also show that such an approximate Nash equilibrium can be found in polynomial time. Our proof is based on the Lovász local lemma and Talagrand inequality, a proof technique that can be useful in showing similar existence results for pure (approximate) Nash equilibria also in other classes of games.