ZERO Regrets Algorithm: Optimizing over Pure Nash Equilibria via Integer Programming

In Algorithmic Game Theory ($AGT$), designing efficient algorithms to compute Nash equilibria poses considerable challenges. We make progress in the field and shed new light on the intersection between Algorithmic Game Theory and Integer Programming. We introduce $ZERO$ $Regrets$, a general cutting plane algorithm to compute, enumerate, and select Pure Nash Equilibria ($PNEs$) in Integer Programming Games, a class of simultaneous and non-cooperative games. We present a theoretical foundation for our algorithmic reasoning and provide a polyhedral characterization of the convex hull of the Pure Nash Equilibria. We introduce the concept of $equilibrium$ $inequality$, and devise an $equilibrium$ $separation$ $oracle$ to separate non-equilibrium strategies from $PNEs$. We test $ZERO$ $Regrets$ on two paradigmatic classes of games: the Knapsack Game and the Network Formation Game, a well-studied game in $AGT$. Our algorithm successfully solves relevant instances of both games and shows promising applications for equilibria selection.