Computing equilibria for integer programming games

The recently-defined class of integer programming games (IPG) models situations where multiple self-interested decision makers interact, with their strategy sets represented by a finite set of linear constraints together with integer requirements. Many real-world problems can suitably be cast in this way, hence anticipating IPG outcomes is of crucial value for policy makers. Nash equilibria have been widely accepted as the solution concept of a game. Thus, their computation provides a reasonable prediction of games outcome. In this paper, we start by showing the computational complexity of deciding the existence of a Nash equilibrium for an IPG. Then, using sufficient conditions for their existence, we develop a general algorithmic approach that is guaranteed to return a Nash equilibrium when the game is finite and to approximate an equilibrium when payoff functions are Lipschitz continuous. We also showcase how our methodology can be changed to determine other types of equilibria. The performance of our methods is analyzed through computational experiments on knapsack, kidney exchange and a competitive lot-sizing games. To the best of our knowledge, this is the first time that equilibria computation methods for general IPGs have been designed and computationally tested.