Problem definition: Integer programming games provide a framework to model (integer) decisions under competition. This has several applications ranging from decisions on competitive production volumes, over discrete capacity decisions among competitors to location selection under competition. The predominant solution concept for integer programming games in the literature is the identification of a Nash equilibrium; a strategy combination where no player benefits from unilaterally changing his/her strategy. While there are algorithms capable of identifying singular equilibria, they do not enumerate the full set of equilibria and thus cannot select the single most likely equilibrium. Methodology: Focusing on separable integer programming games, we propose a solution method in which we combine sampling techniques and equilibrium selection theory within an algorithm that integrates the determination of all equilibria and the subsequent identification of the most probable equilibrium among said equilibria. For this process, we use a column-generation approach, in which we divide the n-player integer programming game into a MIP-master problem capable of identifying equilibria in a sample, as well as column-generating subproblems tasked with the sampling of best responses and additional solution candidates. Results and managerial implications: We showcase algorithmic performance in various instances of knapsack games and large scale facility location and design games, including 3-player formulations. We highlight differences in solution quality between the proposed approach and the state of the art, enabling decision makers in competitive scenarios to base their actions on the most probable equilibrium rather than any identifiable equilibrium.