Proximal-like algorithms for equilibrium seeking in mixed-integer Nash equilibrium problems

We consider potential games with mixed-integer variables, for which we propose two distributed, proximal-like equilibrium seeking algorithms. Specifically, we focus on two scenarios: i) the underlying game is generalized ordinal and the agents update through iterations by choosing an exact optimal strategy; ii) the game admits an exact potential and the agents adopt approximated optimal responses. By exploiting the properties of integer-compatible regularization functions used as penalty terms affecting the local cost function of each agent, we are able to show that both algorithms converge to either an exact or an $\epsilon$-approximate equilibrium. We corroborate our findings on a numerical instance of a Cournot oligopoly model.