  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, we 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.
