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

TL;DR

This paper introduces two distributed proximal-like algorithms for finding equilibria in potential games with mixed-integer variables, ensuring convergence to exact or approximate solutions, demonstrated through a Cournot oligopoly example.

Contribution

It proposes novel distributed algorithms tailored for mixed-integer potential games, handling both exact and approximate equilibrium seeking with convergence guarantees.

Findings

Algorithms converge to exact or ε-approximate equilibria.

Numerical validation on a Cournot oligopoly model.

Use of integer-compatible regularization functions enhances convergence.

Abstract

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.