Best-response algorithms for a class of monotone Nash equilibrium problems with mixed-integer variables

TL;DR

This paper analyzes the convergence of best-response algorithms for mixed-integer Nash equilibrium problems, providing theoretical insights, solution existence conditions, and practical applications in smart building control.

Contribution

It introduces new convergence and complexity results for best-response algorithms applied to monotone NEPs with mixed-integer variables, and offers conditions for solution existence.

Findings

Best-response sequences approach a bounded region containing the solution set.

Data-dependent convergence to the unique solution of the relaxed NEP.

Provided conditions for the existence of solutions to MI-NEPs.

Abstract

We characterize the convergence properties of traditional best-response (BR) algorithms in computing solutions to mixed-integer Nash equilibrium problems (MI-NEPs) that turn into a class of monotone Nash equilibrium problems (NEPs) once relaxed the integer restrictions. We show that the sequence produced by a Jacobi/Gauss-Seidel BR method always approaches a bounded region containing the entire solution set of the MI-NEP, whose tightness depends on the problem data, and it is related to the degree of strong monotonicity of the relaxed NEP. When the underlying algorithm is applied to the relaxed NEP, we establish data-dependent complexity results characterizing its convergence to the unique solution of the NEP. In addition, we derive one of the very few sufficient conditions for the existence of solutions to MI-NEPs. The theoretical results developed bring important practical benefits, illustrated on a numerical instance of a smart building control application.