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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 aJacobi/Gauss-Seidel BR method always approaches a bounded region containing theentire 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. Whenthe underlying algorithm is applied to the relaxed NEP, we establishdata-dependent complexity results characterizing its convergence to the uniquesolution of the NEP. In addition, we derive one of the very few sufficientconditions for the existence of solutions to MI-NEPs. The theoretical resultsdeveloped bring important practical advantages that are illustrated on anumerical instance of a smart building control application.

On best-response algorithms for monotone Nash equilibrium problems with mixed-integer variables