  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.
