Generalizations of weighted matroid congestion games: pure Nash equilibrium, sensitivity analysis, and discrete convex function

Abstract

Congestion games provide a model of human’s behavior of choosing an optimal strategy while avoiding congestion. In the past decade, matroid congestion games have been actively studied and their good properties have been revealed. In most of the previous work, the cost functions are assumed to be univariate or bivariate. In this paper, we discuss generalizations of matroid congestion games in which the cost functions are n-variate, where n is the number of players. First, motivated from polymatroid congestion games with $\mathrm {M}^\natural$-convex cost functions, we conduct sensitivity analysis for separable $\mathrm {M}^\natural$-convex optimization, which extends that for separable convex optimization over base polyhedra by Harks et al. (SIAM J Optim 28:2222–2245, 2018. https://doi.org/10.1137/16M1107450). Second, we prove the existence of pure Nash equilibria in matroid congestion games with monotone cost functions, which extends that for weighted matroid congestion games by Ackermann et al. (Theor Comput Sci 410(17):1552–1563, 2009. https://doi.org/10.1016/j.tcs.2008.12.035). Finally, we prove the existence of pure Nash equilibria in matroid resource buying games with submodular cost functions, which extends that for matroid resource buying games with marginally nonincreasing cost functions by Harks and Peis (Algorithmica 70(3):493–512, 2014. https://doi.org/10.1007/s00453-014-9876-6).