Monotonicity of Equilibria in Nonatomic Congestion Games

This paper studies the monotonicity of equilibrium costs and equilibrium loads in nonatomic congestion games, in response to variations of the demands. The main goal is to identify conditions under which a paradoxical non-monotone behavior can be excluded. In contrast with routing games with a single commodity, where the network topology is the sole determinant factor for monotonicity, for general congestion games with multiple commodities the structure of the strategy sets plays a crucial role. We frame our study in the general setting of congestion games, with a special attention on multi-commodity singleton congestion games, for which we establish the monotonicity of equilibrium loads with respect to every demand. We then provide conditions for comonotonicity of the equilibrium loads, i.e. we investigate when they jointly increase or decrease after variations of the demands. We finally extend our study from singleton congestion games to the larger class of product-union congestion games, whose structure is reminiscent of the concept of a series-parallel network.