Monotonicity of Equilibria in Nonatomic Congestion Games

TL;DR

This paper investigates how equilibrium costs and loads in nonatomic congestion games respond to demand changes, identifying conditions that prevent non-monotone behaviors and focusing on singleton and series-parallel game structures.

Contribution

It establishes monotonicity of equilibrium loads in singleton congestion games and extends the analysis to constrained series-parallel congestion games, highlighting the role of strategy set structures.

Findings

Equilibrium loads are monotone in singleton congestion games.

Conditions for joint monotonicity of equilibrium loads are provided.

Extension of monotonicity results to series-parallel congestion games.

Abstract

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 to 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 focus on 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 constrained series-parallel congestion games, whose structure is reminiscent of the concept of a series-parallel network.