Monotonicity of Equilibria in Nonatomic Congestion Games

TL;DR

The paper tackles the problem of whether equilibrium costs and loads in $nonatomic$ congestion games respond monotonically to demand changes, addressing classic paradoxes like Braess's and Fisk's. It leverages the Beckmann convex formulation to establish continuity and monotonicity properties, first proving a monotone equilibrium selection ($MES$) for singleton congestion games and then introducing comonotonicity regions $\Gamma^{\precsim}$ and active-regime partitions $\Gamma^{\precsim}_{\boldsymbol{\varrho}}$ to characterize joint load movements. The authors extend these monotonicity properties to the larger class of constrained series-parallel congestion games (CSP) by showing MES is preserved under series and parallel constructions and by embedding CSPs into constrained routing frameworks, including a common-OD representation. They also demonstrate that topology alone does not guarantee monotonicity in multi-OD routing, illustrating the critical role of path structure across commodities. Overall, the work advances understanding of when and how equilibrium loads behave monotonically with demand, with implications for network design, optimization, and algorithmic tracking of equilibria under varying demands.

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.

Figures (7)

• Figure 1: Fisk's network.
• Figure 2: In the Wheatstone network with three paths and a single OD pair, the equilibrium load on the vertical edge $(v_{1},v_{2})$ equals the equilibrium flow on the path $\mathsf{O}\,v_{1}\,v_{2}\mathsf{D}$ and is decreasing for $\mu\in[1,2]$.
• Figure 3: A singleton congestion game with affine costs.
• Figure 4: An example with quadratic costs. The first commodity uses the two top edges, and the second commodity uses the bottom two. The three colors represent the regions $\Gamma^\precsim$ for the possible orders $\precsim$ of the equilibrium costs. The straight lines within each region separate sub-regions corresponding to different active regimes. The regions $\Gamma^{\precsim}$ are not convex, but the boundary between sub-regions is still affine.
• Figure 5: Fisk's multi-commodity network can be embedded in a SP network with a common-OD, by adding two edges with zero cost.

Theorems & Definitions (39)

• Proposition 1
• Remark 2
• Example 3
• Example 4
• Definition 5
• Theorem 6
• proof
• Remark 7
• Remark 8
• Definition 9