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Phase Transitions of the Price-of-Anarchy Function in Multi-Commodity Routing Games

Roberto Cominetti, Valerio Dose, Marco Scarsini

TL;DR

This paper analyzes how Wardrop equilibria and the price of anarchy ($PoA$) in nonatomic multi-commodity routing respond to varying demands, emphasizing smoothness and phase transitions when the set of minimum-cost paths changes. It proves that, under proportional demand scaling across all OD pairs, the largest one-sided derivatives of the social cost at equilibrium and of the $PoA$ align with the smaller active regime at a breakpoint, providing a rigorous confirmation of a prior simulation-based conjecture; it also presents counterexamples showing the conjecture can fail without proportional scaling. The authors develop a robust differentiability framework using Beckmann duality and the Moore–Penrose pseudoinverse to handle fixed-regime analyses and derive derivative relationships via convex quadratic programs, including a restricted linear-demand version with strict or weak inequalities depending on regime inclusions. Together, these results connect observed kinks in PoA in real networks to underlying phase transitions and clarify when one-sided derivatives can be ordered, offering guidance for traffic planning and policy design in multi-commodity contexts.

Abstract

We consider the behavior of the price of anarchy and equilibrium flows in nonatomic multi-commodity routing games as a function of the traffic demand. We analyze their smoothness with a special attention to specific values of the demand at which the support of the Wardrop equilibrium exhibits a phase transition with an abrupt change in the set of optimal routes. Typically, when such a phase transition occurs, the price of anarchy function has a breakpoint, \ie is not differentiable. We prove that, if the demand varies proportionally across all commodities, then, at a breakpoint, the largest left or right derivatives of the price of anarchy and of the social cost at equilibrium, are associated with the smaller equilibrium support. This proves -- under the assumption of proportional demand -- a conjecture of O'Hare et al. (2016), who observed this behavior in simulations. We also provide counterexamples showing that this monotonicity of the one-sided derivatives may fail when the demand does not vary proportionally, even if it moves along a straight line not passing through the origin.

Phase Transitions of the Price-of-Anarchy Function in Multi-Commodity Routing Games

TL;DR

This paper analyzes how Wardrop equilibria and the price of anarchy () in nonatomic multi-commodity routing respond to varying demands, emphasizing smoothness and phase transitions when the set of minimum-cost paths changes. It proves that, under proportional demand scaling across all OD pairs, the largest one-sided derivatives of the social cost at equilibrium and of the align with the smaller active regime at a breakpoint, providing a rigorous confirmation of a prior simulation-based conjecture; it also presents counterexamples showing the conjecture can fail without proportional scaling. The authors develop a robust differentiability framework using Beckmann duality and the Moore–Penrose pseudoinverse to handle fixed-regime analyses and derive derivative relationships via convex quadratic programs, including a restricted linear-demand version with strict or weak inequalities depending on regime inclusions. Together, these results connect observed kinks in PoA in real networks to underlying phase transitions and clarify when one-sided derivatives can be ordered, offering guidance for traffic planning and policy design in multi-commodity contexts.

Abstract

We consider the behavior of the price of anarchy and equilibrium flows in nonatomic multi-commodity routing games as a function of the traffic demand. We analyze their smoothness with a special attention to specific values of the demand at which the support of the Wardrop equilibrium exhibits a phase transition with an abrupt change in the set of optimal routes. Typically, when such a phase transition occurs, the price of anarchy function has a breakpoint, \ie is not differentiable. We prove that, if the demand varies proportionally across all commodities, then, at a breakpoint, the largest left or right derivatives of the price of anarchy and of the social cost at equilibrium, are associated with the smaller equilibrium support. This proves -- under the assumption of proportional demand -- a conjecture of O'Hare et al. (2016), who observed this behavior in simulations. We also provide counterexamples showing that this monotonicity of the one-sided derivatives may fail when the demand does not vary proportionally, even if it moves along a straight line not passing through the origin.
Paper Structure (11 sections, 7 theorems, 58 equations, 6 figures)

This paper contains 11 sections, 7 theorems, 58 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Our Results
  3. Related Work
  4. Organization of the paper
  5. Network games with variable demand
  6. Equilibria and price of anarchy
  7. Continuity and smoothness of equilibria and PoA.
  8. Behavior around breakpoints
  9. Proof of \ref{['th:WatlingSC']}
  10. Missing proofs
  11. List of symbols

Key Result

Theorem 3.2

Let $({G},\boldsymbol{c},\mathcal{H},\boldsymbol{\mu}(\,\cdot\,))$ be a nonatomic routing game with a continuously differentiable demand function $\boldsymbol{\mu}(\,\cdot\,)$. Let the cost functions $c_{e}$ be $\mathcal{C}^{1}$ with strictly positive derivative, and suppose that the active regime $

Figures (6)

  • Figure 1: A single-commodity example of the behavior of the PoA as a function of the demand. This function is nonsmooth at demands $1$, $3$, $4$, $6$ and $27/2$. Between these breakpoints the set of optimum paths remains stable and the PoA behaves smoothly. At demand $4$ the set of optimum paths exhibits a contraction and the derivative jumps up. At all the other breakpoints the set of optimum paths undergoes an expansion and the derivative jumps down.
  • Figure 2: Plot of the cost functions in \ref{['eq:c1']} (in blue) and \ref{['eq:c2']} (in dotted red). The two functions are $\mathcal{C}^1$ but not convex.
  • Figure 3: Plot of the cost functions $c_3$ (in red) and $c_4$ (in blue) in \ref{['ex:non-differentiable-2']}. The two functions are $\mathcal{C}^1$ but not convex.
  • Figure 4: On the left we have the classical Wheatstone network with the cost functions defined in \ref{['ex:non-differentiable-2']}. The plot on the right shows the load on the vertical link with demand varying in the interval $[1,3]$. Notice that at demand $2$ the load is zero and the load function is not differentiable.
  • Figure 5: An example where \ref{['conj:Watling']} does not hold.
  • ...and 1 more figures

Theorems & Definitions (30)

  • Definition 2.1
  • Definition 2.2
  • Definition 3.1
  • Theorem 3.2
  • Lemma 3.3
  • Lemma 3.4
  • proof : Proof of \ref{['th:differentiability-on-curve']}
  • Remark 3.5
  • Example 3.6
  • Example 3.7
  • ...and 20 more