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Selfish routing games with priority lanes

Yang Li, Alexander Skopalik, Marc Uetz

TL;DR

The paper analyzes selfish routing with voluntary priority lanes by introducing a two-tier model where each edge offers regular and priority options, with priority users paying a fee that also raises regular-user delays. Existence of equilibria is established for linear latency functions via Kakutani fixed-point theory, and edge-latencies are shown to be unique at equilibrium even when per-edge splits are not. A central result is that marginal-externality pricing, where each edge price equals $\omega_e=f_e^*\ell'_e(f_e^*)$, yields socially optimal flows and PoA $=1$, effectively internalizing congestion externalities through voluntary upgrades. However, uniform pricing across edges does not generally achieve the optimum, with Pigou-type examples showing PoA can approach $\frac{4}{3}$, highlighting limitations of uniform schemes. Overall, the work provides a theoretical foundation for using voluntary priority options to mimic marginal-cost pricing and improve efficiency in tiered service networks, while delineating practical and theoretical limits for broader applicability.

Abstract

We study selfish routing games where users can choose between regular and priority service for each network edge on their chosen path. Priority users pay an additional fee, but in turn they may travel the edge prior to non-priority users, hence experiencing potentially less congestion. For this model, we establish existence of equilibria for linear latency functions and prove uniqueness of edge latencies, despite potentially different strategic choices in equilibrium. Our main contribution demonstrates that marginal cost pricing achieves system optimality: When priority fees equal marginal externality costs, the equilibrium flow coincides with the socially optimal flow, hence the price of anarchy equals $1$. This voluntary priority mechanism therefore provides an incentive-compatible alternative to mandatory congestion pricing, whilst achieving the same result. We also discuss the limitations of a uniform pricing scheme for the priority option.

Selfish routing games with priority lanes

TL;DR

The paper analyzes selfish routing with voluntary priority lanes by introducing a two-tier model where each edge offers regular and priority options, with priority users paying a fee that also raises regular-user delays. Existence of equilibria is established for linear latency functions via Kakutani fixed-point theory, and edge-latencies are shown to be unique at equilibrium even when per-edge splits are not. A central result is that marginal-externality pricing, where each edge price equals , yields socially optimal flows and PoA , effectively internalizing congestion externalities through voluntary upgrades. However, uniform pricing across edges does not generally achieve the optimum, with Pigou-type examples showing PoA can approach , highlighting limitations of uniform schemes. Overall, the work provides a theoretical foundation for using voluntary priority options to mimic marginal-cost pricing and improve efficiency in tiered service networks, while delineating practical and theoretical limits for broader applicability.

Abstract

We study selfish routing games where users can choose between regular and priority service for each network edge on their chosen path. Priority users pay an additional fee, but in turn they may travel the edge prior to non-priority users, hence experiencing potentially less congestion. For this model, we establish existence of equilibria for linear latency functions and prove uniqueness of edge latencies, despite potentially different strategic choices in equilibrium. Our main contribution demonstrates that marginal cost pricing achieves system optimality: When priority fees equal marginal externality costs, the equilibrium flow coincides with the socially optimal flow, hence the price of anarchy equals . This voluntary priority mechanism therefore provides an incentive-compatible alternative to mandatory congestion pricing, whilst achieving the same result. We also discuss the limitations of a uniform pricing scheme for the priority option.
Paper Structure (7 sections, 7 theorems, 31 equations, 2 figures)

This paper contains 7 sections, 7 theorems, 31 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The selfish routing model with priority lanes
  3. Existence of equilibria.
  4. Discussion on equilibrium uniqueness
  5. Optimal pricing yields socially optimal outcomes
  6. The effects of uniform pricing
  7. Conclusion

Key Result

Theorem 1

Given a price vector $\omega$, an instance $({G},{r},\tilde{\ell},\omega)$ admits at least one equilibrium flow.

Figures (2)

  • Figure 1: The perceived cost in equilibrium as a function of the total flow $r$ on the single edge $e$ of Example \ref{['ex:1']}.
  • Figure 2: The network for instance $(G_k,r_k,\tilde{\ell}_k,\omega_k)$ illustrating the structural inefficiency of uniform pricing.

Theorems & Definitions (14)

  • Definition 1
  • Example 1
  • Theorem 1
  • proof
  • Lemma 1: Variational inequality characterization
  • proof
  • Theorem 2
  • proof
  • Corollary 1
  • Theorem 3
  • ...and 4 more