Public Signals in Network Congestion Games

TL;DR

This work studies how information signaling can steer Wardrop equilibria in nonatomic congestion games with affine edge costs that include state-based offsets. The authors prove that the equilibrium cost as a function of belief is piecewise linear and characterize when full information revelation is optimal: precisely for series-parallel, single-commodity networks. They develop a general LP framework that reduces signaling to optimal support selection and show how to compute optimal schemes efficiently when the number of states or commodities is limited, including a polynomial-time algorithm for parallel-edge networks with a fixed number of commodities and states. Complementary results establish that the number of Wardrop-supports can be exponential in general, but is manageable in realistic networks, as evidenced by computational studies on CSData22 networks where full information revelation is often optimal or near-optimal. Overall, the paper combines structural, algorithmic, and empirical insights to quantify the value and computation of information design in traffic networks, with implications for mobility services and congestion management.

Abstract

We consider a largely untapped potential for the improvement of traffic networks that is rooted in the inherent uncertainty of travel times. Travel times are subject to stochastic uncertainty resulting from various parameters such as weather condition, occurrences of road works, or traffic accidents. Large mobility services have an informational advantage over single network users as they are able to learn traffic conditions from data. A benevolent mobility service may use this informational advantage in order to steer the traffic equilibrium into a favorable direction. The resulting optimization problem is a task commonly referred to as signaling or Bayesian persuasion. Previous work has shown that the underlying signaling problem can be NP-hard to approximate within any non-trivial bounds, even for affine cost functions with stochastic offsets. In contrast, we show that in this case, the signaling problem is easy for many networks. We tightly characterize the class of single-commodity networks, in which full information revelation is always an optimal signaling strategy. Moreover, we construct a reduction from optimal signaling to computing an optimal collection of support vectors for the Wardrop equilibrium. For two states, this insight can be used to compute an optimal signaling scheme. The algorithm runs in polynomial time whenever the number of different supports resulting from any signal distribution is bounded to a polynomial in the input size. Using a cell decomposition technique, we extend the approach to a polynomial-time algorithm for multi-commodity parallel link networks with a constant number of commodities, even when we have a constant number of different states of nature.

Figures (9)

• Figure 1: Instance in Example \ref{['ex:concave']}: (a) cost functions for state $\theta_1$; (b) cost functions for state $\theta_2$; (c) cost $C(\mu)$ of the Wardrop equilibrium as a function of the belief, described by parameter $\mu_{\theta_2} = 1-\mu_{\theta_1} \in [0,1]$.
• Figure 2: Instance in Example \ref{['ex:3d']}: (a) cost functions for state $\theta_1$; (b) cost functions for state $\theta_2$; (c) cost functions for state $\theta_3$; (d) cost $C(\mu)$ of a Wardrop equilibrium as a function of the belief, described by the parameters $\mu_{\theta_2}$ and $\mu_{\theta_3}$ with $\mu_{\theta_2} + \mu_{\theta_3} \leq 1$.
• Figure 3: Braess network: (a) cost functions for state $\theta_1$; (b) cost functions for state $\theta_2$; (c) cost of a Wardrop equilibrium as a function of the belief, described by parameter $\mu_{\theta_2} = 1 - \mu_{\theta_1} \in [0,1]$, the costs are dashed where the supports are infeasible for the given $\mu_{\theta_2}$.
• Figure 4: Example of an instance on parallel edges with two commodities where full information revelation is suboptimal.
• Figure 5: A parallel edges network where full information revelation is suboptimal. The orange line in (c) indicates the optimal signaling scheme.

Theorems & Definitions (41)

• Proposition 1
• Example 1
• Example 2
• Definition 1: Support
• Lemma 1
• Corollary 1
• Lemma 2
• Lemma 3
• proof
• Theorem 1