Approximately Optimal Toll Design for Efficiency and Equity in Arc-Based Traffic Assignment Models

Abstract: Congestion pricing policies have emerged as promising traffic management tools to alleviate traffic congestion caused by travelers' selfish routing behaviors. The core principle behind deploying tolls is to impose monetary costs on frequently overcrowded routes, to incentivize self-interested travelers to select less easily congested routes. Recent literature has focused on toll design based on arc-based traffic assignment models (TAMs), which characterize commuters as traveling through a traffic network by successively selecting an outgoing arc from every intermediate node along their journey. However, existing tolling mechanisms predicated on arc-based TAMs often target the design of a single congestion-minimizing toll, ignoring crucial fairness considerations, such as the financial impact of high congestion fees on low-income travelers. To address these shortcomings, in this paper, we pose the dual considerations of efficiency and equity in traffic routing as bilevel optimization problems. Since such problems are in general computationally intractable to solve precisely, we construct a linear program approximation by introducing a polytope approximation for the set of all tolls that induce congestion-minimizing traffic flow patterns. Finally, we provide numerical results that validate our theoretical conclusions.

• The paper introduces a bilevel optimization framework that integrates efficiency and equity in toll design for arc-based traffic models.
• The methodology uses a polytope approximation to reduce toll design into a tractable linear program while achieving similar congestion reductions as marginal tolls.
• Empirical results on Sioux Falls and Atlanta networks show significant reductions in toll revenue and user cost compared to classical tolling approaches.

Approximately Optimal Toll Design for Efficiency and Equity in Arc-Based Traffic Assignment Models

Overview

This paper addresses the dual objectives of efficiency and equity in congestion pricing for arc-based traffic assignment models (TAMs). The central contribution is a bilevel optimization framework for toll design that induces congestion-minimizing equilibrium flows while minimizing financial burdens on low-income travelers. The authors introduce a polytope approximation for the set of congestion-minimizing tolls, enabling the reduction of the bilevel problem to a tractable linear program (LP) under certain fairness objectives. Empirical results on realistic network topologies demonstrate that the proposed tolling strategies achieve the same congestion reduction as classical marginal tolls, but with significantly lower total toll revenue and maximum user cost.

Arc-Based Traffic Assignment Model and Equilibrium

The arc-based TAM models travelers as making sequential arc choices at each node, rather than selecting entire routes upfront. Each arc $a$ is associated with a strictly convex, increasing latency function $s_a(w_a)$, and a non-negative toll $p_a$. The perceived cost incorporates stochasticity via a Gumbel-distributed noise term, leading to a Markovian traffic equilibrium (MTE) characterized by logit choice probabilities. The equilibrium flow $\bar w^\beta(p)$ is the unique fixed point of the induced routing probabilities and flow conservation constraints.

The total network latency is defined as:

$$L(w) = \sum_{a \in A} w_a s_a(w_a) + \frac{1}{\beta} \left[ \sum_{a \in A_i^+} w_a \ln w_a - \left(\sum_{a \in A_i^+} w_a\right) \ln \left(\sum_{a \in A_i^+} w_a\right) \right]$$

where the second term regularizes for noisy user behavior.

Bilevel Optimization for Efficiency and Equity

The toll design problem is posed as a bilevel optimization:

• Inner level: Find tolls $p$ that induce the congestion-minimizing equilibrium flow $w^\star = \arg\min_{w \in W} L(w)$.
• Outer level: Among such tolls, minimize a fairness objective $F(w^\star, p)$.

Three fairness objectives are considered:

1. Minimum Revenue ($F_1$): Minimize total toll revenue $\sum_{a} w_a p_a$.
2. Minimum Max-Cost ($F_2$): Minimize the worst-case total cost (latency + toll) over all routes.
3. Minimum Relative Entropy of Tolls ($F_3$): Minimize the dispersion of tolls across routes, regularized by $\|p\|_2^2$.

The marginal toll $p^\star$ (generalizing the classical Pigovian toll) is shown to induce the congestion-minimizing flow, but may impose unnecessarily high tolls on some arcs, exacerbating equity concerns.

Polytope Approximation and LP Formulation

The set of all congestion-minimizing tolls $P$ is difficult to characterize directly. The authors prove that $P$ contains a $(|I|-1)$-dimensional polytope $\tilde P$ parameterized by node potentials $\tau_i$:

$$\tilde P = \{ (p_a^\star + \tau_{i_a} - \tau_{j_a})_{a \in A} : \tau_d = 0, p_a \geq 0 \}$$

This structure arises from the invariance of equilibrium flows under certain affine transformations of tolls.

By restricting the outer optimization to $\tilde P$, the bilevel problem reduces to a convex program, and for $\lambda_3 = 0$ (i.e., omitting the entropy term), to a linear program:

$\min_{p \in \tilde P} F_\lambda(w^\star, p)$

where $F_\lambda$ is a non-negative linear combination of $F_1$, $F_2$, and $F_3$.

Empirical Results

Simulations on two real-world networks (Sioux Falls and downtown Atlanta) demonstrate the practical impact of the proposed method. For both networks, the LP-derived tolls induce the same congestion-minimizing equilibrium flows as the marginal tolls, but with substantially lower total toll revenue and maximum user cost. For example, in the Sioux Falls network, the marginal toll yields $F_1 = 101.68$ and $F_2 = 27.21$, while the LP-derived tolls achieve $F_1 = 28.88$ and $F_2 = 20.66$ for several choices of $\lambda$. Similar improvements are observed in the Atlanta network.

These results validate the claim that the polytope-based LP approach can achieve efficiency-equity trade-offs unattainable by classical marginal tolling.

Theoretical and Practical Implications

The polytope approximation provides a rigorous foundation for tractable toll design in arc-based TAMs, extending prior work on route-based models. The framework enables explicit control over equity metrics, which is critical for public acceptance and policy compliance. The invariance of equilibrium flows under affine toll transformations suggests new directions for second-best pricing, where only a subset of arcs are tolled.

From a computational perspective, the reduction to LPs and convex programs allows for scalable deployment in large networks, provided the set of feasible routes and arc-node incidence relations are efficiently enumerated.

Future Directions

Key open problems include:

• Quantifying the tightness of the polytope approximation $\tilde P$ relative to the full set $P$.
• Extending the framework to multi-origin, multi-destination, and cyclic networks.
• Designing second-best tolling schemes where only a subset of arcs are tolled.
• Integrating dynamic or time-varying tolls and user heterogeneity in value-of-time.

Conclusion

This work establishes a principled, computationally efficient approach for designing congestion pricing schemes in arc-based traffic assignment models that jointly optimize for efficiency and equity. The polytope approximation enables the use of LPs to find tolls that minimize congestion and financial burden, outperforming classical marginal tolls in both theoretical and empirical evaluations. The framework is extensible to more complex network topologies and fairness objectives, providing a foundation for future research in equitable congestion management.