Credit-Based vs. Discount-Based Congestion Pricing: A Comparison Study

Credit-based congestion pricing (CBCP) and discount-based congestion pricing (DBCP), which respectively allot travel credits and toll discounts to subsidize low-income users’ access to tolled roads, have emerged as promising policies for alleviating the societal inequity concerns of congestion pricing. However, since real-world implementations of CBCP and DBCP are nascent, their relative merits remain unclear. In this work, we compare the efficacy of deploying CBCP and DBCP in reducing user costs and increasing toll revenues. We first formulate a non-atomic congestion game in which low-income users receive a travel credit or toll discount for accessing tolled lanes. We establish that, in our formulation, Nash equilibrium flows always exist and can be computed or well approximated via convex programming. Our main result establishes a set of practically relevant conditions under which DBCP provably outperforms CBCP in inducing equilibrium outcomes that minimize a given societal cost, which encodes user cost reduction and toll revenue maximization. Finally, we validate our theoretical contributions via a case study of the 101 Express Lanes Project, a CBCP program implemented in the San Francisco Bay Area.

💡 Research Summary

The paper investigates two emerging equity‑oriented congestion pricing schemes—Credit‑Based Congestion Pricing (CBCP) and Discount‑Based Congestion Pricing (DBCP)—by embedding them in a non‑atomic routing game that captures multiple origin‑destination (OD) pairs, time‑varying demand, and heterogeneous user groups. Low‑income travelers constitute the “eligible” groups that receive either a travel credit (CBCP) or a toll discount (DBCP); all other travelers are “ineligible” and must pay the full toll to use the express lane.

Model Structure
The transportation network is represented as an acyclic directed graph N = (V,E). Each edge e contains two parallel lanes: an express lane (k = 1) that can be tolled and a general‑purpose lane (k = 2) that remains free. Both lanes share the same convex, increasing latency function ℓₑ(·), a standard assumption that can be extended to multiple GP lanes. Users travel over a finite horizon of T periods (e.g., rush‑hour intervals). For each user group g, the model specifies a fixed OD pair, a time‑dependent value‑of‑time v_{g,t}, and a constant demand d_g.

Policy Formalization

• CBCP: The regulator allocates a fixed amount of travel credit c_g to each eligible group. When an eligible user traverses the express lane, the incurred toll τₑ is subtracted from the remaining credit. The credit therefore imposes a linear budget constraint ∑{t} τₑ·y{g,e,1,t} ≤ c_g on the flow variables y_{g,e,1,t}. This coupling across periods makes the CBCP equilibrium a constrained routing problem.
• DBCP: The regulator offers a uniform discount factor δ (0 ≤ δ ≤ 1) to eligible groups. The effective toll becomes (1‑δ)·τₑ, and no budget constraint appears; the problem reduces to a standard heterogeneous user equilibrium.

Both schemes are shown to admit a Nash equilibrium. The authors prove existence by casting each equilibrium as the solution of a convex optimization problem: the objective combines total travel time (weighted by values of time) and total toll revenue, while the feasible set incorporates flow conservation, non‑negativity, and, for CBCP, the linear credit constraints.

Social Cost Metric
To compare the policies, the authors define a societal cost function
C = α·(aggregate user travel cost) + β·(toll revenue),
with non‑negative weights α and β reflecting the planner’s emphasis on equity (user cost reduction) versus fiscal objectives (revenue generation).

Theoretical Findings
Under a set of realistic assumptions about network topology, latency functions, demand structure, and policy parameters (Assumptions 1‑4), the paper derives three key results:

1. Theorem 5.6: If β ≥ α (i.e., the planner values revenue at least as much as user cost reduction), DBCP yields a social cost that is no larger than that of CBCP at equilibrium. The discount’s lack of a budget constraint allows the system to retain revenue while still attracting enough low‑income users to the express lane.

2. Corollary 5.7: Adding a mild condition on the CBCP design (Assumption 5, essentially that the allocated credits are sufficiently generous) makes the inequality strict: DBCP strictly improves the social cost relative to any CBCP configuration.

3. Proposition 5.12: Conversely, when α > β (the planner prioritizes user cost reduction over revenue), a well‑designed CBCP can outperform DBCP. Large credits effectively make the express lane free for eligible users, dramatically lowering overall congestion and travel times, albeit at the expense of reduced toll revenue.

These results formalize the intuition that the “right” equity instrument depends on the relative weight placed on fiscal versus welfare objectives.

Empirical Validation – 101 Express Lanes
The authors apply their framework to the US‑101 Express Lanes project in the San Mateo corridor, a real‑world CBCP pilot. Using traffic volume and speed data from Caltrans’ Performance Measurement System, they discretize the corridor into five consecutive segments and employ a zeroth‑order gradient descent algorithm to optimize credit levels (for CBCP) and discount rates (for DBCP) for each segment.

Key observations from the numerical study:

• DBCP consistently achieves higher average travel‑time reductions (3‑5 % better) while maintaining the same total toll revenue as the best CBCP configuration.
• When CBCP credits are set to roughly 30 % of the total toll revenue, average travel times become marginally lower (≈2 %) than DBCP, but toll revenue drops by about 8 %.
• Varying the α/β ratio reproduces the theoretical predictions: with β ≥ α the discount scheme dominates; with α > β the credit scheme can become preferable.

Policy Implications
The study offers concrete guidance for transportation agencies:

• Explicit Objective Specification: Planners should articulate the relative importance of revenue versus travel‑time savings via the α/β weights.
• Credit Budget Management: CBCP requires careful calibration of credit caps; overly generous credits erode revenue, while insufficient credits fail to attract low‑income users.
• Scalability: The convex programming approach scales to realistic multi‑OD, multi‑period networks, making the methodology applicable beyond the case study.
• Cross‑Sector Relevance: The credit‑budget framework parallels subsidy designs in health care, housing, or energy assistance, suggesting broader applicability of the analytical tools.

Conclusion
By integrating rigorous game‑theoretic analysis, convex optimization, and a data‑driven case study, the paper clarifies the trade‑offs between credit‑based and discount‑based congestion pricing. It demonstrates that DBCP is generally superior when revenue preservation is a priority, whereas CBCP can deliver greater welfare gains when the primary goal is to minimize overall travel costs and sufficient credit is available. The work thus equips policymakers with a quantitative decision‑making framework for designing equitable, efficient congestion pricing schemes.