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

arXiv:2602.11077 · eess.SY, cs.SY · Submitted 2026-02-11 · Read on arXiv

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Introduction to the show: ident: Robotics Radio. Generated commentary on the latest robotics and control papers.

Rosa: Today's paper: "Credit-Based vs. Discount-Based Congestion Pricing".

Dev: Discount-Based Congestion Pricing:

Rosa: First, who's behind it and why it matters.

Title and authors: Rosa: So we're talking about this paper now titled Credit-Based vs. Discount-Based Congestion Pricing: A Comparison Study by Chiu, Jalota, and Pavone. We looked at how these two different ways of helping low-income users access tolled roads affect travel costs and how much revenue the system makes.

Dev: It’s about comparing CBCP, which gives people travel credits, to DBCP, which gives them a discount on the toll itself. The authors set up this whole thing as a non-atomic congestion game to see which one actually does better for society under different priorities.

Taro: So the big question here is what happens when you weigh reducing someone’s travel cost against making sure the road system collects enough money to keep running. It sounds like they are trying to find a clear answer for transportation agencies that are currently stuck between two options.

Rosa: Right, exactly. The paper lays out this comparison using a set of weights called Pareto weights—things like how much you care about revenue versus how much you care about making sure the people who can't afford high tolls can still get around.

Dev: Their main finding is that DBCP often comes out ahead if the goal is maximizing toll revenue, but it depends on exactly how those priorities are weighted. They show that under certain conditions, DBCP policies actually generate a lower total societal cost at equilibrium than CBCP policies >

Taro: Wait, so it's not just one policy that wins; it depends entirely on the setup of the game and what society decides is most important in that moment. That’s interesting when you think about real-world deployment because agencies don't always have a single, fixed priority.

Rosa: That’s right. They establish these practically relevant conditions where DBCP provably outperforms CBCP if the societal cost function strictly prioritizes toll revenue maximization over minimizing eligible users’ travel costs, which they call Theorem five point six > <ref:2602.11077#pg3,strictly prioritizes toll revenue maximization over>

Dev: And there's a stronger result if you get very strict about that revenue focus; if the weight for revenue is higher than the weight for user cost, then DBCP can achieve a strictly lower minimum societal cost compared to any CBCP policy at equilibrium >

Title and authors: Taro: That suggests that if an agency’s main pressure is keeping the toll stream healthy, they have a mathematical reason to look at discount-based options first, provided they meet those structural assumptions >

Rosa: Exactly. But the authors also show a case where things flip; if the societal cost function is defined to prioritize minimizing eligible users’ travel costs over toll revenue maximization, then CBCP can actually outperform DBCP in minimizing that societal cost >

Dev: That means it's not always a simple choice between the two mechanisms depending on which one you pick based on the weights >

Taro: So for someone just listening to this show, what does that mean practically? It means if you're designing a toll system, you need to be really honest about whether your primary job is revenue collection or social equity in defining your cost function >

Rosa: Precisely. And they tested this with a numerical study of the San Mateo one hundred one Express Lanes Project, using existing zeroth-order gradient descent methods to compute both first-order stationary CBCP and DBCP policies > <ref:2602.11077#pg2>

Dev: The experiment results confirmed their theory: under the conditions they set, DBCP policies outperformed CBCP policies in terms of user travel costs and generated toll revenues across different weight sets >

Taro: The paper notes that they used existing methods, adapting zeroth-order gradient descent to compute these stationary policies for consecutive segments of the highway >

Rosa: And this is where the engineering side gets important, Dev. They used different computational approaches for each policy: CBCP used a centralized approach in Algorithm one while DBCP leveraged Lemma five point eight to compute equilibria separately across edges and time periods >

Dev: That separation for DBCP was key because it allowed them to handle things in a way that matched the structure of the discount mechanism better than the centralized method for CBCP >

Taro: So, when you look at how they computed these, it shows that for DBCP, you can break down the problem into smaller pieces and solve them individually before putting them back together >

Title and authors: Rosa: And this whole comparison framework is designed to show how different choices of access mechanisms—credits versus discounts—impact the overall system performance based on those weights >

Dev: The paper points out that they have to work within Assumptions one through four which define the network structure, user attributes, and cost environment for their results to hold true >

Taro: That’s a limitation you have to keep in mind. The theoretical guarantees only hold if those underlying assumptions about the road network and user behavior are actually true in real life >

Rosa: And they also flag that when it comes to computation, they have to use specific algorithms like Algorithm one for CBCP and Algorithm two for DBCP, which is necessary because of how the optimization problems are structured >

Dev: It’s a solid numerical study, but you have to remember the context. They compare these two policies in a non-atomic congestion game setting where users choose between credits or discounts >

Taro: The paper does suggest that hybrid DBCP-CBCP policies might be effective if you want to push the societal cost down even further, which is a nice idea for future policy design >

Rosa: Yeah, so to wrap up on this comparison study, the main point is that DBCP is provably better than CBCP when revenue maximization is the goal and certain structural assumptions are met >

Dev: We’ve covered how they set up the game, how they found these results numerically on projects like San Mateo one hundred one and what those findings mean for policy choice based on your weightings <ref:2602.11077#pg2>.

Taro: I just want to say that the main thing this paper does is give us a clear mathematical tool to decide which pricing mechanism is theoretically superior depending on the societal priorities we choose >

Rosa: So that's how we wrap up this discussion on Credit-Based vs. Discount-Based Congestion Pricing: A Comparison Study. We’ll keep an eye out for more research on how these tools might be applied in other mixed economy settings next time >

The paper's summary: Rosa: So, we just talked about how these two pricing options—credits versus discounts—can perform differently depending on what society cares about most.

Dev: Right, and now we’re going to break down the actual summary of this paper, "Credit-Based vs. Discount-Based Congestion Pricing: A Comparison Study," and what that actually means for us when we’re thinking about real infrastructure.

Rosa: Basically, the core idea is they built a mathematical framework to compare Credit-Based Congestion Pricing, or CBCP, against Discount-Based Congestion Pricing, or DBCP.

Dev: They set up this non-atomic congestion game because it lets them test how users choose between those two access methods while keeping an eye on the flow dynamics.

Rosa: The key finding is that the paper proves a very specific condition under which DBCP actually wins out over CBCP in terms of minimizing societal cost.

Dev: It boils down to prioritizing toll revenue maximization over minimizing the travel costs for eligible users. That’s when DBCP provably achieves a lower total societal cost at equilibrium compared to CBCP policies.

Rosa: The authors show that if you set your weights in a certain way—if revenue is more important than user savings—then discounting the toll is the mathematically superior choice for running the whole system efficiently.

Dev: But they also showed that if you flip those weights and prioritize making sure eligible users have as low a cost as possible, then CBCP might actually be better at keeping that specific travel cost down.

Rosa: So, it’s not a simple 'always use DBCP' scenario; the advantage shifts completely based on how much money society puts into revenue versus how much it puts into equity.

Dev: And they validated this theory using real-world data from a project in San Mateo, confirming that when revenue is the main driver, DBCP policies performed better in those numerical tests.

Rosa: What this means for us listening right now is that if you’re designing a new toll system or looking at an existing one, you can’t just pick one mechanism blindly based on intuition.

Dev: You have to look at your priorities—your weights—and see which theoretical condition the system fits into, because that dictates whether the discount or the credit structure is better for your goals.

Rosa: It’s about moving away from just picking a policy and starting to understand how your societal goals translate into mathematical performance under different constraints.

Dev: And this sets up a pretty interesting challenge for us next, because if we want to figure out how to make these dynamic choices in real-time, that’s what we need to tackle next.

The paper's improvements: Tom: So, we just talked about how these two pricing options—credits versus discounts—can perform differently depending on what society cares about most.

Rosa: Right, and now we’re going to break down the actual summary of this paper, "Credit-Based vs. Discount-Based Congestion Pricing: A Comparison Study," and what that actually means for us when we’re thinking about real infrastructure.

Dev: Basically, the core idea is they built a mathematical framework to compare Credit-Based Congestion Pricing, or CBCP, against Discount-Based Congestion Pricing, or DBCP.

Rosa: They set up this non-atomic congestion game because it lets them test how users choose between those two access methods while keeping an eye on the flow dynamics.

Dev: The key finding is that the paper proves a very specific condition under which DBCP actually wins out over CBCP in terms of minimizing societal cost.

Rosa: It boils down to prioritizing toll revenue maximization over minimizing the travel costs for eligible users. That’s when DBCP provably achieves a lower total societal cost at equilibrium compared to CBCP policies.

Dev: But they also showed that if you flip those weights and prioritize making sure eligible users have as low a cost as possible, then CBCP might actually be better at keeping that specific travel cost down.

Rosa: So, it’s not a simple 'always use DBCP' scenario; the advantage shifts completely based on how much money society puts into revenue versus how much it puts into equity.

Dev: And they validated this theory using real-world data from a project in San Mateo, confirming that when revenue is the main driver, DBCP policies performed better in those numerical tests.

Rosa: What this means for us listening right now is that if you’re designing a new toll system or looking at an existing one, you can’t just pick one mechanism blindly based on intuition.

Dev: You have to look at your priorities—your weights—and see which theoretical condition the system fits into because that dictates whether the discount or the credit structure is better for your goals.

Rosa: It’s about moving away from just picking a policy and starting to understand how your societal goals translate into mathematical performance under different constraints.

Dev: And this sets up a pretty interesting challenge for us next, because if we want to figure out how to make these dynamic choices in real-time, that’s what we need to tackle next.

Conclusion: Tom: So we’re wrapping up this look at the "Credit-Based vs. Discount-Based Congestion Pricing: A Comparison Study."

Rosa: Basically, we established that DBCP is mathematically better than CBCP when the goal is maximizing revenue over minimizing user travel costs.

Dev: That means if your primary objective is keeping the system financially healthy, you should lean toward discount mechanisms under those specific conditions.

Rosa: And they showed that this isn't just a theory; they tested it on real data from San Mateo, and DBCP policies actually outperformed CBCP ones in their numerical results.

Dev: That’s a big deal for control engineers because it means the structure of how you model the access mechanism really matters for the system’s overall performance metrics.

Taro: I just want to push on what happens when things get messy, like if user behavior misbehaves or the road network changes unexpectedly.

Rosa: That's a fair point, Taro, because they did flag that their theoretical guarantees only hold true as long as the underlying assumptions about user behavior and network structure actually hold up in reality.

Dev: Exactly, and that points to where we need more robust modeling tools like the ones we’ve seen in papers about uncertainty-aware control.

Taro: Right, so if the world misbehaves, we need models that can handle those shocks without breaking the pricing logic entirely.

Rosa: Well said. So for our listeners, this paper means understanding your societal goals is just as important as picking a technology; it’s about aligning your weights with what you actually want to achieve on the road.

Dev: It’s about realizing that these two methods aren't interchangeable tools; they serve different masters depending on whether you prioritize user equity or system revenue.

Taro: We need to keep looking at how these mechanisms can be blended, because maybe a hybrid approach could handle those misbehaviors better than either one alone.

Rosa: That’s the direction we’re heading—looking for ways to combine the best parts of both credit and discount approaches for even better outcomes.

Dev: Next week, we'll be looking at how AI is using these kinds of complex models to predict and manage those flows in real-time.

eess.SY, cs.SY

Submitted: 2026-02-11

Updated: 2026-10-09

Project page: https://portal.311.nyc.gov/article/?kanumber=KA-03612

License: http://creativecommons.org/licenses/by-nc-nd/4.0/

Importance score: 85/100

The gist: Discount-Based Congestion Pricing: A Comparison Study." The material presents a sophisticated comparison between two congestion pricing mechanisms—Credit-Based Congestion Pricing (CBCP) and

Key concepts

Credit-Based Congestion Pricing (CBCP)
This system gives low-income users travel credits to use tolled roads. It's like a voucher that offsets the cost of using the express lanes, aiming to subsidize access for those who cannot afford full tolls.
Discount-Based Congestion Pricing (DBCP)
This policy offers low-income users a direct discount on tolls instead of credits. It's a direct price reduction mechanism, which the research shows is more effective than credits when revenue maximization is prioritized.
Societal Cost Metric
This is a single measure used to judge pricing policies. It combines two competing goals: reducing the travel costs for eligible users and maximizing the revenue collected from tolls. Policies are then evaluated based on which one this combined cost metric minimizes.

Terminology

Summary

Discount-Based Congestion Pricing: A Comparison Study." The material presents a sophisticated comparison between two congestion pricing mechanisms—Credit-Based Congestion Pricing (CBCP) and Discount-Based Congestion Pricing (DBCP)—within a non-atomic congestion game setting, focusing on how these policies affect equilibrium outcomes under various societal cost priorities.

Here is the comprehensive, detailed summary synthesizing the theoretical framework, main results, and methodological approach described in the excerpts:

Comprehensive Research Summary: Credit-Based vs. Discount-Based Congestion Pricing Comparison

This research investigates and rigorously compares Credit-Based Congestion Pricing (CBCP) and Discount-Based Congestion Pricing (DBCP) within a mixed economy model designed to manage congestion on tolled roads. The core objective is to determine which policy induces equilibrium outcomes that minimize a specified societal cost metric, which itself is defined by balancing user cost reduction and toll revenue maximization.

  1. Model and Setup

The study operates within a mixed economy model where eligible users have the choice between two access mechanisms:

  • CBCP: Users receive travel credits.

  • DBCP: Users receive toll discounts.

Eligible users are contrasted with ineligible users who must pay out-of-pocket for express lane access.

The comparison is quantified using a societal cost metric, which is formalized by objective functions f C lambda (for CBCP) and f D lambda (for DBCP). These functions are parameterized by a set of Pareto weights lambda = (lambda E, lambda R, lambda I) in R 3 at least 0, where:

  • lambda E relates to the minimization of eligible users' travel costs.

  • lambda R relates to the maximization of toll revenue.

  • lambda I likely pertains to other societal factors (e.g., related to ineligible user costs or infrastructure).

The comparison hinges on how these policies perform under different prioritization schemes defined by the relationship between lambda R and lambda E.

  1. Theoretical Contributions and Main Results

The paper establishes a set of practically relevant conditions under which DBCP provably outperforms CBCP in achieving lower societal costs at equilibrium, depending entirely on how the societal cost is weighted:

  • Revenue-Prioritized Scenario (lambda R at least lambda E): Under Assumptions 1 through 4 (which define the network structure, user attributes, and cost environment), if the societal cost function prioritizes toll revenue maximization over minimizing eligible users’ travel costs (i.e., lambda R at least lambda E), then DBCP policies induce a lower or equal societal cost at equilibrium compared to CBCP policies (Theorem 5.6).

  • Strict Revenue-Prioritized Scenario (lambda R > lambda E): Furthermore, if this prioritization is strict and an additional mild regularity condition on the CBCP policies (Assumption 5) holds, then DBCP achieves a strictly lower minimum societal cost compared to any CBCP policy (Corollary 5.7).

  • Cost-Prioritized Scenario (lambda E > lambda R): Conversely, if the societal cost is defined to prioritize the minimization of eligible users’ travel costs over toll revenue maximization, it is possible for CBCP policies to outperform DBCP policies in minimizing the equilibrium societal cost (Proposition 5.12).

The central theoretical contribution is thus establishing that DBCP offers a provable advantage over CBCP when maximizing toll revenue is a primary goal, provided certain structural assumptions hold.

  1. Methodological Details and Validation

The comparison relies on comparing the optimal societal costs derived from the respective objective functions:

  • CBCP Optimal Cost (f C lambda): Defined by equation (5.3a) and (5.3b), involving minimization subject to constraints on travel time (tau) and budget allocation (B).

  • DBCP Optimal Cost (f D lambda): Defined by equation (5.4a) and (5.4b), involving minimization subject to constraints on toll/discount parameters (tau, alpha) and user access conditions.

The comparison is fundamentally driven by the relationship: Under Assumptions 1-4, if lambda R at least lambda E, then f D lambda at most f C lambda.

These theoretical results are empirically validated through a case study of the 101 Express Lanes Project in the San Francisco Bay Area (specifically San Mateo). Numerical experiments confirm that DBCP policies yield lower societal costs than CBCP policies when toll revenue generation is favored over minimizing eligible users' travel costs.

  1. Algorithmic Implementation

The computational aspect involves iterative optimization techniques:

  • CBCP Computation: Utilizes a zeroth-order optimization method (Algorithm 1) to compute tolls (tau) and budgets (B) for the stationary policy, starting from random initial iterates.

  • DBCP Computation: Employs a similar zeroth-order method (Algorithm 2), but crucially leverages Lemma 5.8 to compute DBCP equilibria separately across edges and time periods, contrasting this with the centralized approach used for CBCP in Algorithm 1.

The study presents comparative results in Table 3, which summarizes key performance indicators—including total travel costs, toll revenues, societal costs, express lane usage, and average travel times—for both optimal CBCP and DBCP policies across different weight sets S(1) (lambda R at least lambda E) and S(2) (lambda R < lambda E).

Conclusion

In summary, the paper provides a rigorous mathematical proof that DBCP is superior to CBCP when the societal objective prioritizes toll revenue maximization over minimizing eligible user costs (lambda R at least lambda E). The findings are robustly supported by numerical experiments on real-world infrastructure projects and offer crucial guidance for policymakers seeking optimal congestion pricing strategies. The study also suggests that hybrid DBCP-CBCP policies may be effective in further minimizing societal cost.

Improvements for AI systems

  1. Bold Header: Equitable Congestion Pricing Policy Optimization

The improved system can dynamically select between Credit-Based Congestion Pricing (CBCP) and Discount-Based Congestion Pricing (DBCP) by optimizing a societal cost metric, specifically choosing DBCP when the societal cost is defined to prioritize maximizing toll revenues over the minimization of eligible users’ travel costs (Thm. 5.6).

  1. Bold Header: Revenue Maximization Under High Weight

The system can generate strictly lower societal costs compared to CBCP policies if "λR > λE, and a mild regularity assumption on express lane flows (Assumption 5) holds in addition to Assumptions 1-4" (Cor. 5.7).

  1. Bold Header: Sensitivity Analysis for Optimal Policy Selection

The system can determine the optimal policy by evaluating the relative magnitudes of toll revenue weight versus eligible user cost weight, as "if λR < λE, CBCP may strictly outperform DBCP in minimizing the societal cost" (Prop. 5.12).

  1. Bold Header: Robust Equilibrium Flow Computation

The system can compute computationally tractable equilibrium flows for both policies by utilizing convex optimization methods, such as the program in Proposition 4.3 for DBCP equilibria and Proposition 4.4 for CBCP equilibria, even when considering time-varying user Value of Time (VoT).

Abstract

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 deployments of CBCP and DBCP remain limited, transportation agencies lack clear guidance on which policy better achieves travel cost reduction and revenue generation objectives. 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. Our results provide theoretical and computational foundations for evaluating equity-oriented congestion pricing policies and offer practical guidance for transportation agencies seeking to design congestion pricing programs that better balance travel cost reduction and revenue generation objectives.

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