Reinforcement Learning-based Sequential Route Recommendation for System-Optimal Traffic Assignment

Cheng Lyu; Leizhen Wang; Peibo Duan; Zhenliang Ma

REVIEW 4 major objections 6 minor 26 references

Reinforcement Learning-based Sequential Route Recommendation for System-Optimal Traffic Assignment

T0 review · 4 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A central reinforcement-learning agent that sequentially assigns routes to travelers can converge to the system-optimal traffic assignment, reaching the theoretical optimum on a small paradox-prone network and coming within 0.35% on a…

desk verdict A clever reformulation of SO assignment as sequential RL, but the discount factor makes the stated objective inconsistent with the reported results, and the performance doesn't beat classical MSA. read the letter →

arxiv 2505.20889 v1 pith:OA6EC352 submitted 2025-05-27 cs.AI

Leizhen Wang , Peibo Duan , Cheng Lyu , Zhenliang Ma This is my paper

classification cs.AI

keywords reinforcementlearningrouterecommendationsystemoptimumtrafficassignmentdeepQ-learningmethodofsuccessiveaveragesactionspacedesignsequentialdecisionmaking

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper asks whether route recommendations made one traveler at a time, the way a navigation service works, can collectively produce the system-optimal traffic state rather than a merely self-interested one. It reformulates the static system-optimal traffic assignment problem as a single-agent deep reinforcement learning task: one central agent picks a route for each incoming traveler, and the reward encodes the marginal travel time of that choice, meaning the traveler's own time plus the extra delay imposed on everyone already on the route. To make learning efficient, the agent's route choices are guided by the method of successive averages, the classical iterative-averaging procedure from traffic assignment. Under an explicit full-compliance assumption, the agent converges to the theoretical optimum on a small diamond-shaped network and lands 0.35% above the analytical system-optimal benchmark on a larger 13-node, 48-link network. The practical point is that sequential, personalized routing can approach centralized efficiency, provided the candidate route set contains the routes the optimum uses.

What carries the argument

The load-bearing mechanism is a Markov decision process whose reward and action selection are both aligned with the system-optimum objective. The reward for assigning a traveler to route $p$ is built from the marginal travel time $\sum_{e\in p}\left(c_e(v_e)+v_e\frac{dc_e}{dv_e}\right)$, the traveler's own link travel time plus the extra delay imposed on everyone already using the route; maximizing this quantity aligns the sequential policy with system-optimal assignment. The MSA guidance works on the action set: at the end of each episode the agent computes an all-or-nothing assignment from the current flows and averages it into the route-choice distribution, so the candidate route set grows and the sampling distribution is steered toward good assignments. The route set is the feasible region of the policy: Algorithm 1's incremental addition of routes is what lets the agent discover routes that a fixed k-shortest-path set would miss, and the RL-10-SP ablation shows what happens when such a route is absent.

What would settle it

Inspect the final route set produced by Algorithm 1 on the 13-node network and check whether every route carrying positive flow in the SO-MSA solution is present; the route-completeness premise would be overturned if the set omits an SO-used route while the agent still lands near the 0.35% gap, and the convergence claim would fail if the gap grows whenever such a route is absent. The RL-10-SP ablation already provides a partial check, since its route set omits SO-used routes and the gap jumps to 3.49%.

Watch

Extended reading notes

Core claim

On its own terms, the paper's central claim is that a static system-optimal traffic assignment can be reached by a single deep-Q-learning agent acting sequentially, one traveler at a time, when the agent's exploration is steered by the method of successive averages. In the small diamond-shaped network, the trained agent stabilizes at a total travel time of 498 minutes, with three travelers on each of the two uncongested routes and none on the added route, which is exactly the theoretical system-optimal solution. In the larger 13-node, 48-link network with 1,700 travelers, the MSA-guided agent reaches 55,000 minutes of total travel time, a 0.35% deviation from the 54,809.8-minute SO-MSA benchmark. The ablation results make the mechanism explicit: a 10-shortest-path action set omits routes used by the optimum and leaves a 3.49% gap, a 15-shortest-path set cuts this to 0.75%, an SO-derived route set achieves 0.26%, and the MSA-guided agent achieves its 0.35% without knowing the SO route set in advance. The authors therefore claim that action-set design, not the RL exploration schedule, is the dominant factor separating sequential assignment from system optimum.

Load-bearing premise

The central agent's candidate route set, expanded incrementally by the MSA step, must contain the routes the system-optimal solution actually uses; the RL-10-SP ablation shows that when a route is missing, the agent cannot reach the optimum.

Editorial extensions

If this is right

Sequential, one-at-a-time route recommendations can reproduce a one-shot centralized system-optimal assignment, so the online interaction itself is not a barrier to system efficiency under full compliance.
A route recommendation system must treat its candidate route set as part of its solution: expanding the set from 10 to 15 shortest paths cuts the deviation from the system optimum from 3.49% to 0.75%.
An MSA-guided learner can approach the system optimum without knowing the optimal route set in advance, reaching 0.35% instead of the 0.26% gap obtained with an oracle route set.
Because the agent converges quickly on the small network, in roughly 240 episodes, and avoids the paradox-inducing route, the approach offers a template for steering demand away from configurations that are good individually but bad collectively.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

An extension the paper leaves for future work is partial compliance: if travelers sometimes ignore recommendations, the 0.35% gap could degrade, and the paper's own framing suggests this is the first stress test worth running.
The marginal-cost reward is the same signal used in marginal-cost pricing, so the learned policy is implicitly computing flows that should match a tolled user equilibrium; a direct comparison of the RL assignment against the toll pattern on the same networks would be a cheap, decisive check.
The MSA-guided route expansion functions as a curriculum over actions; comparing it against other route-generation heuristics, such as randomized path sampling or column generation, on city-scale networks would show whether route-set completeness remains the controlling factor as the network grows.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

Quick take: the paper reformulates static system-optimal traffic assignment as a single-agent RL problem where a central controller assigns routes to travelers sequentially, and it wraps DQN in an MSA loop to guide exploration. That framing is new, and the idea to use marginal-cost pricing as the reward is a sensible bridge to the classical SO literature. The experiments on Braess and the OW network are cleanly described, and comparing against MSA/FW benchmarks is good practice.

What the paper does well: the ablation of action sets shows that the route choice set matters more than the RL machinery; RL-SO with the analytic route set achieves 0.26% deviation while RL-10-SP is 3.49%. That's an honest and useful observation. The Braess example is a nice sanity check.

The soft spots are real, though. The most serious is the discount factor. Eq. (3) defines the return as a discounted sum of rewards, and Table 2 says gamma=0.95. With 1700 travelers, the weight on the 100th reward is about 0.006, so the agent effectively only cares about the first ~20–50 travelers. The reward in Eq. (9) is the negative marginal cost, so the learned policy optimizes a heavily time-weighted sum of marginal costs, not the total system travel time. The reported 0.35% deviation on OW is hard to reconcile with that objective. Either gamma was effectively 1 in the experiments, or the reward was not what Eq. (9) says, or the evaluation policy was not the greedy Q policy. Without code or data, we can't tell. This is a load-bearing inconsistency.

Second, the performance claim is modest compared to its classical parent. Pure MSA reaches a relative gap of 6e-5, while MSA-guided RL is 0.35% from SO. So the contribution is the reformulation, not better solution quality. There's also no comparison to a simple greedy marginal-cost policy, and no error bars.

Third, the paper says 'theoretically grounded' but offers no proof that sequential minimization of marginal cost, even undiscounted, converges to SO in this setting. It's a plausible heuristic, not a theorem.

All that said, the paper is worth a serious referee: the question is meaningful, the experiments are transparent in design, and the issues I raised can be fixed or clarified. I'd accept it for peer review but with the expectation of major revisions.

Regards.

Referee Report

4 major / 6 minor

Summary. The paper reformulates the static system-optimal (SO) traffic assignment problem as a single-agent Markov decision process in which a central agent sequentially recommends routes to travelers, and proposes an MSA-guided deep Q-learning algorithm that uses the Method of Successive Averages both to guide action selection and to incrementally expand the route set. Experiments on the Braess network and the Ortuzar-Willumsen (OW) network report convergence to the theoretical SO solution on Braess and a 0.35% deviation from an SO-MSA benchmark on OW, together with ablations comparing k-shortest-path and SO-informed action sets.

Significance. If the results hold, the paper would offer a practical bridge between sequential personalized route recommendation and classical system-optimal traffic assignment, and would provide an analytical testbed for RL in a well-studied transportation problem. The paper deserves credit for using analytical networks with known link performance functions, for comparing against classical MSA and Frank-Wolfe benchmarks, and for explicitly isolating the effect of action-set design through ablations. However, several load-bearing inconsistencies in the objective specification and the close coupling between the proposed algorithm and the classical MSA solution mean that the central empirical claims are not yet established as reported.

major comments (4)

[§III-D, Eq. (9), and Table 2] The reward in Eq. (9) is written as the negative of the marginal social cost, but the surrounding text calls it the 'marginal travel time' and does not explain the sign. If the intended equivalence is that maximizing cumulative reward minimizes total system travel time, that equivalence holds for the undiscounted sum of these rewards, not for the discounted return in Eq. (3). With γ=0.95 in Table 2 and N=1700 travelers, γ^99≈0.006 for the 100th traveler and later travelers receive essentially zero weight in the return, so the algorithm as specified optimizes a heavily time-weighted sum of marginal costs rather than total system travel time. The reported 0.35% gap on the OW network therefore cannot be explained by the stated objective. The authors should either set γ=1 (or otherwise formulate a total-return objective), report experiments with that objective, or provide a proof or controlled experiment showing that the discounting does not affect the final assignment.
[§III, §IV, and Abstract] The central claim that the RL agent 'converges to the theoretical SO solution' is supported only by empirical training curves (Figures 3 and 5). No proof or formal argument is given that sequential minimization of marginal travel time, with or without MSA guidance, converges to SO for discrete travelers and non-linear BPR cost functions. Given the title and abstract, the paper should either supply a convergence theorem (for example, showing that the undiscounted reward sum telescopes to total system travel time and that the MSA-guided action-selection scheme preserves the optimality of the fixed point) or explicitly limit the claim to an empirical approximation.
[§IV-B, Table 4, and Algorithm 1] The comparison is confounded by the fact that the SO-MSA benchmark is itself computed by MSA, while the proposed MSA-guided RL uses MSA for both route-set expansion (Algorithm 1, lines 9-10) and action sampling (Algorithm 2, line 3). The RL-SO ablation, which uses the route set extracted from the analytical SO solution, achieves the best result (0.26% gap), so part of the performance of MSA-guided RL may be inherited from the classical MSA solution rather than from the learned policy. To identify the marginal contribution of RL, the paper should add ablations with pure MSA using the same route-set update but no RL, with a fixed non-MSA action set, and should report the fraction of actions selected by argmax Q versus by sampling from M, along with the evolution of M over training.
[§IV-B-3 and Algorithm 1] The paper's own ablation RL-10-SP shows a 3.49% deviation from SO because the 10-shortest-path action set omits routes that appear in the SO solution. This makes the central result depend on the MSA-guided incremental route expansion in Algorithm 1, lines 9-10, producing an action set that contains all optimal routes for each OD pair. No evidence is provided that this occurs on the OW network. The authors should report the final route set generated by the MSA-guided method, verify that it contains the routes used in the SO-MSA solution, and discuss conditions under which the incremental all-or-nothing update is guaranteed (or empirically shown) to add the missing optimal routes. If the action set is incomplete, even an optimal RL policy cannot reach SO.

minor comments (6)

[§IV-A] The text says 'the network provides three routes from A to B, namely ACB and ADB,' but before the dashed link is added there are only two routes; the third route, ACDB, appears only after the new segment is introduced.
[§V] The conclusion contains a typo: 'Baress paradox network' should read 'Braess paradox network.'
[§III-B, Eq. (8)] The notation in Eq. (8), 'tte^t', is confusing; it would be clearer to use a consistent notation such as c_e(x_e) for the link travel time function and to define whether x_e is the flow before or after the current traveler is added.
[§III-D, Eq. (9)] In the text following Eq. (9), 'link a' should be 'link e', and the formula should make explicit that v_e is the flow before the marginal traveler is assigned; the current typesetting of v_e dce(ve)/dve omits a parenthesis.
[§IV-B, Table 4] The RL results are reported as single numbers without standard deviations or confidence intervals across random seeds; given the stochastic nature of DQN training, this makes it difficult to judge whether the small differences between MSA-guided RL, RL-SO, and RL-15-SP are significant.
[General] No code or data availability statement is provided. For an empirical RL paper, releasing the implementation and the exact network data would substantially improve reproducibility and would help resolve the ambiguities about the training objective and hyperparameters.

Circularity Check

2 steps flagged · score 5.0 of 10

The MSA-guided RL result is partially circular: the action distribution is updated by the same MSA used to create the SO-MSA benchmark, so the 0.35% gap is partly an MSA-variant self-comparison.

fitted input called prediction [Algorithm 1 lines 9-10; Section III-F; Section IV-B3, Tables 3-4]
"M∗ ← all or nothing(V); M←(1− 1/i )·M+ 1/i M∗ ... The utilization of the MSA method effectively guides the RL agent to explore towards SO traffic assignment and update the action (route) set."

The route-choice distribution M is updated by the classical MSA all-or-nothing operator, and the SO-MSA benchmark (Table 3: 54809.8) is computed by the same MSA. In Algorithm 2, actions are sampled from M with probability epsilon, so the MSA solution is injected directly into the policy's action distribution. The reported 0.35% gap is therefore a comparison of an MSA-guided iterate against an MSA fixed point, not an independent RL prediction. The ablations confirm the injection: removing MSA guidance gives RL-10-SP 3.49% and RL-15-SP 0.75% gaps, while injecting the SO route set gives 0.26%. The central quantitative claim is largely inherited from the embedded classical solver.
fitted input called prediction [Section IV-B2 (Compared Methods) and Table 4]
"RL with system-optimum route paths (RL-SO): RL model using action sets derived from the analytical SO solution, i.e., the route set generated from a traditional traffic assignment algorithm."

RL-SO is constructed from the very SO solution used as the performance benchmark (SO-MSA), so its 0.26% deviation is not a prediction but a constrained construction: the action set already contains the optimal routes, leaving only flow proportions to be learned. The paper uses this result to support the claim that 'SO-informed route sets lead to faster learning and better outcomes,' which holds by construction and is presented as an ablation, not as independent validation of the sequential-learning approach. Because this is an ablation rather than the main claim, it contributes to but does not fully determine the circularity score.

full rationale

The paper's main claim has independent content, so this is only partial circularity. RL-15-SP, which does not use MSA guidance or SO-informed routes, still reaches a 0.75% gap, showing that the RL formulation can learn a reasonable SO approximation on its own. The central circularity is the MSA-guided variant: its action-selection distribution is an MSA iterate (Algorithm 1 lines 9-10), and the SO-MSA benchmark is also an MSA solution. Measuring one MSA-based procedure against another is a self-comparison rather than a test of whether sequential RL recovers the theoretical optimum from first principles. The strongest independent evidence for the paper's thesis is therefore the no-MSA RL-15-SP ablation, which is not circular. No load-bearing self-citations or imported uniqueness theorems are present; Refs. [16] and [9] are background and unrelated to the central derivation. The discount-factor inconsistency (gamma=0.95 with a 1700-traveler horizon) is a correctness/consistency concern, not a circularity, and is not scored here. Overall, the result is partially fitted by construction via the embedded MSA, but the existence of a non-MSA RL variant that approaches SO keeps the score below 6.

Assumptions & free parameters 6 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new entities. It relies on classical traffic assignment assumptions (full compliance, static network, known demand) and standard RL hyperparameters. The only free choices are the action set design and hyperparameters, both of which are shown to affect performance.

free parameters (6)

Learning rate = 2e-5
Chosen by grid search (Table 2); affects convergence speed and final performance.
Discount factor gamma = 0.95
Grid-searched; standard DQN hyperparameter.
Mini-batch size = 128
Grid-searched; standard DQN hyperparameter.
Hidden layer sizes = 512, 256
Grid-searched over two hidden layers.
k in k-shortest paths = 10 and 15
Chosen for action set construction; RL-10-SP failure demonstrates sensitivity.
Epsilon in MSA-guided selection = not specified
Algorithm 2 uses epsilon but the value is omitted.

assumptions (4)

domain assumption All travelers follow the recommended route (full compliance)
Remark 1 in Section II; isolates policy performance but removes a key real-world constraint.
domain assumption Network is static with known link performance functions and known sequential OD demand
Section II; the benchmark SO solution is computed for the same static setting.
ad hoc to paper Minimizing the negative marginal travel time reward at each step leads to a system-optimal assignment
The reward in Eq. (9) is the standard marginal cost pricing rule, but no proof is given that the sequential greedy policy converges to SO; it is validated empirically on two networks.
ad hoc to paper The MSA update and incremental route expansion produce an action set that contains the optimal routes
Algorithm 1 lines 9-10; the failure of RL-10-SP shows the action set is load-bearing.

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Cite this review

Pith. "Pith review of Reinforcement Learning-based Sequential Route Recommendation for System-Optimal Traffic Assignment." pith.science (2026). https://pith.science/paper/OA6EC352

@misc{pith2026250520889,
  author       = {Pith},
  title        = {Pith review of: Reinforcement Learning-based Sequential Route Recommendation for System-Optimal Traffic Assignment},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OA6EC352}},
  note         = {Machine review of arXiv:2505.20889}
}

read the original abstract

Modern navigation systems and shared mobility platforms increasingly rely on personalized route recommendations to improve individual travel experience and operational efficiency. However, a key question remains: can such sequential, personalized routing decisions collectively lead to system-optimal (SO) traffic assignment? This paper addresses this question by proposing a learning-based framework that reformulates the static SO traffic assignment problem as a single-agent deep reinforcement learning (RL) task. A central agent sequentially recommends routes to travelers as origin-destination (OD) demands arrive, to minimize total system travel time. To enhance learning efficiency and solution quality, we develop an MSA-guided deep Q-learning algorithm that integrates the iterative structure of traditional traffic assignment methods into the RL training process. The proposed approach is evaluated on both the Braess and Ortuzar-Willumsen (OW) networks. Results show that the RL agent converges to the theoretical SO solution in the Braess network and achieves only a 0.35% deviation in the OW network. Further ablation studies demonstrate that the route action set's design significantly impacts convergence speed and final performance, with SO-informed route sets leading to faster learning and better outcomes. This work provides a theoretically grounded and practically relevant approach to bridging individual routing behavior with system-level efficiency through learning-based sequential assignment.

Figures

Figures reproduced from arXiv: 2505.20889 by the authors.

**Figure 1.** illustrates the RL-based framework. At each time step t, the agent extracts real-time traffic information st from the road network environment. The agent recommends a route, or action, at based on state st. The environment returns a reward rt as feedback to the agent and transitions to a new state st+1. A large amount of paired data ⟨st, at, rt, st+1⟩ is generated by repeating the above process and stored in the re… view at source ↗

**Figure 2.** FIGURE 2 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** depicts the training progression of the RL agent. After approximately 240 episodes, the total travel time (objective value) fundamentally stabilizes at 498 minutes (with a cost of 83 minutes for each traveler), achieving the SO traffic assignment. In this configuration, three travelers are recommended to ACB, and the remaining three to ADB. Notably, no traveler utilizes the route ACDB. This suggests that the RL agen… view at source ↗

Figures from the paper (1 more)

**Figure 5.** Figure 5: FIGURE 5 [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

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Reference graph

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Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Patriksson,The traffic assignment problem: models and methods

M. Patriksson,The traffic assignment problem: models and methods. Courier Dover Publications, 2015

work page 2015

[2] [2]

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J. G. Wardrop, “Road paper. some theoretical aspects of road traffic research.”Proceedings of the institution of civil engineers, vol. 1, no. 3, pp. 325–362, 1952

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[3] [3]

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M. S. Daskin, “Urban transportation networks: Equilibrium analysis with mathematical programming methods,” 1985

work page 1985

[4] [4]

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V . Morandi, “Bridging the user equilibrium and the system optimum in static traffic assignment: a review,”4OR, vol. 22, no. 1, pp. 89–119, 2024

work page 2024

[5] [5]

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M. Beckmann, C. B. McGuire, and C. B. Winsten, “Studies in the economics of transportation,” Tech. Rep., 1956

1956

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H. Bar-Gera, “Origin-based algorithm for the traffic assignment prob- lem,”Transportation Science, vol. 36, no. 4, pp. 398–417, 2002

work page 2002

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Z. Ke, Q. Zou, J. Liu, and S. Qian, “Real-time system optimal traffic routing under uncertainties—can physics models boost reinforcement learning?”Transportation Research Part C: Emerging Technologies, vol. 173, p. 105040, 2025

work page 2025

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