{ "schemaVersion": "1.6", "slug": "wardrop-smooth-route-regimes", "title": "Smoothness within and across Wardrop route regimes", "shortTitle": "When a new route opens without a kink", "url": "https://evidencepress.org/releases/wardrop-smooth-route-regimes/", "oneLine": "Answers to two open questions of Cominetti, Dose and Scarsini on selfish traffic routing: equilibrium link flows stay smooth while the set of cheapest routes is fixed, even for standard BPR costs, but a new route can switch on with no visible kink in total travel time or the price of anarchy; five links are the minimum for this with a common BPR exponent, and such hidden switches are exceptional.", "abstract": "In a static traffic assignment every traveller uses a cheapest route (Wardrop equilibrium), and the price of anarchy compares the resulting total travel time with the best achievable. Cominetti, Dose and Scarsini (Transportation Research Part B 182, 102922, 2024; arXiv:2305.03459) studied how these quantities change as demand is scaled and left two questions open. First, while the set of cheapest routes stays fixed, are equilibrium link flows continuously differentiable when link costs have zero slope at zero load, as standard BPR costs do? This paper proves that they are, at every interior demand level and on any network; a general version fails at zero demand, and the BPR case there remains open. Second, must the slope of total travel time and of the price of anarchy jump when a new route becomes cheapest, for convex or BPR costs? It need not. A five-link network with ordinary quartic BPR costs has a genuine new route at which total travel time and the price of anarchy have four continuous derivatives; five links are the fewest possible with a common BPR exponent, and larger networks hide any finite number of derivatives. The slope never jumps the wrong way for BPR costs, even with zero free-flow times, and for almost every choice of free-flow times it jumps strictly at nondegenerate breakpoints, so hidden switches are exceptional. When they occur the new route carries flow far below ordinary assignment tolerance; link flows reveal the change earlier than totals. An exploratory Sioux Falls scan finds only transverse activations with visible kinks. AI-assisted, unrefereed candidate; no external peer review, independent reproduction or formal verification is claimed.", "datePublished": "2026-10-22", "dateModified": "2026-10-22", "version": "1.0.0-candidate", "doi": "10.5281/zenodo.23552630", "doiUrl": "https://doi.org/10.5281/zenodo.23552630", "conceptDoi": "10.5281/zenodo.23552629", "pdfUrl": "https://github.com/ipitchford/wardrop-smooth-route-regimes/releases/download/v1.0.0-candidate/wardrop-smooth-route-regimes-1.0.0-candidate.pdf", "altPdfUrl": null, "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/23552630", "repoUrl": "https://github.com/ipitchford/wardrop-smooth-route-regimes", "releaseUrl": "https://github.com/ipitchford/wardrop-smooth-route-regimes/releases/tag/v1.0.0-candidate", "markdownUrl": "https://evidencepress.org/releases/wardrop-smooth-route-regimes/index.md", "bibtexUrl": "https://evidencepress.org/releases/wardrop-smooth-route-regimes/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/wardrop-smooth-route-regimes.mp3", "imageUrl": "https://evidencepress.org/assets/og/wardrop-smooth-route-regimes.png?v=d4c62d8c61", "coverArtUrl": "https://evidencepress.org/assets/art/wardrop-smooth-route-regimes.svg?v=17fd308172", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/wardrop-smooth-route-regimes.mp3", "name": "Plain-English audio briefing", "description": "Communication aid; not additional research evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/wardrop-smooth-route-regimes.txt" } ], "authors": [ "Anonymous" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Unrefereed AI-assisted mathematical candidate. Every theorem has a written proof; exact, numerical and exploratory results are labelled. Internal AI-assisted review, AI-assisted adversarial review and a verification re-review only. External specialist review, independent reproduction and formal verification have not been established." }, "assurance": [ { "dimension": "availability", "label": "Availability and archiving", "question": "Is the evidence package publicly retrievable from an archive under a persistent identifier?", "state": "passed", "evidenceUrl": "https://zenodo.org/records/23552630", "note": "The versioned research package and manuscript are publicly archived on GitHub and Zenodo. Provider asset parity was checked against frozen local files; availability does not establish correctness or peer review." }, { "dimension": "internalReplay", "label": "Internal replay", "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?", "state": "passed", "note": "verify_all.py regenerates every saved output byte for byte. A clean full replay of the final package in a separate copy reproduced every output; the frozen archive was bound to it by hash, and the checks affected by publication changes were rerun." }, { "dimension": "independentRerun", "label": "Independent rerun", "question": "Has someone else run the supplied implementation and obtained the stated result?", "state": "not-assessed", "note": "No unaffiliated rerun has been established." }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "not-assessed", "note": "No independent reimplementation; the exact and numerical checks are producer-side." }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "not-assessed", "note": "No proof-assistant formalisation." }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed", "note": "AI-assisted internal and adversarial review is not external specialist review." }, { "dimension": "editorialPeerReview", "label": "Editorial peer review", "question": "Has a journal or venue run peer review to a decision?", "state": "not-assessed", "note": "No external editorial peer review." }, { "dimension": "dataEnvironmentReproducibility", "label": "Data and environment reproducibility", "question": "Are data and computational environment pinned well enough to rebuild?", "state": "partial", "note": "Exact checks need only the Python standard library and SymPy; numerical checks need mpmath, NumPy, SciPy and Matplotlib (tested versions listed, not locked in a container). The Sioux Falls test-network files are fetched from their source repository and checked by SHA-256." } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "AI-assisted mathematical research", "AI-assisted internal review, verification re-review and adversarial review (including OpenAI gpt-5.6-sol)", "Claude Code: publication packaging and release" ], "humanRole": "Publication authority only; no human research contribution inferred.", "reviewStatus": "Internal AI-assisted and adversarial review, not external peer review." }, "attributionBoundary": null, "problem": null, "orderedProductAverage": null, "corrections": [], "publicCorrections": [], "publicCorrectionIndexes": null, "pageStructureVersion": "2.0", "pageStructureVariant": "canonical-theorem", "pageStructureWaivers": [], "recordMaturity": "current", "metadataProvenance": "Prepared from the approved final manuscript and its exact check outputs. Versioned GitHub and Zenodo research archives are checked against the frozen release assets on 22 October 2026.", "grandfatheredAtSchemaVersion": null, "keywords": [ "Wardrop equilibrium", "price of anarchy", "traffic assignment", "BPR cost functions", "selfish routing", "sensitivity analysis", "phase transitions", "nonatomic congestion games" ], "keyResults": [ "Question 1, yes at interior points: if every link cost is C¹ with positive slope at positive load (BPR costs included), equilibrium link flows are C¹ in demand on every open interval with a fixed set of cheapest routes, on any network (Theorem 3.1, proof). A general extension to zero demand fails (Proposition 3.3, proof); the BPR case there is open.", "Question 2, no: a five-link network with common quartic BPR costs has a genuine expansion of the cheapest-route set at which total cost E and the price of anarchy P are C⁴ but not C⁵ (§5.2, proof; exact). Three links suffice with strictly convex or mixed-exponent costs (§5.1).", "Five links are the minimum for a common BPR exponent, for expansions and contractions (Theorem 6.1, proof; finite enumeration as a check); m + 2 links give E, P ∈ C^(2m−2) but not C^(2m−1) for every p > 1 (Theorem 5.3, proof).", "The slope never jumps the wrong way for BPR costs, even with zero free-flow times (Theorem 4.1, proof), and jumps strictly for almost every free-flow vector at nondegenerate nested breakpoints (Theorem 4.5, proof): hidden switches are exceptional.", "Practice: after a hidden switch the new route carries about 0.01% of its commodity after 10% more demand, below ordinary assignment tolerance; link flows reveal the change two derivative orders earlier than totals (numerical). A Sioux Falls scan finds only transverse activations with visible kinks (exploratory)." ], "reviews": [], "evidencePackage": "Written proofs of every theorem; exact rational checks of the small examples, the five-link series constants, the contact and transversality identities and a finite enumeration supporting the minimality theorem; 50–60-digit numerical computations for the practice section; an exploratory Sioux Falls scan; every saved output regenerated byte for byte by verify_all.py.", "openProblems": [], "mathObjects": [], "relatedWorks": [ { "citation": "Cominetti, Dose and Scarsini (2024), Phase transitions of the price-of-anarchy function in multi-commodity routing games. Transportation Research Part B 182, 102922 (arXiv:2305.03459v2; source of both questions).", "url": "https://doi.org/10.1016/j.trb.2024.102922", "doi": "10.1016/j.trb.2024.102922" }, { "citation": "Wang, Shao and Wu (2026), Regularity of the price of anarchy in routing games over the multivariate demand space (arXiv:2610.16185).", "url": "https://arxiv.org/abs/2610.16185", "doi": "10.48550/arXiv.2610.16185" }, { "citation": "O'Hare, Connors and Watling (2016), Mechanisms that govern how the price of anarchy varies with travel demand. Transportation Research Part B 84, 55–80.", "url": "https://doi.org/10.1016/j.trb.2015.12.005", "doi": "10.1016/j.trb.2015.12.005" }, { "citation": "Cominetti, Dose and Scarsini (2024), The price of anarchy in routing games as a function of the demand. Mathematical Programming 203, 531–558.", "url": "https://doi.org/10.1007/s10107-021-01701-7", "doi": "10.1007/s10107-021-01701-7" }, { "citation": "Colini-Baldeschi, Cominetti, Mertikopoulos and Scarsini (2020), When is selfish routing bad? The price of anarchy in light and heavy traffic. Operations Research 68, 411–434.", "url": "https://doi.org/10.1287/opre.2019.1894", "doi": "10.1287/opre.2019.1894" }, { "citation": "Wu and Möhring (2023), A sensitivity analysis of the price of anarchy in nonatomic congestion games. Mathematics of Operations Research 48, 1364–1392.", "url": "https://doi.org/10.1287/moor.2022.1292", "doi": "10.1287/moor.2022.1292" }, { "citation": "D'Ambrosio (2026), An exact counterexample to affine-like price-of-anarchy shape in quartic BPR routing (arXiv:2609.26179).", "url": "https://arxiv.org/abs/2609.26179", "doi": "10.48550/arXiv.2609.26179" }, { "citation": "Hall (1978), Properties of the equilibrium state in transportation networks. Transportation Science 12, 208–216.", "url": "https://doi.org/10.1287/trsc.12.3.208", "doi": "10.1287/trsc.12.3.208" }, { "citation": "Patriksson (2004), Sensitivity analysis of traffic equilibria. Transportation Science 38, 258–281.", "url": "https://doi.org/10.1287/trsc.1030.0043", "doi": "10.1287/trsc.1030.0043" }, { "citation": "LeBlanc, Morlok and Pierskalla (1975), An efficient approach to solving the road network equilibrium traffic assignment problem. 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