{"id":"5397ffa1-1980-468e-8fe4-b9f045de59e0","arxiv_id":"2506.22966","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Fleet vehicle flows can be recovered from total flows exactly when the fleet strategy is more selfish than altruistic, and the paper shows failure cases for social and altruistic fleets.","lead":"This mathematics paper proves that a city can recover how many self-routing fleet vehicles take each road from total traffic counts, but only when the fleet is more selfish than altruistic. It matters because coordinated autonomous fleets may one day secretly steer traffic, and this result says whether such fleets are detectable.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 18 is internally sound, but the detection claim inherits a load-bearing behavioral premise: the fleet must know the day's HDV flow exactly, so the fixed-point identity can fail under the paper's own day-to-day prediction suggestion.","rationale":"I checked the proof of Theorem 18 carefully; the directional-derivative argument is valid for global and local minimizers, and the feasible-direction issue at boundary points is handled by the convex-combination feasibility of g and -g. Proposition 27 correctly connects positive definiteness to linear independence. Thus I do not see a flaw that would reject the main theorem. The reader's CONDITIONAL verdict is appropriate: the theorem is proven, but the abstract's 'no otherwise' is not proven (only examples), and Proposition 11(i) overstates uniqueness of the forward minimizer. The most load-bearing concern for the detection claim, however, is the behavioral premise: exact myopic knowledge of HDV flows. The paper itself flags this as a baseline assumption and Appendix A suggests day-to-day prediction, which cannot be exact. Because the fixed-point identity is the link between observed q and the fleet's first-order condition, any prediction error breaks that link. The proposed simulation would quantify whether continuity (Theorem 22) makes the recovery error negligible in the realistic prediction regime, or whether the identification guarantee is an artifact of the oracle assumption. Since this concern is already the reader's weakest assumption and is acknowledged by the authors, it does not change the verdict; it sharpens it.","tokens_in":26088,"tokens_out":20763,"duration_ms":222350,"concrete_test":"Simulate a two-route network with convex BPR delays, fixed q_CRV, and a day-to-day myopic fleet that predicts tomorrow's HDV flow as today's realized HDV flow (and in a second run, with Gaussian noise of increasing variance). Each day: the fleet minimizes (2) given its prediction, HDVs respond (e.g., logit adjustment) to the resulting travel times, and totals q_t are recorded. For each t, compute the unique fixed-point q_CRV from Theorem 18 for the observed q_t and compare to the true q_CRV,t. If the median relative error exceeds 10% while the noise is small, the identification guarantee cannot be exported from the exact-information model to the paper's own suggested day-to-day implementation.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The fixed-point uniqueness in Theorem 18 is proven correctly: for two solutions f*, f~, the first-order inequalities (5)-(6) sum to -(λ_CRV-λ_HDV) g·∇t(q)g ≥ 0, which contradicts positive definiteness unless g=0. So the algebra is not the weak point. The load-bearing condition is the behavioral premise in Section 1.2 remark i: the fleet controller minimizes the one-day objective with exact knowledge of the realized HDV flow before assigning vehicles. The observed total q is presumed to satisfy q_CRV ∈ argmin F(q-q_CRV, ·). If the controller instead predicts HDV flows from previous days (the implementation suggested in Appendix A) or with any noise, the realized q need not satisfy this fixed point. Theorem 22 only controls the error for exact fixed points; it does not bound the recovery error when q is off the image of H. Hence the central 'city can determine fleet route flows' claim is only as secure as the exact-information myopic model, which the paper itself calls 'hardly fully realistic.' This does not invalidate Theorem 18 as a mathematical statement, but it is the assumption on which the detection interpretation rests.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces and analyzes the inverse fleet assignment problem: given the total route (or link) flows of a mixed human/fleet system, the fleet size, and the fleet's objective weights (λ_HDV, λ_CRV), can one uniquely recover the fleet's route flows? The main result, Theorem 18, states that if λ_HDV < λ_CRV and the travel-time gradient matrix ∇t(q) is positive definite, then the observed total flow can arise from at most one fleet flow. The authors extend this to link-additive systems with linearly dependent routes (Theorem 28), dependent link-delay functions (Theorem 31), and multiple OD pairs (Theorem 38), and they prove a continuity estimate for the inverse operator in Theorem 22. They also give a discrete rounding pipeline (Proposition 40), counterexamples for social and altruistic fleets, and a comparison of myopic routing with Stackelberg and Nash routing in Appendix A.","tokens_in":26270,"tokens_out":12309,"duration_ms":124631,"significance":"If the behavioral premises are granted, the paper provides a self-contained, parameter-free uniqueness theorem for a novel inverse problem: no fitted parameters are introduced, and the proof follows directly from first-order optimality conditions of the fleet's stated objective. The extensions to link flows, dependent routes, and multiple OD pairs broaden the applicability considerably, and the continuity estimate is useful for stability statements. The paper also makes a helpful conceptual contribution by showing that recoverability is tied to the fleet being more selfish than altruistic. However, the central detection claim inherits two load-bearing qualifications: the exact-information myopic behavioral model and the fact that the 'no otherwise' side of the dichotomy is not fully proved. With those qualifications made explicit, the mathematical contribution is valuable and publishable.","major_comments":[{"comment":"The stated dichotomy—'yes' for myopic strategies that are more selfish than altruistic and 'no' otherwise—is stronger than what is proved. Theorem 18 establishes the positive direction for λ_HDV < λ_CRV. The negative direction is supported only by Examples 2 and 3, which cover the social (λ_HDV = λ_CRV) and altruistic (λ_HDV > 0, λ_CRV = 0) cases; no argument is given for the full parameter region λ_HDV ≥ λ_CRV. In fact, when λ_HDV = λ_CRV, the objective in (4) equals λ q·t(q), independent of f, so non-uniqueness is immediate, but this still does not prove 'no' for every strategy in that region. Moreover, in degenerate networks such as a single-route network, the inverse is trivially unique for every λ, so 'no otherwise' cannot be a universal statement. The abstract and Corollary 19 should be rephrased as a conditional or existence-of-failure statement, or supplemented by a complete characterization.","section":"Abstract; §1.3; Corollary 19"},{"comment":"The detection interpretation rests on the assumption that the fleet controller knows the realized HDV flow exactly before assigning fleet vehicles and then minimizes the one-day objective. The authors themselves call this 'hardly fully realistic' in Section 1.2, remark i, and Appendix A suggests that the controller may predict HDV flows from previous days. If the controller acts on predicted or noisy HDV flows, the observed total flow q need not satisfy the fixed-point inclusion (4) used in Theorem 18, and Theorem 22 bounds the recovery error only for exact fixed points. Thus the central claim that a city can determine fleet route flows is conditional on this exact-information myopic behavior. The paper should state this limitation in the abstract and conclusions, or extend the analysis (e.g., Theorem 22 or Proposition 40) to provide an error bound under prediction error.","section":"§1.2, remark i; Appendix A; Theorem 18"},{"comment":"Proposition 11(i) is false as stated. For an altruistic fleet (λ_HDV > 0, λ_CRV = 0) with independent links and at least two links carrying zero HDV flow, any distribution of the fleet among those zero-HDV links yields the same objective value, so Γ(η) is not a singleton. The conclusion 'Consequently, Γ(ηHDV) is a one-element set' therefore needs an additional hypothesis, for example λ_CRV > 0 or strict convexity of the objective in the relevant variables. This error does not affect the proof of Theorem 18, but it is a technical mistake in a stated result and should be corrected.","section":"Proposition 11(i)"}],"minor_comments":[{"comment":"The line 'Setting g := −g# = g* = f# − f*' is inconsistent; the intended definition appears to be g# = f* − f# so that −g# = g* = f# − f*. Also, since ∇t(q) need not be symmetric (as in the dependent-link example of Appendix C.3), ρ(∇t) should be defined as the smallest eigenvalue of the symmetric part of ∇t(q), consistently with Definition 17.","section":"Theorem 22 proof"},{"comment":"Step 5 of Proposition 40 assumes that a point qcity minimizing the distance from q to the image of H exists; since the image of the continuous fleet assignment operator need not be closed, the existence of such a projection should be justified or an additional assumption stated.","section":"Proposition 40"},{"comment":"The notation qHDV is used both for the vector of HDV flows and for the scalar total HDV flow in expressions such as qHDV = (qHDV/2, qHDV/2); this should be disambiguated.","section":"Appendix A, Example 47"},{"comment":"The displayed 4x4 matrix for ∇t(q) in the network '8' example is garbled, with unclear alignment of the entries; please reformat it for readability.","section":"Appendix C.2"},{"comment":"A proofreading pass is needed for typographical errors, including 'Defintion' (Definition 8), 'funtion' (Definition 8), 'finite-dimentional' (Definition 14), 'sastisfy' (Theorem 22 proof), 'reserach' (§3.4), and 'continous' (Example 42).","section":"Various"}],"recommendation":"major_revision","confidential_remarks":"The mathematical core, Theorem 18 and its link-flow extensions, appears sound and is a worthwhile contribution to the inverse-optimization literature in transportation. The main concerns are the overstatement of the 'no' side of the dichotomy in the abstract and the exact-information myopic premise that underlies the detection interpretation; both are fixable with more careful framing or additional analysis. I also note that the false statement in Proposition 11(i) should be corrected, even though it is not load-bearing for the main theorem."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main result here is real. The inverse fleet assignment problem – recovering fleet route or link flows from aggregate counts given fleet size and behavior – is new, and Theorem 18 is proved cleanly from first-order conditions. The positive-definiteness condition on the travel-time gradient is the right hypothesis, and the extensions to link-additive systems with dependent routes (Theorem 28) and dependent link delays (Theorem 31) are natural and useful. There are no fitted parameters and no dependence on the authors' earlier work, so the circularity burden is low. The continuity estimate (Theorem 22) is plausible and well motivated. The Stackelberg/Nash comparison in Appendix A is thoughtful, and Example 47 nicely shows that optimal malicious Stackelberg routing can be mixed and destabilizing. This deserves a serious referee.\n\nThe soft spots are real but mostly correctable. First, the abstract says the answer is 'yes' for selfish-than-altruistic strategies and 'no' otherwise. The 'no' is not a theorem; it is illustrated by social and altruistic examples. A single-route network is a trivial counterexample to the claim that uniqueness fails whenever λ_HDV ≥ λ_CRV. The authors should either prove a converse or soften the abstract. Second, Proposition 11(i) states that convexity of the objective implies the fleet assignment is a one-element set. That is false without strict convexity; the altruistic case in Example 3 is convex yet admits multiple minimizers. The proof needs an extra strict-convexity argument or a corrected statement. Third, the load-bearing premise for the detection interpretation is exact myopic knowledge: the fleet controller must know the day's HDV flow before assigning vehicles. The paper calls this 'hardly fully realistic' and Appendix A suggests using previous-day flows as a prediction. If the prediction is noisy, the observed total flow need not lie on the image of the forward operator, and Theorem 22 only controls error for exact fixed points. The theorem stands as mathematics, but the 'city can determine fleet flows' claim is contingent on that behavioral assumption. A robustness analysis for off-image q would strengthen the applied reading.\n\nWho should read this: transportation researchers working on CAV fleets and detection, and math-OC readers interested in inverse problems for nonconvex or unusual objectives. It is a solid contribution that needs revision toning down the abstract, fixing Proposition 11, and discussing the exact-information assumption more prominently. I would send it out; with those changes it can be a useful part of the literature.","headline":"The core inverse fleet-assignment theorem is correct and genuinely new, but the abstract oversells the 'no otherwise' claim and the detection story rests on an exact-information myopic assumption that the paper itself admits is not fully realistic.","tokens_in":26815,"tokens_out":2447,"would_cite":true,"duration_ms":28780,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["90B20","90C25","91A10"],"pacs":[],"model":"deepseek-v4-flash","headline":"Selfish coordinated fleets can be detected from total route flows.","keywords":["coordinated fleets of vehicles","collective routing","inverse problems","detection","route assignment","autonomous vehicles","Stackelberg and Nash equilibrium"],"falsifier":"Take a two-route network with strictly increasing convex BPR delay functions, fix a fleet size and a selfish objective $(\\lambda_{\\mathrm{HDV}}, \\lambda_{\\mathrm{CRV}}) = (0,1)$, and scan all decompositions of a fixed total flow $q$ into HDV and CRV parts. If any two distinct CRV flows both satisfy the first-order conditions for minimizing $F$ and give the same total $q$, the theorem's injectivity claim is false; the theorem predicts no such pair exists. A numerical continuation in $q$ would also test the Lipschitz bound, since the ratio of changes in recovered CRV flow to changes in total flow should stay bounded by the constant in inequality (8).","tokens_in":25855,"feed_emoji":"🚗","tokens_out":5131,"duration_ms":54961,"temperature":0.7,"pith_summary":"The paper asks whether a city that observes only the total number of vehicles on each route can work out how many of those vehicles belong to a centrally coordinated fleet, assuming it knows the fleet size and the fleet's routing objective. The authors prove an inverse fleet assignment theorem: when the fleet's objective weights its own travel time more heavily than human drivers' travel time ($\\lambda_{\\mathrm{HDV}} < \\lambda_{\\mathrm{CRV}}$), and the gradient of the travel-time map is positive definite, the total route flow determines at most one fleet flow. This makes fleet flows identifiable for selfish, malicious, and disruptive fleet behaviours, but not for social or altruistic ones. The result matters because coordinated fleets can degrade traffic for everyone while being invisible in aggregate counts, and the theorem shows detection is theoretically possible before any vehicle-level identification.","feed_headline":"Selfish robot fleets can be spotted in traffic counts","feed_subtitle":"A new inverse theorem says a city can recover fleet route choices from aggregate counts, except when the fleet is altruistic.","key_machinery":"The load-bearing object is the Route Fleet Assignment Operator $G$, which sends each human-driven flow $q_{\\mathrm{HDV}}$ to the set of fleet flows of fixed size that minimize the objective $F(h,f) = (\\lambda_{\\mathrm{HDV}} h + \\lambda_{\\mathrm{CRV}} f)\\cdot t(h+f)$. The proof of invertibility compares two candidate fleet flows $f^*$ and $\\tilde f$ at the same total flow $q$; subtracting their first-order optimality conditions yields $-(\\lambda_{\\mathrm{CRV}} - \\lambda_{\\mathrm{HDV}})\\, g \\cdot (\\nabla t(q) g) \\ge 0$ for $g = \\tilde f - f^*$. Since $\\nabla t(q)$ is positive definite and $\\lambda_{\\mathrm{CRV}} - \\lambda_{\\mathrm{HDV}} > 0$, the inequality forces $g = 0$. Thus the mechanism is a coercive quadratic form supplied by the travel-time gradient, combined with the sign condition on the fleet's behavioural weights.","core_discovery":"The central result is Theorem 18. For continuously differentiable route delay functions with positive definite gradient matrix, if $\\lambda_{\\mathrm{HDV}} < \\lambda_{\\mathrm{CRV}}$ then for every total flow $q$ there is at most one fleet flow $q_{\\mathrm{CRV}}$ minimizing the one-day fleet objective. Equivalently, the forward fleet assignment operator $H = \\mathrm{Id} + G$ is injective on its domain, so the inverse map from total counts to fleet route flows is single-valued; Theorem 22 adds that this inverse is continuous, hence stable under small measurement errors. For link-additive networks the same conclusion holds for fleet link flows even when routes are linearly dependent (Theorem 28), and a version holds for dependent link delay functions (Theorem 31) and for multiple origin–destination pairs (Theorem 38).","pith_inferences":["If fleet controllers anticipate detection, the myopic strategy may become a deliberate choice: Appendix A's observation that myopic routing avoids the chaotic mixed strategies of optimal Stackelberg routing suggests a detection-averse fleet would stay inside the regime where the inverse theorem works, not escape it.","The theory could plausibly extend to multiple fleets with distinct objectives, but the paper leaves that open; a natural test is whether the injectivity argument survives when $G$ is replaced by a sum of fleet-specific assignment operators with different $(\\lambda_{\\mathrm{HDV}}, \\lambda_{\\mathrm{CRV}})$.","The positive-definiteness condition is the same one that guarantees uniqueness of user equilibrium, so empirical networks with separable BPR-like delays will typically satisfy it; the practical bottleneck is not the mathematics but the assumption that the city knows the fleet's objective exactly.","A concrete simulation test: in a day-to-day route choice model with adaptive HDVs, estimate fleet flows by the inverse theorem while the fleet uses predicted rather than actual HDV flows; the degradation with prediction error would quantify how far the baseline assumption can be pushed."],"forward_implications":["Cities can in principle recover CRV route flows from aggregate loop or count data for selfish, malicious, disruptive, and competitive fleet objectives, without identifying individual vehicles.","The recovery is stable: the Lipschitz continuity of the inverse means small counting errors translate to small errors in the estimated fleet flow, under the theorem's assumptions.","For networks with linearly dependent routes, the city can still recover fleet link flows uniquely, even when multiple route-flow decompositions give the same link counts.","Social and altruistic fleets are inherently undetectable from total flows: the same total assignment can arise from many different fleet splits.","The discrete version holds approximately: if the fleet assigns by rounding a continuous local minimizer, the recovered HDV flow is close to the true one, with error controlled by the rounding size and the inverse's Lipschitz constant."],"supporting_citations":[{"why":"Shows that myopic coordinated routing by CAVs can degrade travel conditions, the phenomenon the city wants to detect.","marker":"[25]"},{"why":"Supplies the standard traffic assignment framework and convex optimization algorithms used throughout the paper.","marker":"[6]"},{"why":"Cited in Remark 33 for the classical result that positive definiteness of the delay gradient guarantees existence and uniqueness of user equilibrium.","marker":"[14]"},{"why":"Cited for existence, uniqueness and stability of traffic equilibria, underpinning the positive-definiteness assumption.","marker":"[39]"},{"why":"Cited alongside [39] for the same positive-definiteness condition in asymmetric and stochastic-process traffic assignment models.","marker":"[48]"},{"why":"Provides the Stackelberg scheduling framework used in Appendix A to contrast myopic routing.","marker":"[37]"},{"why":"Supplies the Stackelberg routing model for atomic network games used in the comparison of routing types.","marker":"[10]"},{"why":"Supplies Stackelberg-game results and algorithms for multiple equilibrium behaviours on networks, used in the appendices.","marker":"[54]"}],"fun_headline_variants":["Selfish fleets expose route choices in total traffic flows","Inverse theorem: selfish fleet routes can be deduced from counts","Proving urban fleets that act selfish are traceable in traffic","Mathematical proof shows selfish fleets leave recoverable traces","Detection theorem: aggregate counts reveal selfish fleet routing"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The theorem assumes the fleet controller knows that day's human-driven flows exactly before assigning its vehicles and optimizes only that one day's objective; if the fleet plans over multiple days, anticipates human reactions, or acts on imperfect predictions, the identification result no longer applies.","fun_headline_variants_meta":{"raw":{"variants":["Selfish fleets expose route choices in total traffic flows","Inverse theorem: selfish fleet routes can be deduced from counts","Proving urban fleets that act selfish are traceable in traffic","Mathematical proof shows selfish fleets leave recoverable traces","Detection theorem: aggregate counts reveal selfish fleet routing"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000198,"raw_usage":{"total_tokens":1351,"prompt_tokens":914,"completion_tokens":437,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":530,"completion_tokens_details":{"reasoning_tokens":365}},"tokens_in":530,"tokens_out":437,"duration_ms":5801,"temperature":1.0,"reasoning_tokens":365,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T21:55:32.128013+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a two-route network with strictly increasing convex BPR delay functions, fix a fleet size and a selfish objective $(\\lambda_{\\mathrm{HDV}}, \\lambda_{\\mathrm{CRV}}) = (0,1)$, and scan all decompositions of a fixed total flow $q$ into HDV and CRV parts. If any two distinct CRV flows both satisfy the first-order conditions for minimizing $F$ and give the same total $q$, the theorem's injectivity claim is false; the theorem predicts no such pair exists. A numerical continuation in $q$ would also test the Lipschitz bound, since the ratio of changes in recovered CRV flow to changes in total flow should stay bounded by the constant in inequality (8).","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows that myopic coordinated routing by CAVs can degrade travel conditions, the phenomenon the city wants to detect."},{"cited_title":"D., Lownes, N","cited_arxiv_id":null,"evidence_quote":"Supplies the standard traffic assignment framework and convex optimization algorithms used throughout the paper."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Cited in Remark 33 for the classical result that positive definiteness of the delay gradient guarantees existence and uniqueness of user equilibrium."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Cited for existence, uniqueness and stability of traffic equilibria, underpinning the positive-definiteness assumption."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Cited alongside [39] for the same positive-definiteness condition in asymmetric and stochastic-process traffic assignment models."},{"cited_title":"(2001, July)","cited_arxiv_id":null,"evidence_quote":"Provides the Stackelberg scheduling framework used in Appendix A to contrast myopic routing."},{"cited_title":"Columbia Working Paper No","cited_arxiv_id":null,"evidence_quote":"Supplies the Stackelberg routing model for atomic network games used in the comparison of routing types."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies Stackelberg-game results and algorithms for multiple equilibrium behaviours on networks, used in the appendices."}],"review_version":1}