{"id":"943dc30e-1a3f-4182-a7f6-5141bc3048bc","arxiv_id":"2501.18050","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":3.0,"correctness_risk":"high","formal_verification":"none","parameter_count":10,"one_line_summary":"The paper derives a closed-form expression for a soccer player's optimal stubbornness from a custom stochastic model, but the derivation is internally inconsistent and unvalidated.","lead":"This paper derives a formula for the optimal level of stubbornness a soccer player should show, using stochastic calculus and path integral methods on a model of scoring chances. The derivation is not numerically tested, rests on arbitrary constants, and contains internal errors, so the formula is not a usable prediction.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (71) is derived by setting f_xxu=0, but f_xxu is not zero for the paper's own f in Eq. (74), so the explicit u* is not the stationary condition of the stated control problem.","rationale":"The reader's REJECT verdict is correct, but the most load-bearing defect is not only the d\\lambda(s)\\to 0 limit; it is the silent use of f_xxu=0 in deriving Eq. (71). The paper's own f in Eq. (74) has f_xxu\\neq 0, so Eq. (71) is not the first-order condition of the path-integral objective. This is an internal inconsistency, not a disagreement with consensus. The d\\lambda(s)\\to 0 step compounds the problem by discarding the constraint terms, and the subsequent algebra contains concrete errors. The paper provides no data, no calibration, and no independent verification, and the parameter-rich payoff is invented. The central claim, a closed-form optimal stubbornness, therefore lacks support. I agree with the reader's rejection, though I would emphasize the f_xxu error as the primary internal flaw.","tokens_in":31664,"tokens_out":8802,"duration_ms":91450,"concrete_test":"Symbolically recompute Eq. (70) using the f from Eq. (74), retaining both the f_xxu term and all d\\lambda(s) terms, and solve the resulting equation for u*(s). If the condition obtained is not equivalent to Eq. (71), or the solution differs from the nested-root formula, then Proposition 11's optimality condition is invalid and the explicit result is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The explicit solution in Section 3 depends on Proposition 11's first-order condition Eq. (71). That condition is obtained from Eq. (70) by setting f_xxu = 0. For the f defined in Eq. (74), however, f_xxu is not zero: the π-term contributes -3c u exp(-rs)/(2(r-\\bar\\mu)x^{5/2}), and the h-term contributes -(\\sigma_2)^2 e^{\\sigma_2 x} d\\lambda(s) from h_{xx}\\mu_u. Substituting this f into Eq. (70) gives a different stationarity equation, so Eq. (71) does not follow. The example then further drops all d\\lambda(s) terms (\"Assume the effect of d\\lambda(s) is very small\"), which removes the SDE constraint from the stochastic Lagrangian (49); the resulting condition is for an unconstrained problem, not for the constrained optimal-control problem (47)-(1). Even within that reduced problem, the algebra from Eq. (82) to the nested-root formula contains sign/exponent errors, e.g., after dividing by exp(-2rs), the A2 term should carry exp(+rs), not exp(-3rs). Thus the claimed closed-form u* is not the optimum of the model as stated.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript formulates stubbornness as a continuous control u(s) in a stochastic goal-dynamics equation (Eq. (1)), with a payoff (48) depending on injury risk, assist rate, pass accuracy, dribbling ability, and a quadratic cost of stubbornness. It introduces the label BPPSDE for this SDE, proves existence and uniqueness for a related backward SPDE (Proposition 6), then constructs a stochastic Lagrangian (49) and uses a Wick-rotated Schrödinger equation to derive a first-order condition (71) for optimal feedback stubbornness. The main advertised result is the explicit nested-root formula for u*(s) in Section 3 for a modified Ornstein-Uhlenbeck example (Eq. (73)). No empirical data, calibration, or comparison with match outcomes is provided.","tokens_in":32119,"tokens_out":8564,"duration_ms":96011,"significance":"If the derivation were correct, the paper would supply a closed-form feedback policy for a quantitative model of stubbornness and would demonstrate a path-integral route around the HJB equation. The manuscript is transparent about its functional-form choices, states explicit assumptions, and includes a formal existence-and-uniqueness argument, which is helpful. However, all parameters are free placeholders, the objective is chosen by the author rather than derived from match data, and the result is only a stationary condition for that invented objective; the significance is therefore methodological and conditional, not a demonstrated statement about winning soccer matches. There is no code, data, or external benchmark, so the paper cannot be validated against actual match outcomes.","major_comments":[{"comment":"The proof of Proposition 11 states f_xxu = 0 immediately before Eq. (70), but this does not hold for the paper's own example. Differentiating Eq. (79) once more in x and differentiating the u-linear term σ2 exp(σ2 x) u dλ in Eq. (74) gives f_xxu = 3c exp(-rs) u/(2(r-μbar)x^{5/2}) - (σ2)^3 exp(σ2 x) dλ, which is nonzero. Consequently Eq. (71), f_u (f_xx)^2 = 2 f_x f_xu, is not the first-order condition of the stated problem, and the explicit u*(s) in Section 3 is not supported.","section":"Section 3, Eqs. (70)-(71) and (74)"},{"comment":"After Eq. (80), the paper sets dλ(s) → 0 to remove terms A1, A3, and the u-dependent dλ terms, saying the effect is very small. This is not a harmless simplification: λ(s) is the multiplier that enforces the SDE constraint in the Lagrangian (49), and the removed terms include all dependence of f on the drift and diffusion of Eq. (73). The resulting Eq. (81) is the first-order condition for maximizing exp(-rs)π + Mbar without the dynamic constraint. No limiting argument or verification theorem is supplied to show that the dλ → 0 limit preserves the optimum of (72)-(73), so the final u*(s) solves an unconstrained problem, not the stated stochastic control problem.","section":"Section 3, Eqs. (80)-(81)"},{"comment":"After dividing Eq. (81) by exp(-2rs), the term -2 A2 c u(s) exp(-rs)/((r-μbar)x(s)^{3/2}) must become -2 A2 c u(s) exp(+rs)/((r-μbar)x(s)^{3/2}); the manuscript instead writes exp(-3rs). This exponent error is carried into the coefficient k4 and into the final nested-root formula, so the explicit expression for u*(s) is algebraically inconsistent with the preceding equation.","section":"Section 3, Eq. (82)"},{"comment":"Assumption 3 states c + σ1^2 + σ2^2 ≤ 2σ1σ2 ≤ C with c ∈ (0,1). The left inequality implies (σ1-σ2)^2 ≤ -c < 0, which is impossible for real σ1 and σ2. Since this assumption is used for the existence and uniqueness result in Proposition 6, the well-posedness framework of the model is inconsistent as stated.","section":"Section 2.2, Assumption 3"},{"comment":"Lemma 10 establishes only a necessary first-order condition; the paper never verifies that a solution of Eq. (71) is a global or local maximum of J. The second-derivative remark in Lemma 10 is not applied to Eq. (71) or to the nested-root formula, and no HJB or other verification argument appears. Therefore the central claim that u*(s) is optimal stubbornness is not established even under the paper's own objective.","section":"Section 3, Lemma 10 and final formula"}],"minor_comments":[{"comment":"Calling the ordinary SDE in Eq. (1) a backward parabolic partial stochastic differential equation is misleading; no actual SPDE is formulated or solved in the paper.","section":"Section 2.1, Eq. (2)"},{"comment":"The spike variation in Eq. (43) refers to optimal output share of the firm and market share, which are leftovers from a different model; these terms should be removed or adapted to the soccer setting.","section":"Section 2.3, Eq. (43)"},{"comment":"The Itô rules in the proof of Lemma 14 are misstated: ds^2 = 0, not ds, and the product rule should be ds dWs = 0 rather than dWs = 0.","section":"Appendix, Lemma 14"},{"comment":"The proof of Proposition 6 is largely an invocation of Du and Meng (2010); it would be clearer to state their theorem and verify their hypotheses directly rather than re-derive many estimates with frequent notational inconsistencies.","section":"Section 2.2, Proposition 6"},{"comment":"The constants θ, α_i, c, r, and μbar in Eq. (48) are introduced without calibration, dimensional consistency, or empirical motivation; at minimum their units and intended interpretation should be discussed.","section":"Section 3, Eq. (48)"},{"comment":"The paper repeatedly claims a feedback Nash equilibrium, but no game, set of players, strategy space, or equilibrium definition is formally given; the analysis is single-player optimal control.","section":"General"}],"recommendation":"reject","confidential_remarks":"The manuscript has multiple load-bearing technical errors: the f_xxu term is not zero for the paper's own example, the dλ(s) → 0 step removes the dynamic constraint, Assumption 3 is impossible as stated, and the algebra leading to the explicit formula contains exponent errors. Beyond correctness, the framing as a way to win a soccer match is not supported by any data or calibration; all parameters are free, and no comparison with actual match outcomes or existing soccer analytics is provided. I would not request a revision, because the errors are in the central derivation and would require rebuilding the model."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Reject this one. Proposition 11's Eq. (71) is derived by setting f_xxu = 0, but the paper's own f in Eq. (74) has f_xxu ≠ 0, so the explicit u*(s) in Section 3 is not the optimum of the stated problem.\n\nWhat the paper does well: the modeling idea is sensible—stubbornness as a continuous control in a dynamic game, with a path integral / BPPSDE setup, is a legitimate way to frame the question. The Lagrangian (49) correctly includes the SDE constraint, and the existence-uniqueness material, though standard, is not egregiously wrong.\n\nThe problems are in the derivation. Eq. (70) is the correct derivative of the Wick-rotated Schrödinger equation, but the paper then asserts f_xxu = 0. For the f in Eq. (74), f_xxu = -3c u exp(-rs)/(2(r-μ)x^{5/2}) - (σ₂)^2 exp(σ₂ x) dλ(s), which is not zero. So Eq. (71) does not follow. Then Section 3 assumes dλ(s) → 0, which removes the SDE constraint from the Lagrangian entirely; that turns the problem into an unconstrained optimization, not the constrained control problem of Eq. (47)-(1). On top of that, the algebra from Eq. (82) to the nested-root formula has an exponent error: after dividing by exp(-2rs), the A2 term should carry exp(+rs), not exp(-3rs). There is also no second-order condition or verification argument showing the root is a maximum, so even if the FOC were right, we wouldn't know it's a maximizer.\n\nThe paper has no data, no calibration, no external validation; the parameters are placeholders, and the payoff (48) is chosen so the first-order condition produces the formula. That circularity is real but secondary; the primary reason to reject is the internal inconsistency.\n\nThis might be useful as a classroom example of how path integral control can go wrong, but it is not a reliable result. I would not cite it, and I would not send it to peer review—desk reject.","headline":"Internal contradiction in the main derivation: Eq. (71) assumes f_xxu=0 when the paper's own f violates it, so the explicit u* is not the optimum of the stated problem.","tokens_in":32541,"tokens_out":4862,"would_cite":false,"duration_ms":48996,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["49N70","60H15","49K45","91A80"],"pacs":[],"model":"deepseek-v4-flash","headline":"A formula finds how stubborn a soccer player should be.","keywords":["stubbornness","soccer","stochastic differential game","backward parabolic partial stochastic differential equation","path integral control","feedback Nash equilibrium","Ornstein-Uhlenbeck process","sports analytics"],"falsifier":"Compute the same optimal control with the constraint term retained, either by solving the constrained Hamilton-Jacobi-Bellman equation or by numerically maximizing the expected payoff under the modified Ornstein-Uhlenbeck dynamics, and check whether the resulting policy equals the paper's nested-root $u^*(s)$. A mismatch at any parameter set would show that the $d\\lambda(s)\\to 0$ step changes the answer.","tokens_in":31463,"feed_emoji":"⚽","tokens_out":9499,"duration_ms":109826,"temperature":0.7,"pith_summary":"This paper tries to establish that stubbornness in soccer can be treated not as a fixed trait but as a real-time control variable with a unique optimal value. The author models goal-scoring probability as a backward parabolic partial stochastic differential equation, couples it to a payoff that includes injury risk, assist rate, passing accuracy, dribbling skill, and a terminal bonus, and then derives a Feynman-type path-integral condition for optimal feedback stubbornness. The central output is a first-order equation, $f_u(f_{xx})^2 = 2 f_x f_{xu}$, plus an explicit nested-root formula for a modified Ornstein-Uhlenbeck version of the goal dynamics. A sympathetic reader would care because a closed-form stubbornness policy would convert an intangible personality trait into a measurable input for coaching, analytics, and match prediction.","feed_headline":"Formula finds how stubborn a soccer player should be","feed_subtitle":"A path-integral model turns goal dynamics into a closed-form policy balancing injury risk, passing, dribbling, and assists.","key_machinery":"The central object is the augmented Lagrangian density $f(s,x,u)$ defined in Eq. (51): it combines the discounted payoff, a terminal bonus, an integrating-factor function $h(s,x)$, and Lagrange-multiplier terms from the BPPSDE constraint. The path-integral control machinery translates the stochastic control problem into a Wick-rotated Schr\\\"odinger-type equation for the transition density, and differentiating that equation with respect to $u$ yields the first-order identity $f_u(f_{xx})^2 = 2 f_x f_{xu}$, the backbone of Proposition 11. In the example, $h(s,x) = \\exp(\\sigma_2 x)$ is the chosen integrating factor, and the identity reduces to a quadratic in $z = u^2$, producing the nested-root formula for $u^*(s)$.","core_discovery":"The paper argues that in a stochastic model of goal-scoring dynamics, stubbornness has an optimal feedback form rather than a binary temperament. The defining result is Proposition 11: for the augmented Lagrangian density $f(s,x,u)$, the optimal feedback stubbornness $u^*(s,x)$ must satisfy $f_u(f_{xx})^2 = 2 f_x f_{xu}$. In the Section 3 example with dynamics $dx = (a\\sqrt{x}-\\sigma_2 x-u)\\,ds + (\\sigma_1-\\sigma_2 x)\\,dB$, this condition reduces to a quadratic in $z = u^2$, giving the explicit nested-root formula for $u^*(s)$; the nonnegative branch is selected. The claim is that this $u^*(s)$ maximizes the player's expected discounted payoff, which includes injury risk, assist rate, passing accuracy, dribbling skill, performance cost, and a terminal bonus.","pith_inferences":["The same first-order condition $f_u(f_{xx})^2 = 2 f_x f_{xu}$ is not soccer-specific: it follows from the path-integral differentiation step whenever a scalar control enters the drift of a controlled diffusion and the diffusion coefficient is control-independent, so the policy formula could be transplanted to other settings where persistence is a control variable.","Because the explicit formula drops $d\\lambda(s)$ before solving, a natural check is to compute the constrained optimum numerically for the same modified Ornstein-Uhlenbeck dynamics; the difference between the two policies would quantify how much the constraint matters.","If fitted to tracking data, the model parameters $\\theta, \\alpha_i, c, a, \\sigma_1, \\sigma_2, r, \\bar{\\mu}$ could turn $u^*(s)$ into a testable prediction of when a player should abandon a rehearsed plan, and observed deviations from team instructions could be scored against this benchmark.","The terminal-bonus structure suggests a multi-stage extension: replacing $\\sqrt{x(t)}$ with a win/loss/draw utility that depends on the final score line would change the boundary condition of the Wick-rotated Schr\\\"odinger equation and hence the formula."],"forward_implications":["In the worked example, optimal stubbornness is explicit: for fixed parameters and current goal-scoring probability $x(s)$, the policy $u^*(s)$ is given by the nested-root formula, so it can be recomputed as a match evolves.","For this class of stochastic control problems, the path-integral route replaces the Hamilton-Jacobi-Bellman equation with the local condition $f_u(f_{xx})^2 = 2 f_x f_{xu}$, which requires only derivatives of the augmented density $f$.","Stubbornness becomes a continuous control in $[0,1]$ rather than a binary attack/defense choice, with $u=0$ representing full adherence to the coach's plan and higher $u$ representing independent decision-making.","The terminal bonus $M(x(t))$ enters $f$ as a constant in the worked example and therefore does not affect the first-order condition or the resulting policy.","The existence and uniqueness of the BPPSDE solution (Proposition 6) is the supporting guarantee that the goal-dynamics constraint is well-posed before the optimization is performed."],"supporting_citations":[{"why":"Supplies the stochastic Lagrangian form (Eq. 49) on which the action and the first-order condition are built.","marker":"Ewald and Nolan (2024)"},{"why":"Provides the weak-solution theory for super-parabolic backward stochastic PDEs used in Proposition 6 to guarantee unique solutions.","marker":"Du and Meng (2010)"},{"why":"Provides the spike-variation strategy and maximum-principle framework behind the penalized payoff and the adjoint equation (Theorem 9).","marker":"Du and Meng (2013)"},{"why":"Introduces the path-integral action formalism that motivates the Euclidean path-integral control derivation.","marker":"Feynman (1948)"},{"why":"Supplies the path-integral representation of stochastic differential equations used in the appendix to rewrite the SDE as an action integral.","marker":"Chow and Buice (2015)"},{"why":"Supplies Itô's formula and the stochastic calculus rules used in the derivations of Lemma 1 and Proposition 11.","marker":"Oksendal (2013)"},{"why":"Supplies the cooperative stochastic differential game payoff structure, including the terminal bonus, used in the objective function.","marker":"Yeung and Petrosyan (2006)"},{"why":"Provides the Euclidean path-integral optimization of a dynamic profit function that the paper extends to stubbornness.","marker":"Pramanik and Polansky (2024b)"}],"fun_headline_variants":["Optimal stubbornness derived from stochastic goal dynamics","Formula finds optimal stubbornness for soccer players","Path integral control yields closed-form stubbornness policy","Optimal stubbornness is a quadratic equation not a trait","Nested-root formula picks a soccer player's stubbornness"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"In the worked example, the Lagrange-multiplier variation $d\\lambda(s)$ is assumed to be very small and then dropped, so the constraint linking the player's goal dynamics to the stochastic equation is removed from the first-order condition; if dropping it changes the optimum, the closed-form formula solves a different, unconstrained problem.","fun_headline_variants_meta":{"raw":{"variants":["Optimal stubbornness derived from stochastic goal dynamics","Formula finds optimal stubbornness for soccer players","Path integral control yields closed-form stubbornness policy","Optimal stubbornness is a quadratic equation not a trait","Nested-root formula picks a soccer player's stubbornness"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000617,"raw_usage":{"total_tokens":2830,"prompt_tokens":879,"completion_tokens":1951,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":495,"completion_tokens_details":{"reasoning_tokens":1891}},"tokens_in":495,"tokens_out":1951,"duration_ms":285694,"temperature":1.0,"reasoning_tokens":1891,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T00:52:24.618130+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the same optimal control with the constraint term retained, either by solving the constrained Hamilton-Jacobi-Bellman equation or by numerically maximizing the expected payoff under the modified Ornstein-Uhlenbeck dynamics, and check whether the resulting policy equals the paper's nested-root $u^*(s)$. A mismatch at any parameter set would show that the $d\\lambda(s)\\to 0$ step changes the answer.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the stochastic Lagrangian form (Eq. 49) on which the action and the first-order condition are built."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the path-integral action formalism that motivates the Euclidean path-integral control derivation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the path-integral representation of stochastic differential equations used in the appendix to rewrite the SDE as an action integral."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the cooperative stochastic differential game payoff structure, including the terminal bonus, used in the objective function."}],"review_version":1}