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REVIEW 5 major objections 6 minor 2 cited by

An optimal level of Stubbornness to win a soccer match

T0 review · 5 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A formula finds how stubborn a soccer player should be.

desk verdict 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. read the letter →

arxiv 2501.18050 v1 pith:RZWI5FPN submitted 2025-01-29 math.OC math.PR

Paramahansa Pramanik This is my paper · ORCID
classification math.OCmath.PR MSC 49N7060H1549K4591A80
keywords stubbornnesssoccerstochasticdifferentialgamebackwardparabolicpartialequationpathintegralcontrolfeedbackNashequilibriumOrnstein-Uhlenbeckprocesssportsanalytics
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 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.

What carries the argument

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 $u^*(s)$.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 6 minor

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.

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 (5)
  1. [Section 3, Eqs. (70)-(71) and (74)] 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.
  2. [Section 3, Eqs. (80)-(81)] 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.
  3. [Section 3, Eq. (82)] 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.
  4. [Section 2.2, Assumption 3] 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.
  5. [Section 3, Lemma 10 and final formula] 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.
minor comments (6)
  1. [Section 2.1, Eq. (2)] 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.
  2. [Section 2.3, Eq. (43)] 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.
  3. [Appendix, Lemma 14] 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.
  4. [Section 2.2, Proposition 6] 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.
  5. [Section 3, Eq. (48)] 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.
  6. [General] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is a formal optimal-control calculation on a self-contained payoff; no fitted data or externally predicted quantity is recycled as an input.

full rationale

The central result u*(s,x) is obtained by differentiating the path-integral integrand f and setting the first-order condition in Eq. (71) to zero. This is the standard stationarity condition for maximizing the model's own expected payoff J in Eq. (47), so the appearance of the chosen payoff in the final formula reflects the definition of the optimization problem, not a circular reuse of an output as an input. There is no data fitting: the constants θ, α_i, c, ω, σ_1, σ_2, a, r, and \bar μ are free placeholders, and no empirically measured quantity is fed back into the derivation. The many self-citations in the introduction and existence/uniqueness sections are not load-bearing for the main optimality calculation, which relies on Itô calculus, Gaussian integration, and the cited external frameworks of Ewald and Nolan (2024) and Du and Meng (2010, 2013); those external results are not replaced by the paper's own assertions. The unsupported step 'Assume the effect of dλ(s) is very small' (Section 3, before Eq. 81) and the setting of f_xxu=0 in Proposition 11 are serious mathematical gaps that remove the SDE constraint and invalidate the explicit formula as a solution of the stated constrained problem; they constitute correctness risks, not circularity. Similarly, the choice h(s,x)=exp(σ2 x) is an arbitrary ansatz, but it is not smuggled in via a self-citation and does not make the argument circular. The paper is therefore best assessed as non-circular, while remaining doubtful on mathematical grounds at those junctures.

Assumptions & free parameters 10 free parameters · 8 assumptions · 2 invented entities

The derivation rests on a hand-selected payoff and SDE, on standard Ito calculus, and on three ad hoc simplifications: f_xxu=0, dλ(s)→0, and du(s)=0. All constants are free placeholders and no empirical anchor exists.

free parameters (10)
  • theta (injury risk)
    Positive constant in payoff Eq. (48); no data or estimation procedure is given.
  • alpha_1 (assist rate)
    Constant in payoff Eq. (48); unspecified and never calibrated.
  • alpha_2 (pass accuracy)
    Constant in payoff Eq. (48); unspecified and never calibrated.
  • alpha_3 (dribbling ability)
    Constant in payoff Eq. (48); unspecified and never calibrated.
  • c (marginal cost of stubbornness)
    Positive constant multiplying u(s)^2 in payoff Eq. (48); controls the curvature of the objective; unestimated.
  • r (discount rate)
    Discount factor in J(u) in Eq. (47); unestimated.
  • mu_bar (average drift)
    Used in the denominator of the payoff in Eq. (48); no value or estimation method is provided.
  • a (drift coefficient)
    Drift parameter in the modified Ornstein-Uhlenbeck goal dynamics Eq. (73); unestimated.
  • sigma_1, sigma_2 (volatilities)
    Diffusion coefficients in Eq. (73); unspecified and never calibrated.
  • omega (terminal bonus weight)
    Weight on terminal sqrt(x(t)) in payoff Eq. (72); unestimated and later absorbed into the constant Mbar.
assumptions (8)
  • standard math Standard Ito calculus and martingale inequalities are valid.
    Ito's formula, BDG inequality, and Gronwall-type arguments are invoked throughout Lemma 1, Proposition 6, and the Appendix.
  • domain assumption Existence and uniqueness of solutions to the superparabolic backward stochastic partial differential equation hold.
    Proposition 6 imports existence and uniqueness results from Du and Meng (2010); the paper's own proof is a Galerkin sketch relying on those external results.
  • ad hoc to paper The path integral representation and Wick-rotated Schrodinger equation are valid for this control problem.
    The bridge from the SDE to Eq. (69) via Gaussian integrals is asserted without proving integrability, compactness, or boundary conditions; Appendix Proposition 12 introduces ad hoc absolute values.
  • domain assumption The diffusion coefficient is independent of the control u (Assumption 7).
    Assumption 7 in Section 2.2 simplifies the optimality condition; if sigma depends on u, Eq. (71) would change.
  • ad hoc to paper The Lagrange multiplier variation dλ(s) can be neglected.
    Section 3 after Eq. (80) assumes dλ(s) is very small; this removes the constraint terms and is load-bearing for the explicit solution.
  • ad hoc to paper The third mixed derivative f_xxu is zero in Eq. (70).
    Needed to reduce Eq. (70) to Eq. (71), but the paper's own f in Eq. (74) has f_xxu nonzero.
  • ad hoc to paper The payoff functional forms in Eq. (48) are the correct model for a soccer player's objective.
    The linear benefit, quadratic cost divided by sqrt(x), and terminal sqrt(x) bonus are chosen by hand; no data or behavioral derivation is given.
  • ad hoc to paper Stubbornness is neutral over time, so du(s)=0.
    Lemma 14 in the Appendix assumes du(s)=0 to apply Ito's formula, which undercuts the dynamic nature of the control variable.
invented entities (2)
  • Stubbornness control u(s)
    purpose: A continuous control in [0,1] representing a player's tendency to ignore team strategy; the object whose optimal value is derived.
    No operational definition, measurement scale, or dataset ties u to observable player behavior; the paper only defines it verbally and symbolically.
  • BPPSDE label for the goal dynamics equation
    purpose: The paper labels the finite-dimensional SDE in Eq. (1) as a backward parabolic partial stochastic differential equation and builds its theory on this object.
    Equation (1) is an ordinary SDE for x(s), not a stochastic PDE; the label adds no independent mathematical content and obscures the actual structure.

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Pith. "Pith review of An optimal level of Stubbornness to win a soccer match." pith.science (2026). https://pith.science/paper/RZWI5FPN

@misc{pith2026250118050,
  author       = {Pith},
  title        = {Pith review of: An optimal level of Stubbornness to win a soccer match},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RZWI5FPN}},
  note         = {Machine review of arXiv:2501.18050}
}
read the original abstract

This study conceptualizes stubbornness as an optimal feedback Nash equilibrium within a dynamic setting. To assess a soccer player's performance, we analyze a payoff function that incorporates key factors such as injury risk, assist rate, passing accuracy, and dribbling ability. The evolution of goal-related dynamics is represented through a backward parabolic partial stochastic differential equation (BPPSDE), chosen for its theoretical connection to the Feynman-Kac formula, which links stochastic differential equations (SDEs) to partial differential equations (PDEs). This relationship allows stochastic problems to be reformulated as PDEs, facilitating both analytical and numerical solutions for complex systems. We construct a stochastic Lagrangian and utilize a path integral control framework to derive an optimal measure of stubbornness. Furthermore, we introduce a modified Ornstein-Uhlenbeck BPPSDE to obtain an explicit solution for a player's optimal level of stubbornness.

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    2006

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.