An optimal level of Stubbornness to win a soccer match
Paramahansa Pramanik
TL;DR
The paper investigates how to determine the optimal level of stubbornness in soccer using a stochastic dynamic-game framework. It formulates goal dynamics as a backward parabolic SPDE (BPPSDE) linked to Feynman-type path-integral control and derives an explicit feedback Nash equilibrium for the stubbornness control $u^*(s,x)$ under a mean-field setting. The payoff combines injury risk, assists, passing accuracy, and dribbling, with a terminal bonus, and the authors establish existence and uniqueness of BPPSDE solutions, plus a tractable path-integral approach to compute the optimal stubbornness via a Wick-rotated Schrödinger-type equation and a first-order optimality condition $f_u (f_{xx})^2=2 f_x f_{xu}$. This framework offers a principled alternative to HJB for dynamic sports analytics and could extend to other score-dynamics domains, with practical implications for modeling player decision-making under strategic uncertainty.
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.