Economic resources, sporting efficiency and uncertainty: a stochastic optimization model of football performance in Africa
We propose a stochastic optimization framework in which football performance is modeled as the result of an efficiency-driven production process under uncertainty. The model incorporates economic inputs, a latent efficiency variable evolving according to a stochastic differential equation, and a control variable representing investment in sports development. We derive the associated discounted Hamilton--Jacobi--Bellman (HJB) equation, characterize the optimal investment policy, prove a verification theorem under explicit growth conditions, and show that a bounded investment budget is required for the control problem to be well posed. We relate our efficiency measure to the stochastic frontier and data envelopment analysis (SFA/DEA) traditions used elsewhere in sports economics, and clarify what a dynamic control formulation adds relative to those static approaches. The theoretical results are illustrated with a numerical solution of the HJB equation -- validated by a grid-refinement convergence study -- and Monte Carlo simulations. A cross-sectional analysis of the 24 AFCON 2025 national teams, based on GDP, population, and FIFA ranking data, estimates country-specific conditional log-efficiency residuals using heteroskedasticity-robust inference. We show that these residuals are mechanically and strongly correlated with FIFA points by construction, quantify this dependence, and interpret the results accordingly. Teams such as Senegal and Morocco consistently outperform their GDP-implied potential, whereas others, including South Africa and Tanzania, tend to underperform. While these patterns are broadly consistent with observed AFCON outcomes, the estimated residual also captures unobserved factors, such as institutional quality, diaspora talent, and demographic structure, that are not included in the model.