经济资源、体育效率与不确定性：非洲足球表现的随机优化模型

中文摘要 AI 辅助

我们提出了一个随机优化框架，其中足球表现被建模为在不确定性下由效率驱动的生产过程的结果。该模型纳入了经济投入、一个根据随机微分方程演化的潜在效率变量，以及一个代表体育发展投资的控制变量。我们推导了相关的贴现Hamilton-Jacobi-Bellman (HJB)方程，刻画了最优投资策略，在显式增长条件下证明了验证定理，并表明需要有限的投资预算才能使控制问题适定。我们将我们的效率度量与体育经济学中其他地方使用的随机前沿和数据包络分析（SFA/DEA）传统联系起来，并阐明了动态控制公式相对于这些静态方法的补充之处。理论结果通过HJB方程的数值解（通过网格细化收敛研究验证）和蒙特卡洛模拟加以说明。基于GDP、人口和FIFA排名数据，对2025年非洲杯（AFCON）24支国家队进行的横截面分析，使用异方差稳健推断估计了国家特定的条件对数效率残差。我们表明，这些残差在构造上与FIFA积分机械地且强烈相关，量化了这种依赖性，并相应地解释了结果。塞内加尔和摩洛哥等球队持续超越其GDP隐含的潜力，而南非和坦桑尼亚等其他球队则往往表现不佳。虽然这些模式与观察到的非洲杯结果大致一致，但估计的残差也捕捉了模型中未包含的未观察因素，如制度质量、侨民人才和人口结构。

英文摘要

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.

发表机构

• University of Cheikh Anta Diop（达喀尔大学）
• Laboratory for Analysis, Research and Study of Development（发展分析、研究与学习实验室）
• Laboratory of Mathematics of Decision and Numerical Analysis（决策与数值分析数学实验室）

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