Complex interactions between two opposing agents frequently occur in domains of machine learning, game theory, and other application domains. Quantitatively analyzing the strategies involved can provide an objective basis for decision-making. One such critical scenario is shot-taking in football, where decisions, such as whether the attacker should shoot or pass the ball and whether the defender should attempt to block the shot, play a crucial role in the outcome of the game. However, there are currently no effective data-driven and/or theory-based approaches to analyzing such situations. To address this issue, we proposed a novel framework to analyze such scenarios based on game theory, where we estimate the expected payoff with machine learning (ML) models, and additional features for ML models were extracted with a theory-based shot block model. Conventionally, successes or failures (1 or 0) are used as payoffs, while a success shot (goal) is extremely rare in football. Therefore, we proposed the Expected Probability of Shot On Target (xSOT) metric to evaluate players' actions even if the shot results in no goal; this allows for effective differentiation and comparison between different shots and even enables counterfactual shot situation analysis. In our experiments, we have validated the framework by comparing it with baseline and ablated models. Furthermore, we have observed a high correlation between the xSOT and existing metrics. This alignment of information suggests that xSOT provides valuable insights. Lastly, as an illustration, we studied optimal strategies in the World Cup 2022 and analyzed a shot situation in EURO 2020.

翻译：两个对立智能体之间的复杂交互在机器学习、博弈论及其他应用领域中频繁出现。定量分析其中涉及的策略可为决策提供客观依据。在足球射门这一关键场景中，攻击者应选择射门还是传球、防守者应尝试封堵射门还是放弃封堵等决策，对比赛结果具有决定性影响。然而，目前尚缺乏有效的数据驱动和/或基于理论的方法来分析此类情境。针对这一问题，我们提出了一种基于博弈论的新框架，通过机器学习模型估计期望收益，并利用基于理论的射门封堵模型提取机器学习模型的附加特征。传统上，成功或失败（1或0）被用作收益值，但足球中成功射门（进球）极为罕见。为此，我们提出了"射正预期概率"（xSOT）指标来评估球员动作，即使射门未造成进球也能进行评价——这有效区分并比较了不同射门行为，甚至支持反事实射门情境分析。实验中，我们通过与基线模型和消融模型的对比验证了该框架的有效性。此外，我们观察到xSOT与现有指标之间存在高度相关性，这种信息一致性表明xSOT能提供有价值的洞见。最后，作为应用示范，我们研究了2022年世界杯的最优策略，并分析了2020年欧洲杯中的一个射门情境。