  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.
