Monte Carlo Pass Search: Using Trajectory Generation for 3D Counterfactual Pass Evaluation in Football

Pass receiver selection is a fundamental task in football analytics, aiming to predict the intended receiver under a given game state. This task is challenging with event-centered freeze-frame observations, a broadcast-like setting that provides only partial and variable player visibility without complete trajectories or stable player identities. The model must therefore reason over anonymous visible candidates, opponent pressure, and recent context under partial observation. To address this setting, we propose a Hierarchical Possession-aware Graph Pointer Network (HPGPN), which formulates pass receiver selection as variable-size candidate prediction over visible teammates. HPGPN jointly models current player interactions, local event context, and possession-level temporal dynamics. It represents the current pass situation with a graph, incorporates fixed event context, and uses dynamic possession history to capture how the attacking sequence evolves. Candidate representations are refined hierarchically by integrating spatial, contextual, and historical evidence, and a glimpse pointer head scores the receiver candidates. Experiments on public football event and freeze-frame data show that HPGPN improves pass receiver selection performance. Ablation studies demonstrate the effectiveness of graph-based interaction modeling, fixed event context, and dual-branch dynamic possession-history modeling.

The process of decision-making in football is characterized by a complex interplay between spatial positioning, opponent pressure, and player intent. This work introduces a Graph Neural Network (GNN) framework designed to predict Receiver Selection, the optimal passing target, by modeling on-field interactions as dynamic graphs. Each player is represented as a node with positional and contextual features, while potential passing lines form weighted edges characterized by distance, angle, and pressure metrics. A Message-Passing Neural Network (MPNN) has been developed and trained using a combination of tracking data and event data from professional matches, synchronized through a robust pipeline based on an optimized version of the Needleman-Wunsch Algorithm. The model achieves competitive accuracy in identifying the actual chosen receiver and state-of-the-art accuracy within its top three suggestions. Our model further offers quantification of each option's likelihood, threat, and creativity, enabling performance analysts to evaluate over 1,000 passes in seconds.

Repeated-sampling evaluations increasingly extrapolate pass@k far beyond the number n of samples collected per problem. We show that, in the pooled/random-task conditional-Binomial model, fixed-n success counts identify only the n free moments of the latent per-task success distribution. Consequently, direct pass@k is identified for k <= n, but generic extrapolated pass@k, tail exponents, and tail constants are not identified for k > n, even with arbitrarily many exchangeable tasks at the same rollout budget. This is stronger than the observation that the usual estimator is undefined beyond n: it characterizes the information missing from the fixed-depth count-law experiment. We give exact count-law-preserving constructions with incompatible extrapolations, state the exceptional unique-extension case, and compute sharp population identified intervals through Hausdorff principal representations. On the public 10,000-rollout-per-problem release of Brown et al., counterfactual n = 16 evaluations leave failure at k = 1000 ambiguous by factors from 1.5 to over 2,600 across four MATH/GSM8K/CodeContests configurations. The calibration shows that intermediate-scale failure share alone does not determine width. Our result does not reject parametric inference-time scaling laws; it supplies the nonparametric baseline against which their assumptions can be evaluated. We give an exact, conservative one-coordinate finite-task confidence certificate and a reporting standard separating direct estimates, identified sets, and model-conditioned forecasts.