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Real-time Win Probability and Latent Player Ability via STATS X in Team Sports

Yasutaka Shimizu, Atsushi Yamanobe

TL;DR

This study proposes a statistically grounded framework for real-time win probability evaluation and player assessment in score-based team sports, based on minute-by-minute cumulative box-score data, and introduces a continuous dominance indicator that maps final scores to real values consistent with win/lose outcomes.

Abstract

This study proposes a statistically grounded framework for real-time win probability evaluation and player assessment in score-based team sports, based on minute-by-minute cumulative box-score data. We introduce a continuous dominance indicator (T-score) that maps final scores to real values consistent with win/lose outcomes, and formulate it as a time-evolving stochastic representation (T-process) driven by standardized cumulative statistics. This structure captures temporal game dynamics and enables sequential, analytically tractable updates of in-game win probability. Through this stochastic formulation, competitive advantage is decomposed into interpretable statistical components. Furthermore, we define a latent contribution index, STATS X, which quantifies a player's involvement in favorable dominance intervals identified by the T-process. This allows us to separate a team's baseline strength from game-specific performance fluctuations and provides a coherent, structural evaluation framework for both teams and players. While we do not implement AI methods in this paper, our framework is positioned as a foundational step toward hybrid integration with AI. By providing a structured time-series representation of dominance with an explicit probabilistic interpretation, the framework enables flexible learning mechanisms and incorporation of high-dimensional data, while preserving statistical coherence and interpretability. This work provides a basis for advancing AI-driven sports analytics.

Real-time Win Probability and Latent Player Ability via STATS X in Team Sports

TL;DR

This study proposes a statistically grounded framework for real-time win probability evaluation and player assessment in score-based team sports, based on minute-by-minute cumulative box-score data, and introduces a continuous dominance indicator that maps final scores to real values consistent with win/lose outcomes.

Abstract

This study proposes a statistically grounded framework for real-time win probability evaluation and player assessment in score-based team sports, based on minute-by-minute cumulative box-score data. We introduce a continuous dominance indicator (T-score) that maps final scores to real values consistent with win/lose outcomes, and formulate it as a time-evolving stochastic representation (T-process) driven by standardized cumulative statistics. This structure captures temporal game dynamics and enables sequential, analytically tractable updates of in-game win probability. Through this stochastic formulation, competitive advantage is decomposed into interpretable statistical components. Furthermore, we define a latent contribution index, STATS X, which quantifies a player's involvement in favorable dominance intervals identified by the T-process. This allows us to separate a team's baseline strength from game-specific performance fluctuations and provides a coherent, structural evaluation framework for both teams and players. While we do not implement AI methods in this paper, our framework is positioned as a foundational step toward hybrid integration with AI. By providing a structured time-series representation of dominance with an explicit probabilistic interpretation, the framework enables flexible learning mechanisms and incorporation of high-dimensional data, while preserving statistical coherence and interpretability. This work provides a basis for advancing AI-driven sports analytics.
Paper Structure (38 sections, 1 theorem, 86 equations, 4 figures, 9 tables)

This paper contains 38 sections, 1 theorem, 86 equations, 4 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Team Performance Index: T-score
  3. Evaluation of Outcome Superiority: T-score Function
  4. Concrete Examples of T-score Functions
  5. A General Nonlinear Model for the T-score and Statistical Strength Evaluation
  6. Team-wise Modeling of the T-score
  7. Player-wise Modeling of the T-score
  8. Dynamic Modeling of the T-score
  9. On the “Standardization” of Cumulative STATS
  10. Dynamic Modeling of the T-score: T-process
  11. A Modification of the T-process: modified T-process
  12. Future T-scores: predicted T-process
  13. Computation of Real-Time Win Probability
  14. Evaluation of Players’ Latent Ability: STATS X
  15. Real Data Analysis: Basketball
  16. ...and 23 more sections

Key Result

Theorem 5.1

Let $a(t)$ and $b(t)$ denote the points scored and conceded at time $t\in[0,1]$, respectively. Assume the following: Then the win probability is expressed as where $c$ is the draw benchmark value, $\tau^2 = \sum_{i=1}^d \alpha_i^2$, and $\Phi$ denotes the cumulative distribution function of the standard normal distribution.

Figures (4)

  • Figure 1: Deviation of the corrected T-score from the baseline value $1$
  • Figure 2: Scatter plot of $TFS$ and win rate ($\rho=0.989$)
  • Figure 3: Trajectories of ${}_mT$-process and $PW_t$ (vs. Ryukyu Golden Kings)
  • Figure 4: Transition of the ${}_mT$-process and $PW_t$ (vs Alvark Tokyo)

Theorems & Definitions (33)

  • Definition 2.1
  • Remark 2.1
  • Example 2.1: Simple score difference and ratio type
  • Example 2.2: Symmetric correction of score ratio
  • Example 2.3: Log-ratio type
  • Example 2.4: Relative difference type
  • Example 2.5: Normalized type
  • Remark 3.1
  • Definition 3.1
  • Definition 3.2
  • ...and 23 more