## 1 Introduction  
In competitive sports, ranking systems and tournament structures play a critical role in ensuring the procedural legitimacy of outcomes. Whether used for playoff qualification, seed allocation, or athlete selection, rankings are expected to reflect actual team or player performance while minimizing the influence of random variation and structural imbalance. When a ranking system fails to align with observed performance, it can undermine the transparency, credibility, and predictability of the competition framework. The inaccurate evaluation of team strength distorts the interpretation of comparative performance and limits the utility of rankings as a basis for competition outcomes [1]. Particularly in team sports at the school level, the robustness and consistency of ranking models carry broader implications, influencing regional athletic development, resource allocation, and the design of sports governance policies [2, 3]. In most contemporary baseball tournaments, team rankings are typically determined by statistical indicators, such as the number of wins, win percentage (win rate), or run differential. These methods are widely adopted across various levels of competition due to their computational simplicity, interpretability, and institutional transparency [4]. Among them, the most commonly used approach is win percentage, which calculates a team's success rate based on the proportion of games won. This method is intuitive and operationally straightforward, offering a basic performance metric. However, it critically depends on a complete and balanced schedule. When the competition structure is highly incomplete or asymmetric, the win percentage fails to capture qualitative differences in the opponents faced and outcomes achieved [5]. From a representational standpoint, teams with stronger opponents may be penalized in the rankings, while those with weaker schedules may receive disproportionate advantages. Another traditional method is run differential, which quantifies overall team performance by subtracting runs allowed from runs scored across games [6]. Although this approach provides a finer-grained metric of offensive and defensive capability, it is highly sensitive to outliers, such as one-sided games or unpredictable results. In short seasons or elimination-based formats, these fluctuations may distort rankings and place resource-constrained teams, which often lack the opportunity to accumulate consistent long-term results, at a disadvantage. To address the limitations of purely statistical ranking systems, mathematical models such as the Colley and Massey methods have been proposed. These models incorporate opponent strength to mitigate schedule imbalance. The Colley matrix uses systems of linear equations based on pairwise outcomes to generate strength estimates and is noted for its transparency and objectivity [7]. The Massey method minimizes the squared error between observed and predicted score margins, aiming to produce rankings that reflect relative performance more equitably [8]. While both methods have demonstrated efficacy in professional or collegiate baseball leagues with long and relatively complete seasons, their application in high school tournaments or single-elimination settings remains limited due to sparse data and potential ranking distortions [9; 10; 11]. Although these traditional methods remain widely accepted in practice for their simplicity and intuitive logic, they often struggle to account for latent performance differentials and structural biases under incomplete schedules or highly uneven matchups. Thus, there is a growing need to develop or adapt ranking models that can generate more representative and robust outcomes under real-world tournament constraints. The empirical evaluation of such models in authentic competitive environments remains an important direction for further research. This challenge is especially acute in school-level competitions, where structural limitations are more severe.