Volume 13, Issue 3

Since the advent of high-resolution pitch tracking data (PITCHf/x), many in the sabermetrics community have attempted to quantify a Major League Baseball catcher’s ability to “frame” a pitch (i.e. increase the chance that a pitch is a called as a strike). Especially in the last 3 years, there has been an explosion of interest in the “art of pitch framing” in the popular press as well as signs that teams are considering framing when making roster decisions. We introduce a Bayesian hierarchical model to estimate each umpire’s probability of calling a strike, adjusting for the pitch participants, pitch location, and contextual information like the count. Using our model, we can estimate each catcher’s effect on an umpire’s chance of calling a strike. We are then able translate these estimated effects into average runs saved across a season. We also introduce a new metric, analogous to Jensen, Shirley, and Wyner’s Spatially Aggregate Fielding Evaluation metric, which provides a more honest assessment of the impact of framing.

Most professional European football clubs are well-structured businesses. Therefore, the financial performance of investments in players becomes crucial. In this paper, after the problem is discussed and formalized, an optimization model with the objective of maximizing the expected value of the team is presented. The model ensures that the team has the required mix of skills, that competition regulations are met, and that budget limits are respected. The model explicitly takes into account the uncertainty in the career development of football players. A case study based on the English Premier League is presented. Our results show that the model has significant potential to improve current decisions ensuring a steady growth of the value of the team. The team value growth reported is particularly driven by investments in young prospects.