Volume 21, Issue 2

The Hardy distribution, derived by van der Ven (2012. The Hardy distribution for golf hole scores. Math. Gaz. 96: 428–438) and named after an idea by Hardy (1945. A mathematical theorem about golf. Math. Gaz . 29: 226–227), is a discrete probability distribution for modeling golf hole scores. According to the Hardy distribution, a golfer’s score on a hole is determined by the par of the hole, the golfer’s probability of hitting a good shot, and the golfer’s probability of hitting a bad shot. To fit a Hardy distribution to golf scores on a hole, an analyst needs only the scores and a value for the par of the hole. We present four different hierarchical modeling strategies to jointly model the scores of multiple golfers on multiple holes during one golf tournament round using Hardy distributions. We then apply our new modeling strategies to golf hole scores from two very different golfer populations: male professional golfers on the PGA Tour and female high school golfers from the state of Iowa, USA. Probabilities of good and bad shots vary across holes for both the male professional golfers on the PGA Tour and the female high school golfers. We find little variation among the male professional golfers but substantial variation among the female high school golfers.

The progression of a single point in volleyball starts with a serve and then alternates between teams, each team allowed up to three contacts with the ball. Using charted data from the 2022 NCAA Division I women’s volleyball season (4,147 matches, 600,000+ points, more than 5 million recorded contacts), we model the progression of a point as a Markov chain with the state space defined by the sequence of contacts in the current possession. We estimate the probability of each team winning the point, which changes on each contact. We attribute changes in point probability to the player(s) responsible for each contact, facilitating measurement of performance on the same point scale for different skills. Traditional volleyball statistics do not allow apples-to-apples comparisons across skills, and they do not measure the impact of the performances on team success. For adversarial contact groups (serve/reception and set/attack/block/dig), we estimate a hierarchical linear model for the outcome, with random effects for the players involved; and we adjust performance for strength of schedule not only on the conference/team level but on the individual player level. We can use the results to answer practical questions for volleyball coaches.

The masters movement in swimming and running has exploded, resulting in an abundance of data to study the impact of age on performance. Analyzing data from masters events in running and swimming for athletes aged 35 to 80, we model the percentage increase in event time (or decrease in performance) by age and sex via stacked models that combine polynomial models, neural networks, and natural splines. To answer fundamental questions on the nature of performance decline for competitive athletes, we bootstrap the procedure to obtain confidence intervals. Cross-sectional masters data from the past decade are used to construct models, and the model predictions are compared to the trajectory of current world records by age and to estimates of decline using longitudinal data. Furthermore, the study explores the impact of constituent year, birth cohort, and participation effects, emphasizing the challenges in distinguishing age-related decline from factors like evolving training practices and varied participation rates. Our results give evidence that men generally decline more slowly than women, performance declines more rapidly for endurance events, athletes who participate more frequently decline more slowly than others, and masters level runners decline at rates roughly equivalent to world record holders.

This work develops a family of solutions related to Shin’s model in producing accurate probability forecasts. More precisely, closed-form solutions are derived based on an analytical approach to known solutions and evaluated experimentally for sports betting data sets.

Penalty kicks are critical to game outcomes in soccer. The typical quantitative strategy is as follows. First, model such an interaction as a two or three strategy game between the kicker and the goalkeeper. Second, find the mixed-strategy Nash equilibrium (MSNE) to determine the “optimal” probabilities of each player choosing to kick or dive to either side or to the center of the goal. While this is the usual path to a solution, it is also fraught with many assumptions due to the nature of penalty kick data as well as due to implicit assumptions within the game theory model. In this paper, we introduce an alternative set of strategies, known as adversarial risk analysis (ARA), to determine optimal decisions for the penalty kick game. ARA, which is grounded in the principles of Bayesian decision analysis, allows the decision maker to avoid many of the assumptions and limitations encountered when using the game theory approach. By examining 2018–2019 Major League Soccer (MLS) penalty kicks on an individual basis, we show that our most accurate ARA model predicts the correct goalkeeper decision 65 % of the time, while the game-theoretic model predicts correctly 51 % of the time, a statistically significant difference.