arXiv subjects

Nathan Sandholtz

Publications and source records attributed to Nathan Sandholtz.

7 recordsLinked to original sources

arXiv2026

Stated, Realized, and Optimal Aiming Strategies for the Tennis Serve: Experimental Evidence from Collegiate Athletes

We study where, within a chosen Wide or T region, a tennis player should aim a serve. Aiming near a service-box boundary makes a serve harder to return but raises the probability of a fault, creating a trade-off that depends on the player's execution error. Ordinary match data provide limited information about execution error because players' intended targets are unobserved. We therefore conducted an experiment with members of the Brigham Young University men's tennis team. Each player identified targets they believed to be optimal, which we marked on the court. Players then served repeatedly toward each target while we recorded precise bounce locations. From these data, we estimate player-specific serve distributions while accounting for serves censored by net contact, then combine them with an estimated point-win reward surface in a Bayesian, two-period Markov decision process with a continuous spatial action space. We find that the estimated optimal second-serve target is more conservative than its first-serve counterpart in every player-region combination, consistent with prior theory. Also, while players' stated targets were generally more aggressive than optimal, their realized serve centers were more conservative than those targets and markedly closer to the estimated optima.

stat.AP↗

arXiv2026

Uncertainty Quantification in Data-Driven Inverse Optimization via Bayesian Inference

Inverse optimization (IO) is used to estimate unknown parameters of an optimization model from observed decisions. In the data-driven context, the estimated parameters are inherently uncertain, yet quantifying this uncertainty has received limited attention in the literature, where existing methods return a point estimate. In this paper, we propose a hierarchical Bayesian framework for parameter uncertainty quantification in data-driven inverse optimization. Considering two data-generating processes, we develop two Markov chain Monte Carlo algorithms to estimate the posterior distribution of the unknown parameter vector, which is used to construct credible regions. We establish posterior consistency under standard identifiability conditions. Numerical experiments demonstrate near-nominal empirical coverage of the credible regions and show that the regions shrink as the number of observed decisions increases.

math.OC↗

arXiv2023

Learning Risk Preferences in Markov Decision Processes: an Application to the Fourth Down Decision in the National Football League

For decades, National Football League (NFL) coaches' observed fourth down decisions have been largely inconsistent with prescriptions based on statistical models. In this paper, we develop a framework to explain this discrepancy using an inverse optimization approach. We model the fourth down decision and the subsequent sequence of plays in a game as a Markov decision process (MDP), the dynamics of which we estimate from NFL play-by-play data from the 2014 through 2022 seasons. We assume that coaches' observed decisions are optimal but that the risk preferences governing their decisions are unknown. This yields an inverse decision problem for which the optimality criterion, or risk measure, of the MDP is the estimand. Using the quantile function to parameterize risk, we estimate which quantile-optimal policy yields the coaches' observed decisions as minimally suboptimal. In general, we find that coaches' fourth-down behavior is consistent with optimizing low quantiles of the next-state value distribution, which corresponds to conservative risk preferences. We also find that coaches exhibit higher risk tolerances when making decisions in the opponent's half of the field as opposed to their own half, and that league average fourth down risk tolerances have increased over time.

stat.AP↗

arXiv2023

Miss It Like Messi: Extracting Value from Off-Target Shots in Soccer

Measuring soccer shooting skill is a challenging analytics problem due to the scarcity and highly contextual nature of scoring events. The introduction of more advanced data surrounding soccer shots has given rise to model-based metrics which better cope with these challenges. Specifically, metrics such as expected goals added, goals above expectation, and post-shot expected goals all use advanced data to offer an improvement over the classical conversion rate. However, all metrics developed to date assign a value of zero to off-target shots, which account for almost two-thirds of all shots, since these shots have no probability of scoring. We posit that there is non-negligible shooting skill signal contained in the trajectories of off-target shots and propose two shooting skill metrics that incorporate the signal contained in off-target shots. Specifically, we develop a player-specific generative model for shot trajectories based on a mixture of truncated bivariate Gaussian distributions. We use this generative model to compute metrics that allow us to attach non-zero value to off-target shots. We demonstrate that our proposed metrics are more stable than current state-of-the-art metrics and have increased predictive power.

stat.AP↗

arXiv2022

Inverse Bayesian Optimization: Learning Human Acquisition Functions in an Exploration vs Exploitation Search Task

This paper introduces a probabilistic framework to estimate parameters of an acquisition function given observed human behavior that can be modeled as a collection of sample paths from a Bayesian optimization procedure. The methodology involves defining a likelihood on observed human behavior from an optimization task, where the likelihood is parameterized by a Bayesian optimization subroutine governed by an unknown acquisition function. This structure enables us to make inference on a subject's acquisition function while allowing their behavior to deviate around the solution to the Bayesian optimization subroutine. To test our methods, we designed a sequential optimization task which forced subjects to balance exploration and exploitation in search of an invisible target location. Applying our proposed methods to the resulting data, we find that many subjects tend to exhibit exploration preferences beyond that of standard acquisition functions to capture. Guided by the model discrepancies, we augment the candidate acquisition functions to yield a superior fit to the human behavior in this task.

cs.HC↗

arXiv2020

Measuring Spatial Allocative Efficiency in Basketball

Every shot in basketball has an opportunity cost; one player's shot eliminates all potential opportunities from their teammates for that play. For this reason, player-shot efficiency should ultimately be considered relative to the lineup. This aspect of efficiency---the optimal way to allocate shots within a lineup---is the focus of our paper. Allocative efficiency should be considered in a spatial context since the distribution of shot attempts within a lineup is highly dependent on court location. We propose a new metric for spatial allocative efficiency by comparing a player's field goal percentage (FG%) to their field goal attempt (FGA) rate in context of both their four teammates on the court and the spatial distribution of their shots. Leveraging publicly available data provided by the National Basketball Association (NBA), we estimate player FG% at every location in the offensive half court using a Bayesian hierarchical model. Then, by ordering a lineup's estimated FG%s and pairing these rankings with the lineup's empirical FGA rate rankings, we detect areas where the lineup exhibits inefficient shot allocation. Lastly, we analyze the impact that sub-optimal shot allocation has on a team's overall offensive potential, demonstrating that inefficient shot allocation correlates with reduced scoring.

stat.AP↗

arXiv2020

Markov Decision Processes with Dynamic Transition Probabilities: An Analysis of Shooting Strategies in Basketball

In this paper we model basketball plays as episodes from team-specific non-stationary Markov decision processes (MDPs) with shot clock dependent transition probabilities. Bayesian hierarchical models are employed in the modeling and parametrization of the transition probabilities to borrow strength across players and through time. To enable computational feasibility, we combine lineup-specific MDPs into team-average MDPs using a novel transition weighting scheme. Specifically, we derive the dynamics of the team-average process such that the expected transition count for an arbitrary state-pair is equal to the weighted sum of the expected counts of the separate lineup-specific MDPs. We then utilize these non-stationary MDPs in the creation of a basketball play simulator with uncertainty propagated via posterior samples of the model components. After calibration, we simulate seasons both on-policy and under altered policies and explore the net changes in efficiency and production under the alternate policies. Additionally, we discuss the game-theoretic ramifications of testing alternative decision policies.

stat.AP↗