Nathan Sandholtz

Modeling Human Decision-making in Spatial and Temporal Systems.

In this thesis, we analyze three separate applications of human decision-making in spatial and temporal environments. The first two projects are applications to basketball while the third project analyzes an experiment that tests how people balance exploration and exploitation when searching for the maximum of an unknown function.

The first project explores how efficiently players in a basketball lineup collectively allocate shots. We propose a new metric for 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. 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 suboptimal shot allocation has on a team’s overall offensive potential, finding that inefficient shot allocation correlates with reduced scoring.

In the second basketball application, we model basketball plays as episodes from team-specific nonstationary Markov decision processes (MDPs) with shot clock dependent transition probabilities. Bayesian hierarchical models are employed in the 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 nonstationary 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. We also discuss the game-theoretic ramifications of testing alternative decision policies.

For the final project, we take a different perspective on the behavior of the decision-makers. Broadly speaking, both basketball projects assume the agents (players) act suboptimally and the goal of the analyses is to evaluate how far their observed decisions are from optimal. By contrast, in the final project we assume that the agents’ actions are optimal, but that the criteria over which they optimize are unknown. The goal of the analysis is to make inference on these latent optimization criteria. We categorize this type of problem—where decisions are assumed to be optimal and the goal is to reverse engineer the parameters of the optimization such that the optimality assumption holds—an inverse decision problem. Some inverse decision problems have been extensively studied, particularly in the fields of operations research (inverse optimization) and machine learning (inverse reinforcement learning).

Our final project thus explores the inverse problem of Bayesian optimization. Specifically, we seek to estimate an agent’s latent acquisition function based on their observed trajectories. After introducing a probabilistic solution framework for the problem, we illustrate our method by analyzing human behavior from an experiment. The experiment was designed to force subjects to balance exploration and exploitation in search of a global optimum. We find that subjects exhibit a wide array of acquisition preferences but that some subjects do not map well to any of the candidate acquisition functions initially proposed. Guided by the model discrepancies, we augment the candidate acquisition functions to yield a superior fit for the subjects in this task.

Keywords: Basketball, Bayesian hierarchical model, Markov decision process, simulation, ranking and ordering, Bayesian optimization, inverse optimization.