Modeling human decision-making in spatial and temporal systems by Nathan Gene Sandholtz M.Sc., Brigham Young University, 2016 B.Sc., Brigham Young University, 2012 Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in the Department of Statistics and Actuarial Science Faculty of Science ⃝c Nathan Gene Sandholtz 2020 SIMON FRASER UNIVERSITY Summer 2020 Copyright in this work rests with the author. Please ensure that any reproduction or re-use is done in accordance with the relevant national copyright legislation. Approval Name: Nathan Gene Sandholtz Degree: Doctor of Philosophy (Statistics) Title: Modeling human decision-making in spatial and temporal systems Examining Committee: Chair: Jean-François Bégin Assistant Professor Luke Bornn Senior Supervisor Associate Professor Derek Bingham Supervisor Professor Tim Swartz Internal Examiner Professor Department of Statistics and Actuarial Science Robert B. Gramacy External Examiner Professor Department of Statistics Virginia Polytechnic and State University Date Defended: August 20, 2020 ii Abstract The first project in this thesis explores how efficiently players in a basketball lineup col- lectively 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. Lever- aging 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 hierar- chical 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. In the second basketball application, we model basketball plays as episodes from team- specific nonstationary Markov decision processes (MDPs) with shot clock dependent tran- sition 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 us- ing a novel transition weighting scheme. 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. In the final project, we explore the inverse problem of Bayesian optimization. Specifically, we seek to estimate an agent’s latent acquisition function based on their observed search paths. 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 range of acquisition preferences; however, some subject’s behavior does not map well to any of the candidate acquisitions functions we consider. Guided by the model discrepancies, we augment the candidate acquisition functions to yield a superior fit to the human behavior in this task. iii Keywords: Basketball statistics, Bayesian hierarchical model, Bayesian optimization, In- verse optimization, Markov decision process, Rankings and orderings iv Dedication To the future students I’ll teach and mentor, and to my family—those who are already here and those who are yet to come. v Acknowledgements I owe thanks to many people, without whom this work would never have happened. First of all, thank you to the Brigham Young University statistics faculty who helped me prepare me for a PhD program. In particular, I’m grateful to Shane Reese and William Christensen for helping me discover the statistics program at Simon Fraser. I’d like to thank the statistics faculty here at Simon Fraser University. I’d particularly like to thank Tim Swartz, who has been very generous to me with his time and resources since the day I met him. Most of all, I’m indebted to my supervisors, Luke Bornn and Derek Bingham, for their examples, guidance, support, and mentorship. They were always actively engaged and willing to help at a moments notice. I’d particularly like to thank Luke for the rich opportunities he’s provided me. Thanks to him this program has been a unique and singular experience, and I’ve received uncommon opportunities that I never anticipated before I began the program. I’d like to thank my coauthors, Jacob Mortensen, Luke Bornn, Yohsuke Miyamoto, and Maurice Smith. Their contributions to the chapters in this thesis were invaluable. I’d also like to thank the referees and editors who reviewed Chapters 2 and 3 of this thesis. Their input strengthened each project immensely. I’m grateful to my brother Wayne, who was a constant source of support to me during this program. A huge thank you to my friend and fellow student Jacob Mortensen. This program would have been much more difficult and overwhelming without him. More than anyone else, thank you to my wife Christina. I couldn’t have done it without her support. vi Table of Contents Approval ii Abstract iii Dedication v Acknowledgements vi Table of Contents vii List of Tables x List of Figures xi 1 Introduction 1 1.1 Connecting ideas between projects ....................... 2 1.2 Background .................................... 4 1.2.1 Hierarchical modeling .......................... 4 1.2.2 Markov decision processes ........................ 5 1.2.3 Bayesian optimization .......................... 7 2 Measuring Spatial Allocative Efficiency in Basketball 9 2.1 Introduction .................................... 9 2.1.1 Related work ............................... 12 2.1.2 Data and code .............................. 12 2.2 Models ....................................... 13 2.2.1 Estimating FG% surfaces ........................ 13 2.2.2 Determining FGA rate surfaces ..................... 17 2.3 Allocative efficiency metrics ........................... 18 2.3.1 Spatial rankings within a lineup .................... 19 2.3.2 Lineup points lost ............................ 22 2.3.3 Player LPL contribution ......................... 26 2.3.4 Empirical implementation ........................ 27 vii 2.4 Optimality - discussion and implications .................... 29 2.4.1 Do lineups minimize LPL? ....................... 29 2.4.2 Does LPL relate to offensive production? . 31 2.4.3 How can LPL inform strategy? ..................... 34 2.4.4 Is minimizing LPL always optimal? ................... 35 2.5 Conclusion .................................... 36 3 Markov Decision Processes with Dynamic Transition Probabilities: An Analysis of Shooting Strategies in Basketball 37 3.1 Introduction .................................... 37 3.1.1 Related work and contributions ..................... 39 3.1.2 Description of data ............................ 40 3.1.3 Outline .................................. 40 3.2 Decision process framework ........................... 41 3.2.1 Markov decision processes ........................ 41 3.2.2 State and action space .......................... 42 3.2.3 Defining the average chain ........................ 43 3.2.4 Transition and policy tensors ...................... 46 3.3 Hierarchical modeling and inference ...................... 48 3.3.1 Shot policy model ............................ 48 3.3.2 Transition probability model ...................... 51 3.3.3 Transition model two-stage approximation . 52 3.3.4 Reward function ............................. 53 3.3.5 Inference and validation ......................... 55 3.3.6 Model fit ................................. 56 3.4 Simulating plays ................................. 58 3.4.1 Play simulation algorithm ........................ 58 3.4.2 Calibration ................................ 59 3.5 Altering policies ................................. 61 3.5.1 Game theory ............................... 61 3.5.2 Shot policy changes ........................... 62 3.5.3 Passing policy changes .......................... 64 3.6 Conclusion .................................... 65 4 Inverse Decision Problems 66 4.1 Inverse optimization in operations research . 66 4.2 Examples ..................................... 67 4.2.1 Multi-armed bandit ........................... 67 4.2.2 Fourth down decisions in American football . 68 viii 5 Inverse Bayesian Optimization: Learning Human Acquisition Prefer- ences in an Exploration vs. Exploitation Search Task 73 5.1 Introduction .................................... 73 5.1.1 Background ................................ 74 5.1.2 Hotspot search task ........................... 74 5.1.3 Identifying risk preferences ....................... 75 5.2 Bayesian optimization .............................. 79 5.2.1 Choosing a surrogate function ...................... 79 5.2.2 Updating the surrogate via Bayesian inference . 81 5.2.3 Choosing an acquisition function .................... 82 5.3 Inverse Bayesian optimization .......................... 85 5.3.1 IBO under perfect acquisition ...................... 88 5.3.2 IBO under imperfect acquisition .................... 88 5.3.3 Search task implementation ....................... 89 5.3.4 Incorporating human tendencies .................... 91 5.4 Model Extensions ................................. 93 5.4.1 Perception