Making Decisions with Incomplete and Inaccurate Information by Vijay Menon A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Doctor of Philosophy in Computer Science Waterloo, Ontario, Canada, 2021 © Vijay Menon 2021 Examining Committee Membership The following served on the Examining Committee for this thesis. The decision of the Examining Committee is by majority vote. External Examiner: Ioannis Caragiannis Professor Department of Computer Science, Aarhus University Supervisor: Kate Larson Professor Cheriton School of Computer Science, University of Waterloo Internal Members: Shai Ben-David Professor Cheriton School of Computer Science, University of Waterloo Eric Blais Associate Professor Cheriton School of Computer Science, University of Waterloo Internal-External Member: Stanko Dimitrov Associate Professor Department of Management Sciences, University of Waterloo ii Author’s Declaration I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. iii Statement of Contributions The contents of this thesis are based on the following papers that I have co-authored. 1. Text and results in Chapter2 are based on [ML18] Vijay Menon and Kate Larson. “Robust and Approximately Stable Marriages Under Partial Information”. In: Proceedings of the Fourteenth International Conference on Web and Internet Economics (WINE). 2018, pp. 341–355 2. Text and results in Chapter3 are based on [ML19] Vijay Menon and Kate Larson. “Mechanism Design for Locating a Facility Under Partial Information”. In: Proceedings of the Twelfth International Symposium on Algorithmic Game Theory (SAGT). 2019, pp. 49–62 3. Text and results in Chapter4 are based on [MML21] Thomas Ma, Vijay Menon, and Kate Larson. “Improving Welfare in One-Sided Matching Using Simple Threshold Queries”. In: Proceedings of the Thirteeth International Joint Conference on Artificial Intelligence (IJCAI). 2021 4. Text and results in Chapter5 are based on [ML20] Vijay Menon and Kate Larson. Algorithmic Stability in Fair Allo- cation of Indivisible Goods Among Two Agents. 2020. arXiv: 2007.15203 [cs.GT] iv Abstract From assigning students to public schools to arriving at divorce settlements, there are many settings where preferences expressed by a set of stakeholders are used to make decisions that affect them. Due to its numerous applications, and thanks to the range of questions involved, such settings have received considerable attention in fields ranging from philosophy to political science, and particularly from economics and, more recently, computer science. Although there exists a significant body of literature studying such settings, much of the work in this space make the assumption that stakeholders provide complete and accurate preference information to such decision-making procedures. However, due to, say, the high cognitive burden involved or privacy concerns, this may not always be feasible. The goal of this thesis is to explicitly address these limitations. We do so by building on previous work that looks at working with incomplete information, and by introducing solution concepts and notions that support the design of algorithms and mechanisms that can handle incomplete and inaccurate information in different settings. We present our results in two parts. In Part I we look at decision-making in the presence of incomplete information. We focus on two broad themes, both from the perspective of an algorithm or mechanism designer. Informally, the first one studies the following question: Given incomplete preferences, how does one design algorithms that are ‘robust’, i.e., ones that produce solutions that are “good” with respect to the underlying complete preferences? We look at this question in context of two well-studied problems, namely, i) (a version of) the two-sided matching problem and ii) (a version of) the facility location problem, and show how one can design approximately-robust algorithms in such settings. Following this, we look at the second theme, which considers the following question: Given incomplete preferences, how can one ask the agents for some more information in order to aid in the design of ‘robust’ algorithms? We study this question in the context of the one-sided matching problem and show how even a very small amount of extra information can be used to get much better outcomes overall. In Part II we turn our attention to decision-making in the presence of inaccurate information and look at the following question: How can one design ‘stable’ algorithms, i.e., ones that do not produce vastly different outcomes as long as there are only small inaccuracies in a stakeholder’s report of their preferences? We study this in the context of fair allocation of indivisible goods among two agents and show how, in contrast to popular fair allocation algorithms, there are alternative algorithms that are fair and approximately-stable. v Acknowledgements I would first like to thank my advisor, Kate Larson. Kate has always been supportive and encouraging, has always balanced my pessimism when it comes to my work with her optimism, and has always given me the freedom to pursue my own interests. I am grateful to have had the opportunity to work with her over the last several years. I would also like to thank the members of my committee, Shai Ben-David, Eric Blais, Ioannis Caragiannis, and Stanko Dimitrov, for reading my thesis, for providing feedback, and for insightful questions. Finally, I am grateful to my friends for all the good times and their companionship over the years, and thank my family for their support. vi Table of Contents List of Figures xii List of Tables xiii 1 Introduction1 1.1 Some Basic Terms and Concepts........................3 1.2 Overview of the Thesis.............................5 1.2.1 Part I: Making decisions with incomplete information........5 1.2.2 Part II: Making decisions with inaccurate information........8 I Making Decisions with Incomplete Information 10 2 Two-Sided Matching Under Partial Information 11 2.1 Introduction................................... 11 2.1.1 Working with partial information................... 13 2.1.2 Measuring the amount of missing information............ 14 2.1.3 Our contributions............................ 15 2.1.4 Related work.............................. 16 2.2 Preliminaries.................................. 17 vii 2.2.1 Problem definitions........................... 19 2.3 Investigating Weakly-Stable Matchings.................... 22 2.3.1 Using weakly-stable matchings to approximate the δ-min-bp-super- stable-matching problem........................ 22 2.3.2 Can we do better when restricted to weakly-stable matchings?... 24 2.3.3 The case of one-sided top-truncated preferences: An O(n) approxi- mation algorithm............................ 26 2.4 Beyond Weak-Stability............................. 33 2.4.1 Inapproximability of δ-min-bp-super-stable-matching........ 33 2.4.2 A possible general approach for obtaining a near-tight approximation factor for δ-min-bp-super-stable-matching.............. 38 2.5 Discussion.................................... 39 3 Mechanism Design for Locating a Facility Under Partial Information 41 3.1 Introduction................................... 41 3.2 Preliminaries.................................. 44 3.2.1 Mechanisms, solution concepts, and implementation......... 45 3.2.2 Some Q & A on the definitions.................... 47 3.3 Related Work.................................. 49 3.4 Implementing the Average Cost Objective.................. 51 3.4.1 Properties of the minimax optimal solution for avgCost....... 52 3.4.2 Implementation in very weakly dominant strategies......... 52 3.4.3 Implementation in minimax dominant strategies........... 54 3.5 Implementing the Maximum Cost Objective................. 68 3.5.1 Implementation in very weakly dominant strategies......... 69 3.5.2 Implementation in minimax dominant strategies........... 69 3.6 Discussion.................................... 72 viii 4 Improving Welfare in One-Sided Matching Using Threshold Queries 73 4.1 Introduction................................... 73 4.1.1 Our contributions............................ 76 4.1.2 Related work.............................. 78 4.2 Preliminaries.................................. 79 4.2.1 Notions of economic efficiency..................... 80 4.2.2 Going beyond completely ordinal or completely cardinal algorithms 83 4.2.3 Finding welfare-optimal priority-p matchings when utilities are known 84 4.3 Improving Welfare using Threshold Queries................. 86 4.3.1 Adaptive algorithm to achieve (1 + )-approximation........ 86 4.3.2 Non-adaptive algorithms: asking one query per (agent, object) pair 90 4.4 Lower Bounds.................................. 96 4.5 Discussion.................................... 99 II Making Decisions with Inaccurate Information 101 5 Algorithmic Stability in Fair Allocation of Indivisible Goods 102 5.1 Introduction................................... 102 5.2 Preliminaries.................................. 105 5.2.1 Stability................................. 107 5.2.2 Some Q & A on assumptions and definitions............. 111 5.3 Related Work.................................. 112 5.4 Approximate-Stability in Fair Allocation of Indivisible Goods........ 114 5.4.1 How (approximately) stable are the existing algorithms?...... 115 5.4.2 Are