Equity and efficiency in computational social choice Gerdus Benad`e May 2, 2019 Tepper School of Business Carnegie Mellon University Pittsburgh, PA 15213 Thesis Committee: Ariel D. Procaccia John Hooker R.Ravi Jay Sethuraman Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Algorithms, Combinatorics, and Optimization. Copyright © 2019 Gerdus Benad`e . Abstract This thesis studies problems in computational social choice and fair division. Computational social choice ask how to aggregate individual votes and opinions into a joint decision. Participatory budgeting enables the allocation of public funds by collecting and aggregating individual preferences; it has already had a sizable real-world impact. We analytically compare four pref- erence elicitation methods through the lens of implicit utilitarian voting, and find that threshold approval votes are qualitatively superior. This conclusion is supported by experiments using data from real participatory budgeting elections. We also conduct a human subject experiment conducted on Ama- zon Mechanical Turk to study the cognitive burden that different elicitation formats place on voters. Under implicit utilitarian voting we attempt to maximize a utilitarian objective in the presence of uncertainty about voter utility functions. Next, we take a very different approach, and assume votes are noisy estimates of an unknown ground truth. We build on previous work which replaced struc- tural assumptions on the noise with a worst-case approach, and minimize the expected error with respect to a set of feasibly true rankings. We derive (mostly sharp) analytical bounds on the expected error and find that our approach has useful practical properties. Fair division problems involve allocating goods to heterogeneous agents. Motivated by the problem of a food bank allocating donations to their bene- ficiaries without knowledge of future arrivals, we study the online allocation of indivisible items. Our goal is to design allocation algorithms that min- imize the maximum envy, defined as the maximum difference between any agent's overall value for items allocated to another agent and to herself. An algorithm has vanishing envy if the ratio of envy over time goes to zero as time goes to infinity. We find a polynomial-time, deterministic algorithm that achieves vanishing envy, and show the rate at which envy vanishes is asymptotically optimal. Finally, we consider the problem of gerrymandering. We start with an impartial protocol and derive a notion of fairness which provides guidance about what to expect in an impartial districting. Specifically, we propose a party should win a number of districts equal to the midpoint between what they win in their best and worst districtings. We show that this notion of fair- ness has close ties to proportionality and that, in contrast to proportionality, there always exists a districting satisfying our notion of fairness. iv Contents 1 Introduction 1 2 Participatory budgeting 5 2.1 Introduction . .5 2.1.1 Our Approach and Results . .6 2.1.2 Related Work . .9 2.2 The Model . 11 2.3 Theoretical Results . 13 2.3.1 Randomized Aggregation Rules . 13 2.3.2 Deterministic Aggregation Rules . 24 2.4 Computing Worst-Case Optimal Aggregation Rules . 29 2.4.1 Deterministic Rules . 30 2.4.2 Randomized Rules . 32 2.4.3 Scaleability of computing distortion-minimizing sets . 33 2.5 Empirical Results . 35 2.5.1 Is It Useful to Learn the Threshold? . 38 2.6 User study . 40 2.6.1 Experimental Setup . 41 2.6.2 Efficiency . 43 2.6.3 Usability . 45 2.6.4 Discussion of user study . 50 2.7 Discussion . 52 3 Low-distortion rankings 53 3.1 Introduction . 53 3.1.1 Our Approach and Results . 54 3.1.2 Related Work . 55 3.2 The Model . 56 v 3.3 Distortion Bound . 57 3.3.1 Proof of Lemma 3.4 . 64 3.4 Empirical Results . 69 3.5 Discussion . 72 4 Aggregating noisy estimates of a ground truth 73 4.1 Introduction . 73 4.1.1 The Worst-Case Approach . 74 4.1.2 Our Approach and Results . 76 4.2 Preliminaries . 77 4.3 Returning the Right Ranking, in Theory . 79 4.4 Returning the Right Alternatives, in Theory . 84 4.4.1 The KT and Footrule Distances . 87 4.4.2 The Maximum Displacement Distance . 92 4.4.3 The Cayley Distance . 95 4.5 Making the right decisions, in practice . 99 4.6 Discussion . 103 5 Dynamic fair division of indivisible goods 105 5.1 Introduction . 105 5.1.1 Our Results . 106 5.1.2 Related Work . 108 5.2 Model . 109 5.3 Single Arrivals under Full Information . 110 5.3.1 Upper bound via Random Allocation . 112 5.3.2 Derandomization with Pessimistic Estimators . 117 5.3.3 Lower Bound . 122 5.4 Batch Arrivals under Full Information . 127 5.4.1 Upper Bound . 127 5.4.2 Polynomial-Time Special Cases . 135 5.4.3 Lower Bound . 140 5.5 Single Arrivals under Partial Information . 141 5.5.1 Randomized Algorithms . 141 5.5.2 Deterministic Algorithms . 141 5.6 Discussion . 142 vi 6 Political districting 145 6.1 Introduction . 145 6.1.1 Our approach and results . 147 6.1.2 Related work . 148 6.2 Fairness measures . 149 6.2.1 Seats-votes curve . 150 6.2.2 Efficiency gap . 150 6.2.3 Compactness measures . 152 6.2.4 Distributional approaches . 153 6.3 Fair target property . 153 6.3.1 Guaranteeing targets on a plane . 155 6.3.2 Case study . 160 6.4 An exact model for redistricting . 163 6.4.1 Modeling the Objective Function . 165 6.4.2 Other valid inequalities . 167 6.4.3 Preliminary computational results . 167 Bibliography 171 vii viii Chapter 1 Introduction As algorithms start affecting more parts of society, our social structures of democracy and collaborative decision making give rise to interesting ques- tions about how to design interactions to elicit truthful and informative opinions from participants. There is also a growing awareness that mak- ing responsible decisions in dynamic and socially intertwined environments requires novel algorithmic and technical frameworks that balance notions of equality and fairness with efficiency. Many of these problems may be for- mulated as traditional optimization problems with the wrinkle that social considerations lead to new avenues of investigation. I will focus on problems in computational social choice and dynamic fair division. Computational social choice Social choice theory studies how to aggre- gate individual opinions and preferences into collective decisions. Research in computational social choice leverages tools like approximation algorithms and complexity theory to shed new light on traditional social choice problems. One stream of research, labeled implicit utilitarian voting [40] assumes that the vote a voter casts is consistent with his utility function and asks to what extent the social welfare maximizing outcome can be approximated using only these consistent proxies for the voters' utility functions. In chapter 2, we study perhaps the most exciting application of compu- tational social choice, participatory budgeting, through this lens of implicit utilitarian voting. In the participatory budgeting framework a city decides how to spend its budget after allowing the residents of the city to vote over a set of alternatives. Each alternative has a cost and the objective is to maxi- mize social welfare subject to a budget constraint. We make an assumption 1 that voter utilities are additive and analytically compare four preference elic- itation methods | knapsack votes, rankings by value or value for money, and threshold approval votes | and find that threshold approval votes are quali- tatively superior. This conclusion is supported by computational experiments using data from real world-participatory budgeting instances. This problem is challenging because asking voters to report their exact utility functions would be too burdensome, so instead we have to make do with easy-to-cast proxies. We study the cognitive burden associated with each input format through an human subject experiment conducted on Amazon Mechanical Turk. In chapter 3, we extend our results on rankings by value to a more general setting with subadditive utility functions where, instead of returning a set of alternatives satisfying the budget, the question is to return a ranking of the alternatives. The work in these chapters appeared in [18, 19, 21] A second stream of research in computational social choice view votes as estimators of some objective ground truth under some noisy process. In- stead of maximizing social welfare as before, the objective is to recover this unknown ground truth. This is often done by making structural assump- tions about the noise model which allows you return a maximum liklihood estimator. In chapter 4, we tackle this problem while avoiding assumptions about noise models, symmetric noise and large sample sizes which are common in the literature. Following the worst-case approach in Procaccia et al. [104], our only assumption is that the average voter is at bounded distance from the ground truth under some distance metric. This assumption leads to a space of feasible solutions, each of which has the potential to be the ground truth. We deviate from previous work by minimizing the average error with respect to the set of feasible ground truth rankings instead of the worst-case error. We derive (mostly sharp) analytical bounds on the expected error and establish the practical benefits of our approach through experiments. This chapter is based on [] Fair division In fair division problems, a set of agents must be assigned a set of divisible or indivisible goods. In contrast to the traditional assignment problems, the objective is not some measure of cost or profit, instead, the aim is to ensure that the allocation is fair with respect to the agents' heterogenous preferences, under a suitable notion of fairness.