Tradeoffs between efficiency, fairness and information in computational social choice

Abstract

Problems within social choice often involve selecting an outcome that best represents the preferences of a group of agents. In this work we are interested in cases when agents and candidates lie in a shared metric space, and our goal is to select one or multiple winners as efficiently as possible. We show how the additional metric structure can be exploited to improve the quality of the selected outcomes, while also considering the amount of information that is required to do so. This work broadly covers two main questions: (1) How much information is needed to select high-quality outcomes in different models? and (2) How can we ensure that the selected outcomes are fair to different groups of agents while still being efficient? To address the first question, we utilize the framework of metric distortion, which measures the loss in efficiency incurred by only having access to limited information about agent preferences. Formally, we assume that agents and candidates lie in a shared metric space, but the mechanism only has access to ordinal preferences (rankings) or limited cardinal information about agent preferences. To assess the quality of a mechanism we use the distortion framework of \citet{procaccia2006distortion}. The distortion of a mechanism is then defined as the worst-case ratio between the cost of the outcome selected by the mechanism and the cost of the optimal outcome that could be selected if the entire metric space were known. We first study how the addition of threshold approval information, where agents approve all candidates within a ball of radius $\alpha$ times the distance of the closest candidate, can reduce distortion for the social cost objective and the max cost objective. We also study the metric distortion of line-up elections, which are a combination of multi-winner elections and bipartite problems, and give new mechanisms with constant distortion for the social cost. Lastly in the metric distortion setting, we study group-fair objectives, where agents are partitioned into groups and the quality of an outcome is measured using objectives that take into account the partition of the agents into groups to measure efficiency and satisfy other desired properties such as fairness or some form of balance among different groups. We give new mechanisms with improved distortion bounds for these objectives. To answer the second question, we study the problem of selecting committees that are simultaneously low-cost, fair, and representative with full information. We consider fairness axioms such as Proportional Representative Fairness (PRF) \citep{aziz2023proportionallyrepresentativeclustering} that define fairness based on requiring groups of agents of size at least a certain threshold to be adequately close in the selected committee. We also introduce a new fairness axiom, the No Over-representation Principle (NORP), which requires that the set of representatives in the selected committee have a proportional number of agents nearby. We ultimately show that our new axiom is compatible with previous fairness axioms such as PRF, while also having low social cost.