On the Distortion Value of the Elections with Abstention

Abstract

In Spatial Voting Theory, distortion is a measure of how good the winner is. It is proved that no deterministic voting mechanism can guarantee a distortion better than 3, even for simple metrics such as a line. In this study, we wish to answer the following question: how does the distortion value change if we allow less motivated agents to abstain from the election?We consider an election with two candidates and suggest an abstention model, which is a more general form of the abstention model proposed by Kirchgässner [15]. We define the concepts of the expected winner and the expected distortion to evaluate the distortion of an election in our model. Our results fully characterize the distortion value and provide a rather complete picture of the model.

Awareness of Voter Passion Greatly Improves the Distortion of Metric Social Choice

“…In our model, when voter preference strength is less than the smallest threshold (τ 1 > 1), they ef-fectively abstain because their preferred candidate is unknown, and so any reasonable weighted majority rule must assign them a weight of 0. Therefore, our work also bears resemblance to literature on voter abstentions in spatial voting (see [19] and references therein). While there are major technical differences in our model and that of [19], at a high level the model of [19] is similar to a special case of ours with only two candidates and a single threshold on preference strengths (and no knowledge of voter preferences otherwise), which we analyze in Section 5.…”

“…Preferences and a threshold τ costs are normalized in some way, as in e.g., [5,6,7,9,11,12]. The second approach, which we take here, assumes all voters and candidates are points in a metric space [1,2,3,8,13,14,15,17,19,20,22,27,30].…”

“…In particular, we consider voters and candidates which lie in an arbitrary unknown metric space. Our work follows a recent line of research in social choice which considers this setting [1,2,3,8,13,14,15,17,19,20,22,27,30]. The distance between each voter and the winning candidate is interpreted as the cost to that voter.…”

Peeking behind the ordinal curtain: Improving distortion via cardinal queries

“…The distortion framework was introduced by Procaccia and Rosenschein [2006], and has been studied subsequently in a series of papers, most prominently by Boutilier et al [2015], who consider a general social choice setting, under the unit-sum normalization; this general model was also previously studied by Caragiannis and Procaccia [2011] who considered different methods to translate the values of the agents for the alternatives into rankings (embeddings), and more recently by Filos-Ratsikas et al [2019] who bounded the distortion of deterministic mechanisms in district-based elections. A related model is that of distortion of social choice functions in a metric space, which was initiated by , and has since then been studied extensively [Anshelevich and Postl, 2017;Goel et al, 2017;Fain et al, 2019;Goel et al, 2018;Anshelevich and Zhu, 2018;Pierczynski and Skowron, 2019;Gross et al, 2017;Cheng et al, 2017Cheng et al, , 2018Feldman et al, 2016;Ghodsi et al, 2019;Borodin et al, 2019;Munagala and Wang, 2019]. In this setting, there is no normalization of values (or costs), but the valuation (or cost) functions are assumed to satisfy the triangle inequality.…”

Ordinal Approximation for Social Choice, Matching, and Facility Location Problems Given Candidate Positions

“…Discussion and Related Work. Ordinal approximation for the minimum social cost (or maximum social welfare) with underlying utilities/distances between agents and alternatives has been studied in many settings including social choice [4,6,11,13,15,16,19,21,25,28,30,31,33,38,39,41,[43][44][45], matchings [7-9, 12, 18, 22, 29], secretary problems [37], participatory budgeting [10,34], general graph problems [1,7], hierarchical clustering [23], and many other models in recent years. The general assumption of the ordinal setting is that we only have the ordinal preferences of the agents over the alternatives, and the goal is to form a solution that has close to optimal social cost.…”