Comparing Ways of Obtaining Candidate Orderings from Approval Ballots

TL;DR

This paper compares five methods for deriving candidate orderings from approval ballots, analyzing their axiomatic properties and empirical performance on real and synthetic data.

Contribution

It provides an axiomatic comparison of different axis selection rules and characterizes the scoring rule that maximizes interval votes under specific axioms.

Findings

Some axiomatic impossibility results are established.

The scoring rule maximizing interval votes is characterized.

Empirical analysis on French surveys and US Supreme Court data demonstrates rule behaviors.

Abstract

To understand and summarize approval preferences and other binary evaluation data, it is useful to order the items on an axis which explains the data. In a political election using approval voting, this could be an ideological left-right axis such that each voter approves adjacent candidates, an analogue of single-peakedness. In a perfect axis, every approval set would be an interval, which is usually not possible, and so we need to choose an axis that gets closest to this ideal. The literature has developed algorithms for optimizing several objective functions (e.g., minimize the number of added approvals needed to get a perfect axis), but provides little help with choosing among different objectives. In this paper, we take a social choice approach and compare 5 different axis selection rules axiomatically, by studying the properties they satisfy. We establish some impossibility theorems, and characterize (within the class of scoring rules) the rule that chooses the axes that maximize the number of votes that form intervals, using the axioms of ballot monotonicity and resistance to cloning. Finally, we study the behavior of the rules on data from French election surveys, on the votes of justices of the US Supreme Court, and on synthetic data.