Abstract
We characterize elections that are simultaneously single-peaked and single-crossing (SPSC), by establishing a connection between this domain and that of minimally rich elections, i.e., elections where each candidate is ranked first by at least one voter. Specifically, we show that an election is both single-peaked and single-crossing if and only if it can be obtained from a minimally rich single-crossing election by deleting voters.
Highlights
Perhaps the most famous result in social choice is Arrow’s impossibility theorem (Arrow 1951), which establishes that there is no perfect method of aggregating voters’ preferences over three or more candidates into a collective opinion
We show that an election is both single-peaked and single-crossing if and only if it can be obtained from a minimally rich single-crossing election by deleting voters
As the single-peaked and single-crossing (SPSC) domain is larger than the 1-Euclidean domain, we hope that our characterization and algorithms can be useful for dealing with problems for which initial intuitions were obtained in the 1Euclidean setting, but where a combinatorial perspective is necessary; for example, in another paper we used this approach to find an improved algorithm for the egalitarian variant of the Monroe multiwinner rule (Skowron et al 2015)
Summary
Perhaps the most famous result in social choice is Arrow’s impossibility theorem (Arrow 1951), which establishes that there is no perfect method of aggregating voters’ preferences over three or more candidates into a collective opinion. It was recently established that such elections can be characterized in terms of forbidden configurations (Ballester and Haeringer 2011): there are two elections (one containing two voters and four candidates, and the other containing three voters and three candidates) such that an election is single-peaked if and only if it does not contain subelections that are equivalent to one of these two elections Another well-studied restricted domain is that of single-crossing elections (Mirrlees 1971; Roberts 1977). The existing body of work provides us with a good understanding of the properties of elections that are single-peaked or single-crossing Against this background, the goal of this paper is to characterize the elections that belong to the intersection of these two domains; we refer to the resulting class of elections as the SPSC domain. As the SPSC domain is larger than the 1-Euclidean domain, we hope that our characterization and algorithms can be useful for dealing with problems for which initial intuitions were obtained in the 1Euclidean setting, but where a combinatorial perspective is necessary; for example, in another paper we used this approach to find an improved algorithm for the egalitarian variant of the Monroe multiwinner rule (Skowron et al 2015)
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