A Characterization of the Single-Peaked Single-Crossing Domain

We investigate elections that are simultaneously singlepeaked and single-crossing (SPSC). We show that the domain of 1-dimensional Euclidean elections (where voters and candidates are points on the real line, and each voter prefers the candidates that are close to her to the ones that are further away) is a proper subdomain of the SPSC domain, by constructing an election that is single-peaked and singlecrossing, but not 1-Euclidean. We then establish a connection between narcissistic elections (where each candidate is ranked first by at least one voter), single-peaked elections and single-crossing elections, by showing that an election is SPSC if and only if it can be obtained from a narcissistic singlecrossing election by deleting voters. We show two applications of our characterization.