On spatial majority voting with an even (vis-a-vis odd) number of voters: a note

TL;DR

This paper examines how the parity of voters affects the relationship between the core and Condorcet winners in multidimensional spatial voting, revealing that with an even number of voters, the core's singleton element is not a Condorcet winner, unlike the odd case.

Contribution

It demonstrates that in multidimensional spatial voting, an even number of voters with a singleton core does not guarantee the core element is a Condorcet winner, contrasting with the odd voters scenario.

Findings

With an even number of voters, the core's singleton element is never a Condorcet winner.

In the odd voters case, the core's singleton element is always the Condorcet winner.

The result highlights a fundamental difference in voting outcomes based on voter parity.

Abstract

In this note we consider situations of (multidimensional) spatial majority voting. We show that under some assumptions usual in this literature, with an even number of voters if the core of the voting situation is singleton (and in the interior of the policy space) then the element in the core is never a Condorcet winner. This is in sharp contrast with what happens with an odd number of voters: in that case, under identical assumptions, it is well known that if the core of the voting situation is non-empty then the single element in the core is the Condorcet winner as well.