Nonconvergent Electoral Equilibria under Scoring Rules: Beyond Plurality

TL;DR

This paper explores the existence and characteristics of nonconvergent electoral equilibria under scoring rules using spatial models, revealing conditions where such equilibria do or do not exist and analyzing specific candidate scenarios.

Contribution

It extends the analysis of candidate behavior under scoring rules by characterizing nonconvergent equilibria and identifying classes of rules that admit or exclude such equilibria.

Findings

Convex scoring rules generally lack Nash equilibria.

Nonconvergent equilibria can involve candidates spreading out across the issue space.

Complete characterizations are provided for four-, five-, and six-candidate elections.

Abstract

We use Hotelling's spatial model of competition to investigate the position-taking behaviour of political candidates under a class of electoral systems known as scoring rules. In a scoring rule election, voters rank all the candidates running for office, following which the candidates are assigned points according to a vector of nonincreasing scores. Convergent Nash equilibria in which all candidates adopt the same policy were characterised by Cox (1987). Here, we investigate nonconvergent equilibria, where candidates adopt divergent policies. We identify a number of classes of scoring rules exhibiting a range of different equilibrium properties. For some of these, nonconvergent equilibria do not exist. For others, nonconvergent equilibria in which candidates cluster at positions spread across the issue space are observed. In particular, we prove that the class of convex rules does not have Nash equilibria (convergent or nonconvergent) with the exception of some derivatives of Borda rule. Finally, we examine the special cases of four-, five- and six- candidate elections. In the former two cases, we provide a complete characterisation of nonconvergent equilibria.