Nonconvergent Electoral Equilibria Under Scoring Rules: Beyond Plurality

Abstract

We use Hotelling's spatial model of competition to investigate the position-taking behavior of political candidates under a class of electoral systems known as scoring rules, though the model also has a natural interpretation in the firm location context. Candidates choose ideological positions so as to maximize their support in society. Convergent Nash equilibria in which all candidates adopt the same policy were characterized 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. We also look at "two-party" equilibria. Implications for the firm location model are discussed. for helpful discussions and suggestions and also participants of the 1st ATE Symposium held on 12-13 December, 2013, at Massey University, Auckland. We thank John McCabe-Dansted for help with computational experiments. Last but not least we thank the anonymous referees for many useful suggestions.

Electoral competition under best-worst voting rules

“…We include here a number of lemmata that are needed for our main results. Several of these minor results are adapted from results in Cahan and Slinko (2016), though similar conditions have appeared in various form in the previous literature since at least Eaton and Lipsey (1975).…”

“…Cox (1987) characterised all scoring rules that have convergent Nash equilibria, which leads to a straightforward description of all best-worst rules allowing convergent equilibria, as we will describe in Section 4. However, Cox's theorem says nothing about the possibility of divergent equilibria, which is the focus of the paper Cahan and Slinko (2016) and also this paper. Cahan and Slinko (2016) investigate the existence and properties of nonconvergent equilibria under general scoring rules.…”

“…However, Cox's theorem says nothing about the possibility of divergent equilibria, which is the focus of the paper Cahan and Slinko (2016) and also this paper. Cahan and Slinko (2016) investigate the existence and properties of nonconvergent equilibria under general scoring rules. In some subclasses of scoring rules-in particular, those whose score vector is convex-they managed to characterise all rules that allow Nash equilibria.…”

“…The two most relevant papers to this research are Cox (1987) and Cahan and Slinko (2016). Both are concerned with Nash equilibria under the class of voting rules known as general scoring rules, of which the best-worst rules are a subclass.…”

“…6 We do not actually need the distribution to be literally uniform. All we need is that the candidates believe this to be the case or take it as a simplifying assumption in their decisionmaking (Aragonès and Xefteris, 2012;Cahan and Slinko, 2016).…”