Electronic copy available at: http://ssrn.com/abstract=1957790 Nonconvergent Electoral Equilibria under Scoring Rules: Beyond Plurality ∗ Dodge Cahan and Arkadii Slinko 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 candi- dates 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 [4]. 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, nonconver- gent equilibria do not exist. For others, nonconvergent equilibria in which candidates cluster at positions spread across the issue space are observed. Many well-known scoring rules such as plurality, Borda, antiplurality and k-approval fall into the classes considered. Finally, we provide a complete characterisation of nonconvergent equilibria for the special cases of a four- or five-candidate election. 1 Introduction Hotelling’s [11] spatial model of competition, first introduced in 1929, has had a large and varied influence on a number of fields. It has been applied not only in the original context of firms selecting geographic locations along “Main Street” so as to maximise their share of the market, but also to that of producers deciding on how much variety to incorporate into their products [2]. Downs [8] has adapted it with minor modifications to model an election: in particular, the ideological position-taking behaviour of political candidates in their effort to win votes. The model in its simplest form features a number of candidates (firms) adopt- ing positions on a one-dimensional manifold, usually taken to be the interval [0, 1], along which the voters’ ideal positions (consumers) are distributed. The preferences of the voters over the candidates are then determined by the distance between their own ideal positions and those advocated by the candidates. The candidates adopt positions so as to maximise their share of the vote (market). 1 In the economic interpretation of this model where firms compete for market * We also acknowledge Konstantin Tchernov who made important contributions at the initial stages of this paper. 1 See Stigler’s [21] argumentation for this assumption and a discussion of it in [7].