Stability in electoral competition: A case for multiple votes

It is well known that the Hotelling–Downs model generically fails to admit an equilibrium when voting takes place under the plurality rule (Osborne, 1993). This paper studies the Hotelling–Downs model considering that each voter is allowed to vote for up to k candidates and demonstrates that an equilibrium exists for a non-degenerate class of distributions of voters' ideal policies – which includes all log-concave distributions – if and only ifk≥2. That is, the plurality rule (k=1) is shown to be the unique k-vote rule which generically precludes stability in electoral competition. Regarding the features of k-vote rules' equilibria, first, we show that there is no convergent equilibrium and, then, we fully characterize all divergent equilibria. We study comprehensively the simplest kind of divergent equilibria (two-location ones) and we argue that, apart from existing for quite a general class of distributions when k≥2, they have further attractive properties – among others, they are robust to free-entry and to candidates' being uncertain about voters' preferences.