Hotelling-downs competition with free entry when voters have an option to contest

Osborne shows that for almost all distributions of voters' preferences, a pure strategy Nash equilibrium does not exist in the classical Hotelling-Downs model of electoral competition with free entry. We show that equilibrium is generically possible if in addition one allows voters an option to announce their candidacy to compete side-by-side with office-seeking players. The model studied in this paper renders Osborne and the celebrated citizen-candidate model à la Osborne and Slivinski as two extreme cases. We characterize the equilibrium set with two central questions: (i) can there be equilibria where only voters contest? and (ii) are equilibria with contesting office-seeking players possible? We also show that in our general setting, extremists are typically voter-candidates so that in every two-party contest, office-seeking politicians stay out of competition.