A Hotelling–Downs model of electoral competition with the option to quit

We study a variant of the multi-candidate Hotelling–Downs model that recognizes that politicians, even after declaring candidacy, have the option of withdrawing from the electoral contest before the election date and saving the cost of continuing campaign. We find that this natural variant significantly alters equilibrium predictions. We give conditions for the existence of an equilibrium for an arbitrary finite number of candidates and an arbitrary distribution of single-peaked preferences of voters. We also provide a partial characterization of the equilibrium outcomes that addresses whether policy convergence can be a feature of equilibrium outcomes when more than two candidates enter the electoral contest.