Strategy-proofness implies minimal participation under single-peakedness

We study a model in which agents with single-peaked preferences can participate in a costly voting procedure to determine the value of a one-dimensional variable. We show that, for all positive participation costs and all profiles of individual preferences, there exists a unique equilibrium outcome with one single participant whenever the voting rule is strategy-proof, anonymous, and responsive in the sense that the outcome reacts to a unanimous move of the votes of all agents in the same direction; moreover, the single participant is always one of the 'extremist' voters, i.e. either one with the lowest or one with the highest peak. While this uncovers a strong tension between strategy-proofness and participation for all deterministic voting rules on the single-peaked domain (just as in the case of an unrestricted domain), there are simple probabilistic and strategy-proof voting rules that induce full participation in equilibrium.