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.
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Previous versions of this work have been presented at the conference ‘Improving Public Debate and Political Decision Making,’ London, March 2019, the ‘International Conference on Mathematical Programming and Fair Social Decisions,’ Paris, December 2019, in seminars at Université Paris Dauphine, Karlsruhe Institute of Technology (KIT), the University of Vienna, at the COMSOC video seminar series, May 2020, at the COMSOC conference in Haifa, June 2021 and the SAET conference in Seoul, June 2021. We are grateful to the audiences for helpful feedback and comments. Special thanks to Rida Laraki, Reshef Meir, Hervé Moulin, Klaus Nehring, Matías Núñez, Martin Osborne and Marcus Pivato for particularly helpful comments and suggestions. We also thank two anonymous referees of this journal for constructive feedback that helped improving and generalizing the analysis. All errors are our own.
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Müller, M., Puppe, C. Strategy-proofness implies minimal participation under single-peakedness. Econ Theory Bull 11, 131–151 (2023). https://doi.org/10.1007/s40505-023-00247-5