# Condorcet winners on median spaces

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## Abstract

We characterize the outcome of majority voting for single-peaked preferences on median spaces. This large class of preferences covers a variety of multi-dimensional policy spaces including products of lines (e.g. grids), trees, and hypercubes. Our main result is the following: If a Condorcet winner (i.e. a winner in pairwise majority voting) exists, then it coincides with the appropriately defined median (“the median voter”). This result generalizes previous findings which are either restricted to a one-dimensional policy space or to the assumption that any two voters with the same preference peak must have identical preferences. The result applies to models of spatial competition between two political candidates. A bridge to the graph-theoretic literature is built.

## Keywords

Nash Equilibrium Majority Vote Median Voter Political Competition Condorcet Winner## Notes

### Acknowledgments

I thank Veronica Block, Dinko Dimitrov, Jan-Philip Gamp, Anke Gerber, Claus-Jochen Haake, Tim Hellmann, Dominik Karos, Gerd Muehlheusser, Clemens Puppe, Nils Roehl, Walter Trockel, and the participants of the “Public Economic Theory (PET 10)” Conference in Istanbul and the “SING 9” Workshop in Budapest for helpful discussions and comments. Moreover, this paper has benefited from the suggestions of two anonymous referees.

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