# Strategy-proofness of the randomized Condorcet voting system

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## Abstract

In this paper, we study the strategy-proofness properties of the randomized Condorcet voting system (RCVS). Discovered at several occasions independently, the RCVS is arguably the natural extension of the Condorcet method to cases where a deterministic Condorcet winner does not exists. Indeed, it selects the always-existing and essentially unique Condorcet winner of lotteries over alternatives. Our main result is that, in a certain class of voting systems based on pairwise comparisons of alternatives, the RCVS is the only one to be Condorcet-proof. By Condorcet-proof, we mean that, when a Condorcet winner exists, it must be selected and no voter has incentives to misreport his preferences. We also prove two theorems about group-strategy-proofness. On one hand, we prove that there is no group-strategy-proof voting system that always selects existing Condorcet winners. On the other hand, we prove that, when preferences have a one-dimensional structure, the RCVS is group-strategy-proof.

## Keywords

Vote System Median Voter Condorcet Winner Social Choice Theory Social Choice Rule## Notes

### Acknowledgements

I am greatly grateful to Rémi Peyre without whom this paper would not have been possible. He introduced me to social choice theory in popularized articles Peyre (2012b, 2012a, c). Second, he sketched the proof of Theorem 2, and hinted at Theorem 3. Finally, and most importantly, our discussions gave me great insights into the wonderful theory of voting systems. I am also grateful to anonymous referees as well as the managing editor, who provided useful references and remarks that greatly simplified and clarified the exposition of this work.

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