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Cycles in synchronous iterative voting: general robustness and examples in Approval Voting

Abstract

We consider synchronous iterative voting, where voters are given the opportunity to strategically choose their ballots depending on the outcome deduced from the previous collective choices. We propose two settings for synchronous iterative voting, one of classical flavor with a discrete space of states, and a more general continuous-space setting extending the first one. We give a general robustness result for cycles not relying on a tie-breaking rule, showing that they persist under small enough perturbations of the behavior of voters. Then we give examples in Approval Voting of electorates applying simple, sincere and consistent heuristics (namely Laslier’s Leader Rule or a modification of it) leading to cycles with bad outcomes, either not electing an existing Condorcet winner, or possibly electing a candidate ranked last by a majority of voters. Using the robustness result, it follows that those “bad cycles” persist even if only a (large enough) fraction of the electorate updates its choice of ballot at each iteration. We complete these results with examples in other voting methods, including ranking methods satisfying the Condorcet criterion; an in silico experimental study of the rarity of preference profiles exhibiting bad cycles; and an example exhibiting chaotic behavior.

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Notes

  1. A similar argument has been raised by Warren D. Smith and is used by advocates of Approval Voting and Range Voting, see https://www.rangevoting.org/AppCW.html. The argument of Smith is less rigorous than Laslier’s, since the equilibrium assumption is kept implicit.

  2. An example of cycle under the Leader Rule is given as exercise 8.4.4 in Meir (2018), but there are no Condorcet winner.

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Acknowledgements

I am indebted to Adrien Fauré @AdrienGazouille for a long debate on twitter (in French) that lead me to seek and design the examples for Approval Voting presented here, and to François Durand for introducing me to the Social choice theory, for many discussions on Voting Systems, and for many suggestions that helped me improve this article. It also benefited from relevant comments provided by Adrien Fauré, Jean-François Laslier, Reshef Meir and anonymous referees who I warmly thank.

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Correspondence to Benoît R. Kloeckner.

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Kloeckner, B.R. Cycles in synchronous iterative voting: general robustness and examples in Approval Voting. Soc Choice Welf 59, 423–466 (2022). https://doi.org/10.1007/s00355-022-01395-6

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