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            <idno type="halRefHtml">&lt;i&gt;Social Choice and Welfare&lt;/i&gt;, In press</idno>
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            <note type="commentary">v2: added a numerical study of rarity of bad cycles and equilibriums, and a case of chaotic Continuous Polling Dynamics. Many other improvements throughout the text. v3: reorganization and change of order. The robustness result is generalized to all voting systems. v4 Added a simple example to the stability Theorem, various other small modifications.</note>
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              <p>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.</p>
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