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We consider synchronous iterative voting, where voters are given theopportunity to strategically choose their ballots depending on the outcomededuced from the previous collective choices.We propose two settings forsynchronous iterative voting, one of classical flavor with a discrete space ofstates, and a more general continuous-space setting extending the first one. Wegive a general robustness result for cycles not relying on a tie-breaking rule,showing that they persist under small enough perturbations of the behavior ofvoters. Then we give examples in Approval Voting of electorates applyingsimple, sincere and consistent heuristics (namely Laslier's Leader Rule or amodification of it) leading to cycles with bad outcomes, either not electing anexisting Condorcet winner, or possibly electing a candidate ranked last by amajority of voters. Using the robustness result, it follows that those "badcycles" persist even if only a (large enough) fraction of the electorateupdates its choice of ballot at each iteration.We complete these results withexamples in other voting methods, including ranking methods satisfying theCondorcet criterion; an in silico experimental study of the rarity ofpreference profiles exhibiting bad cycles; and an example exhibiting chaoticbehavior.

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