We consider synchronized iterative voting in the Approval Voting system. We
give examples with a Condorcet winner where voters apply simple, sincere,
consistent strategies but where cycles appear that can prevent the election of
the Condorcet winner, or that can even lead to the election of a ''consensual
loser'', rejected in all circumstances by a majority of voters. We conduct
numerical experiments to determine how rare such cycles are. It turns out that
when voters apply Laslier's Leader Rule they are quite uncommon, and we prove
that they cannot happen when voters' preferences are modeled by a
one-dimensional culture. However a slight variation of the Leader Rule
accounting for possible draws in voter's preferences witnesses much more bad
cycle, especially in a one-dimensional culture.Then we introduce a
continuous-space model in which we show that these cycles are stable under
perturbation. Last, we consider models of voters behavior featuring a
competition between strategic behavior and reluctance to vote for candidates
that are ranked low in their preferences. We show that in some cases, this
leads to chaotic behavior, with fractal attractors and positive entropy.Comment: 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 tex