On the Convergence of Iterative Voting: How Restrictive Should Restricted Dynamics Be?

We study convergence properties of iterative voting procedures. Such a procedure is defined by a voting rule and a (restricted) iterative process, where at each step one agent can modify his vote towards a better outcome for himself. It has already been observed in previous works that if voters are allowed to make arbitrary moves (or even only best responses), such processes may not converge for most common voting rules. It is therefore important to investigate whether and which natural restrictions on the dynamics of iterative voting procedures can guarantee convergence. To this end, we provide two general conditions on dynamics based on iterative myopic improvements, each of which is sufficient for convergence. We then identify several classes of voting rules, along with their corresponding iterative processes, for which at least one of these conditions hold. Our work generalizes recent results and relaxes a number of restrictive assumptions made in previous research.