- In multiagent systems, social choice functions can help aggregate the distinct preferences that agents have over alternatives, enabling them to settle on a single choice. Despite the basic manipulability of all reasonable voting systems, it would still be desirable to find ways to reach a stable result, i.e., a situation where no agent would wish to change its vote. One possibility is an iterative process in which, after everyone initially votes, participants may change their votes, one voter at a time. This technique, explored in previous work, converges to a Nash equilibrium when Plurality voting is used, along with a tie-breaking rule that chooses a winner according to a linear order of preferences over candidates. In this paper, we both consider limitations of the iterative voting method, as well as expanding upon it. We demonstrate the significance of tie-breaking rules, showing that when using a general tie-breaking rule, no scoring rule (nor Maximin) need iteratively converge. However, using a restricted tie-breaking rule (such as the linear order rule used in previous work) does not by itself ensure convergence. We demonstrate that many scoring rules (such as Borda) need not converge, regardless of the tie-breaking rule. On a more encouraging note, we prove that Iterative Veto does converge---but that voting rules "between" Plurality and Veto, k-approval rules, do not.