While reading a post on Reddit the other day, I got to thinking about universal domain and IIA. Consider Plurality as a special case of Approval, with a similar ballot: Explicit approval ballot, except you can at most fill out one approval. Then the argument that Approval passes IIA should also be applicable to this variant of Plurality. After all, if the ballots state "vote for X" and X doesn't win and is eliminated, then the ballots that voted for X are now blank, and the ballots that voted for someone else still votes for that someone else. So eliminating X can never change the victory from Y to Z, so the method (with this ballot format) passes IIA. Clearly, unlike in the "standard" (ranked) interpretation of Plurality, Plurality with this ballot format fails universal domain because there's no way to express X>Y>Z. (It's clear that the ranked interpretation is what's actually being used most of the time because otherwise, IRV wouldn't work as Plurality-elimination.) Am I right about this? If so, it suggests that it's rather easy to pass IIA if you give away universal domain. But under the limited expressibility of the restricted ballots, both Approval and Plurality requires a voter with more complex preferences than the ballot allows, to make use of what I've called "manual DSV" to express the preference on that ballot. Thus, saying that manual DSV is not a problem with Approval invites the comparison to Plurality. If manual DSV is not a problem with Approval but is a problem with Plurality, then there must be something particular to the Approval method itself that makes it so. And it would be enlightening to understand just what that is, if it indeed exists. (I would prefer my methods to not have *any* manual DSV for honest voters, even if that means that the method has to fail IIA in consequence.) Are there any methods that are Condorcet under ranked ballot assumptions but pass IIA if we break universal domain? I guess the trivial answer is "yes, every method". Just restrict the domain to the Plurality ballots above, and every Condorcet method will give you the same result as Plurality itself. But are there more complex ballot types that get closer? How close is it possible to get to Condorcet while maintaining IIA by giving up universal domain? I don't know. In any case, if it's universal domain failure that leads to a need for manual DSV even by honest voters, then that is a useful thing to know. Perhaps, then, Approval can be analyzed more honestly by considering it as a ranked method: ER-Plurality. It now fails IIA, and it has a tremenduous compression incentive. However, I don't think that perspective quite captures the nature of the manual DSV. In particular, it's not obvious why the Burr dilemma exists. It also only works for methods like Approval and Plurality, where the domain is more barren than full ranking. It doesn't work for Range, because Range's domain is wider: assuming a sufficiently fine scale, any ranked ballot can be encoded (though not uniquely) as some rated ballot. Perhaps there the problem is more that there are so many honest ballots to choose from, and it's not unambiguously defined which one is "the" honest ballot to choose. At least not in the absence of a common rating or grading standard.