IIA confusion -- did Arrow mean what others mean?

There seems to be some confusion about what is meant by “independence of irrelevant alternatives” (IIA). For example, the Wikipedia article says “Most ranked ballot methods and Plurality voting satisfy the Majority Criterion, and therefore fail IIA automatically by the example above.” But Bordes and Tideman, in a 1991 paper, say that all “real-world” voting systems satisfy IIA.

They state that “IIA means that if the voters’ preferences over the potential-but-not-actual candidates change while their preferences over the actual candidates stay the same, then the choice among the actual candidates stays the same.” They say it is NOT a consistency criterion, as it has commonly been understood.

They go on to write:
"Obviously, no voting rule can make any use of information it does not have, and what transformation occurs in the information a rule does not have, whether that information shrinks (regularity) or changes (IIA), does not matter as long as the information the rule has and makes use of does not change.

“So to ask a voting rule to be a regular or an SCF to satisfy IIA means exactly that we want it to represent a possible real-world voting rule and not some fairy-world voting rule. In fact, in the Arrow Theorem we can do without the conditions of Universal Domain, Choice Consistency, and Unanimity and still have a recognizable voting rule, but we cannot do without IIA. For it is IIA (along with Nondictatorship) that ensures that the Arrow Theorem is a theorem about real-world voting rules. Without IIA it would be a theorem about a fairy-world, and hence quite uninteresting.”

What do y’all make of this? It looks to me like the Wikipedia claim quoted above refers to a version of IIA that differs from Arrow’s.

The concept that I have always seen IIA used to describe is:
For ranked choice voting, it means that if a candidate is removed from the election, but the remaining candidates are kept in the same order as before on each ballot (e.g. if a ballot votes A>B>C and B is removed, it becomes A>C), then the method should return a ranking that keeps the remaining candidates in the same order as before.
For cardinal voting, it’s the same, except rather than the remaining candidates being kept in the same order as before on each ballot, they have the same rating as before on each ballot.

However, in the paper that you linked to, it appears that the ranked choice version of the criterion I just described is what they mean by Consistency. By IIA, they seem to mean: suppose each voter has a ranking of all the potential candidates who could run (e.g. anyone who would be legally allowed to run for the office is included). Suppose that a voter decides to change how they rank some potential candidates, but do not change how they rank the actual candidates. Then the ranking of the actual candidates that the Social Choice Function outputs should not change.

Rated methods also output a ranking. Their raking is just the candidates ranked by their rating. An example of this is minimax, which you can consider the score it outputs for each candidate being each candidate’s worst defeat, and these are ordered from greatest to least to create a ranking. The scores rated voting methods (as well as some ranked voting methods) output are actually irrelevant to almost all voting criteria since they all compare inputs to outputs and to be compatible with all (single winner deterministic party agnostic) voting methods they use the ranking of candidates that they all output. For the methods that only return a single winner then a complete ranking or set of ratings that can be converted into a rating, you can still create such a ranking by making the 1st candidate the winner, the 2nd candidate the one that would win if the 1st were removed, the 3rd the one if the 1st and 2nd were removed, etc.

Yes, I meant in my post that what makes the cardinal version of the criterion that we have always called IIA (but that I guess the paper is calling Consistency) different from the ordinal version is that in the cardinal version, every voter gives each remaining candidate the same rating (which is a different condition than the ordinal IIA). The required consequence remains the same.

1 Like

Sorry my bad. I skimmed your comment and thought you were talking about the output (since if you remove B, if a voting method outputed A>B>C it must now output A>C).

Well it seems a shame that there’s so much ambiguity about such an important property – a pillar of Arrow’s impossibility theorem. The IIA that is most commonly discussed is different from Arrow’s IIA. Several people have commented on the confusion, including William H. Riker in his 1982 book Liberalism Against Populism (p. 101 and in endnotes). Tideman’s 2006 book Collective Decisions and Voting confirms the view expressed in his 1991 paper with Bordes.

Here is Arrow’s definition from a 2008 entry in the New Palgrave Dictionary of Economics:
“Let S be a set of alternatives. Two profiles which have the same ordering of the alternatives in S for every individual determine the same social choice from S.”

Riker writes (1982, p. 271):
The main reason for the dispute is, I believe, the fact that, when it was first presented in Kenneth Arrow, Social Choice and Individual Values, 2nd ed. (New Haven: Yale University Press, 1963), it was discussed (but not used) as if it forbade variations in orderings of alternatives outside the set of alternatives being considered. Thereby it was made into a consistency condition on the way choices from various subsets of alternatives were nested inside each other, a condition that will be discussed in Chapter 5. It is, however, not a consistency condition on choices from sets of different sizes, but simply a consistency condition on the rule, F, of choice.