Irrelevant alternatives aren’t

Every now and then someone discovers Arrow’s Impossibility Theorem and gets all excited. “Democracy is impossible! Let’s have a dictator!” they declare from the rooftops, or words to that effect. I certainly found this fascinating when I first discovered it.

Eventually they calm down. There are a lot of commonly mentioned reasons why Arrow’s impossibility theorem doesn’t have massive real world consequences – it’s not like anyone thought they were using a perfectly fair voting system in the first place, and the mechanism described in the theorem doesn’t actually correspond that closely with real world votes, which are mostly just trying to elect a single winner and don’t require nearly so strong consistency.

One reason I haven’t seen mentioned is the following: If it were possible to create a voting system which satisfied the criteria of Arrow’s impossibility theorem, it would be a bad idea. Independence of irrelevant alternatives, that the ordering of A and B doesn’t depend on the introduction of C, is an appealing condition on the face of it, but it turns out that you don’t actually want real world voting systems to have it. Consider the following set of opinions:

A > B > C
B > C > A
C > A > B
B > C > A

The numbers work out as follows:

There is a 50-50 split on A > B.
75% of people think that B > C
75% of people think that C > A.

Therefore even though there is a tie between A and B, the only fair combined ordering is that B > C > A – any other ordering would make a lot more people unhappy. So the introduction of an apparently irrelevant alternative has taken a tie between A and B and broken it decisively.

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6 thoughts on “Irrelevant alternatives aren’t

  1. Michael Chermside

    Very interesting! I’d never seen this before.

    I had long thought that we could probably live with something that violated independence of irrelevant alternatives, and probably also with something that violated unrestricted domain (sometimes the vote would result in a “can’t decide”. So what? Just re-hold the vote — as long as the “can’t decide” regions are small and unlikely, enough minds will change to produce a result. If this is rare enough, we can live with it). But I had never seen a simple, understandable illustration of why it’s GOOD to violate independence of irrelevant alternatives.

  2. david Post author

    Thanks! Glad you found it interesting.

    I too haven’t seen it anywhere else, and I was a bit surprised by that because it is in some sense “obvious” that such an example has to exist: There is a natural (if inconveniently intractable for large numbers of items) fairest voting algorithm, which is simply to minimize the number of pairwise disagreements (of course this doesn’t produce a unique solution in general, but often it does), and that voting algorithm can’t satisfy independence of irrelevant alternatives by arrow’s theorem. Once you see that it’s easy to extract an example that looks like the above.

    It was however nice to see that the resulting example was so straightforward and intuitive.

  3. Ted Stern

    Try using Range Voting, AKA Score Voting or Cardinal Ratings.

    Because you rate candidates, instead of ranking them, and just sum up the ratings, you are not subject to Arrow’s theorem.

    The winner is still B (assuming some rating such as 10 for the top rank, 9 for second, 0 for third).

    That doesn’t mean that irrelevant alternatives can’t have an effect … it depends on how people set up their ratings. Without sufficient contrast, you won’t get people rating two similar candidates with similar scores. Besides that, having more candidates in the race means that you don’t get an implicit agreement to avoid tough issues, as you do in a two-party duopoly.

  4. Andrew Dale

    Dear David,
    You may be interested to learn that something similar was published by I.J. Good in a short note (C93. On the relevance of imaginary alternatives) in the Journal of Statistical Computation and Simulation 12, No. 3 (1981), pp. 313-5. This has recently been reprinted in The Good Book: Thirty Years of Comments, Conjectures and Conclusions by I.J. Good, edited by David Banks and Eric P. Smith.
    Yours sincerely,
    Andrew Dale.

  5. david Post author

    Hi Andrew,

    That’s good to know! I’m not at all surprised that the observation is not original to me. Thanks for the reference. I’ll check it out.

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