Posted by: courtneyobrien | November 4, 2008

Arrow’s Paradox

Election Day 2008 is now upon us, so I thought it fitting to expand a bit on Sandeep’s post regarding different voting methods.  As Sandeep showed in his example of the Irish Presidential Election of 1990, different voting methods can lead to very different results.  So which voting method is most fair?

Unfortunately, in 1951, the Nobel Prize-winning economist Kenneth Arrow demonstrated that perfect democratic voting is, not just in practice but in principle, impossible.  Originally, Arrow had set out to identify a fair voting system  which would meet the following criteria:

*  unrestricted domain or universality: The vote must have a result.

* non-imposition or citizen sovereignty: Every result must be achievable somehow.

* non-dictatorship: the social choice function should not simply follow the preference order of a single individual while ignoring all others.

* positive association of social and individual values or monotonicity: (An individual should not be able to hurt a candidate by ranking it higher.

* independence of irrelevant alternatives: Removing some candidates should not have an effect on the relative ranking of the remaining candidates.

However, Arrow found that, if there are three or more discrete options to choose from, no voting system can convert the ranked preferences of individuals into a community-wide ranking while also meeting the above criteria . The collective choice of a society therefore cannot be determined by aggregating the votes of individual.  This dilemma became known as Arrow’s paradox.

However, to say that no voting system is fair is an over-simplifications of Arrow’s result, since he required strong assumptions about what makes a voting method “fair”.  Arrow used the term “fair” to refer to his above criteria, but there is no inherent reason that these criteria should be considered a requirement for fairness, especially since he requires a a ranked ballot.

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Responses

  1. Arrow’s Theorem is often paraphrased as “voting can never be fair” so I’m glad you point out the limitation of this interpretation at the end. For example, usual voting where you get only one choice (instead of ranking your choices) works fine according to these axioms (suitably interpreted).

    Too often mathematics is simplified into a false assertion for the layperson in this way! Well meant, I’m sure, but then it joins the list of false urban legends: you can see the great wall from space, 3/4 of everyone who ever lived is alive today, chewing gum takes 7 years to pass through your digestive system, etc. 🙂

    I’m sure there have been further investigations into systems with other types of ballots (weighting instead of ranking perhaps) and other sets of axioms. Anyone know?

  2. Here’s an interesting voting system:

    Wikipedia: Venetian Doge.

    Read the section titled `Selection of the Doge’, specifically the second paragraph. Discover Magazine reports that historians claim this worked fairly well as a way to keep special interests from having too much sway. (!)


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