Thursday, March 15, 2007

Boggling Electoral Outcomes

Sasha Volokh observes a likely violation of Independence of Irrelevant Alternatives (IIA) in the upcoming French election. For those who don’t know, IIA is the idea that if option A beats option B when there are no other alternatives, then A should still beat B when option C is introduced. Recent polls in France indicate that candidate A (Francois Bayrou) would beat candidate B (Nicolas Sarkozy) in a two-way race, and A would also beat candidate C (Segolene Royal) in a two-way race. But A is nevertheless likely to lose the three-way race among A, B, and C. The reason is that A is probably the second choice of many supporters of both B and C (and some other candidates), so he would gain a lot of those votes if either B or C dropped from the race; yet A is not the first choice of enough voters to make the run-off election.

This result is not a fault of the run-off system. If the French held a single election and chose whichever candidate got the plurality of votes, A would still win in a two-way race against either B and C, while he would lose in the three-way race.

This violation of IIA reminded me of another such violation I observed about a month ago, coincidentally while attending a game party hosted by Sasha’s brother Eugene: the game of Boggle. I’m okay at Boggle, but I’m no master. Unfortunately, the people most willing to play against me are masters, so I get beaten a lot. But there’s one circumstance in which I occasionally win (though not this last time, sadly): when I play against two masters. Boggle rules stipulate that when any player finds a word, nobody gets credit for it. As a result, the masters will sometimes wipe out each other’s scores, allowing me to win if I can just find two or three words neither of them found.

Most multi-player games probably violate IIA, because most of them have interactive effects that change with added players. But the IIA-violation is most noticeable in games like Boggle and Scattergories, whose rules makes each player’s score a function of how many unique answers the player has. Note the similarity to most popular electoral systems: what matters most for staying in the race is a candidate’s number of “unique” voters, that is, the voters whose first choice is that candidate.


Ran said...

Maybe I'm missing something, but doesn't Arrow's Theorem guarantee that any decent voting system will violate IIA?

Glen Whitman said...

Ran -- not necessarily. I believe Arrow's Theorem proves that at least one of Arrow's social choice criteria will be violated; it could be IIA, or it could be one of the others, such as the Pareto Principle or Non-dictatorship.

Ran said...

Well, I said decent voting system. I don't think a voting system that violates non-dictatorship can really be considered "decent". Pareto efficiency is maybe less obviously necessary than non-dictatorship, but consider that IIA means a minority can sometimes get what they want, while lack of Pareto efficiency means that sometimes *everyone* agrees the wrong thing was chosen. (I think lack of Pareto efficiency is kind of a theoretical thing; its technically possible to keep IIA and non-dictatorship by giving up Pareto efficiency, but no one could accidentally design such a system and think it made sense.)

Ken Maurer said...

IIA probably still applies to all the individual preferences. The problem is that IIA rarely applies to voting schemes with more than 2 alternatives. The only thing special about the upcoming French election is that the example departs from Arrow, et al's abstract alternatives to concrete French elections, where it was first examined by Condorcet, Borda and others over 200 years ago.