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.