Who's Counting: Non-Transitivity in Baseball, Medicine, Gambling and Politics

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Non-Transitive Dice

Non-transitivity is common in probability and election theory. One well-known example from Bradley Efron involves four dice, A, B, C and D. Being dice, they all have six faces, but these dice are strangely numbered as follows: A has a 4 on four faces and a 0 on two faces; B has 3's on all six faces; C has four faces with a 2 and two faces with a 6; and D has a 5 on three faces and a 1 on three faces.

If die A is rolled against die B, die A will win -- by showing a higher number -- two thirds of the time. Similarly, if die B is rolled against die C, B will win two thirds of the time; and if die C is rolled against die D, it will win two thirds of the time.

Nevertheless -- and here's the punch line -- if die D is rolled against die A, it will win two thirds of the time. A beats B, B beats C, C beats D, yet D beats A, all two thirds of the time.

(You might even profit from this by challenging someone to choose whatever die he or she wanted, and you could then choose a die that would beat it two thirds of the time. If they choose die B, you choose A; if they choose A, you choose D, and so on.)

Politics, Elections, and Non-Transitivity

Versions of non-transitivity crop up in sociological and political discussions where trait X is correlated with Y and Y with Z and so on.

Election preferences provide another example of non-transitivity. A nice illustration of electoral non-transitivity arises if we tweak the recent senatorial election in Alaska, where Lisa Murkowski was just recently declared the winner. Let's imagine that, contrary to fact, the electorate there ranked the three major candidates (all M's, by the way), Lisa Murkowski, Joe Miller, and Scott McAdams.

Let's further imagine that faction A, roughly one third of the electorate, preferred Murkowski to Miller to McAdams; faction B, also about one third of the electorate, preferred Miller to McAdams to Murkowski; and faction C, the remaining one third of the electorate, favored McAdams to Murkowski to Miller.

If this had been the case, a clear majority of the electorate - factions A and C - would have preferred Murkowski to Miller, and a clear majority - factions A and B -- would have preferred Miller to McAdams. Yet a clear majority -- factions B and C -- would have preferred McAdams to Murkowski.

Presidential primary season, say involving Republicans Palin, Romney, Huckabee, and Pawlenty in 2012, might easily produce such non-transitive preference rankings.

Finally, note that understanding why transitivity often fails is correlated with vast personal wealth. Well, maybe not ... but, still, it's important.

John Allen Paulos, a professor of mathematics at Temple University in Philadelphia, is the author of the best-sellers "Innumeracy" and "A Mathematician Reads the Newspaper," as well as, most recently, "Irreligion." He's on Twitter and his "Who's Counting?" column on ABCNews.com usually appears the first weekend of every month.

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