I am a big fan of paradoxes, and I just came across one recently. It’s called voting paradox, first noted by 18th century French philosopher Condorcet. Before getting into the details of this paradox, let’s examine a probability oddity of four special dice.
Die A: {4, 4, 4, 4, 0, 0} [It has 4's on four sides, and 0's on two sides.]
Die B: {3, 3, 3, 3, 3, 3} [It has 3's on all six sides.]
Die C: {2, 2, 2, 2, 6, 6}
Die D: {5, 5, 5, 1, 1, 1}
If die A is rolled against B, die A will win on average. (Because 4 out of 6 times die A will roll a 4 which beats die B.) Similarly, if die B is rolled against die C, die B will win two out of three times; if die C is rolled against die D then die C will win two-thirds of the time. And here’s the kicker: if D is rolled against A, D will win two-thirds of the time!
A beats B, B beats C, C beats D and D beats A!
This kind of non-transitivity is also found in Condorcet’s voting paradox. Here’s a slightly revised version of the original paradox: consider three candidates running for an election – Ina, Mina and Dika. Let’s assume that one-third of the electorate prefers Ina to Mina to Dika, another one-third prefers Mina to Dika and Ina, and remaining one-third prefers Dika to Ina and Mina.
Group #1 = Ina–>Mina–>Dika
Group#2 = Mina–>Dika–>Ina
Group #3 = Dika–>Ina–>Mina
Note that the individual voters (and each of the three groups above) are rational; they have transitive preferences — if they prefer Ina over Mina, and Mina over Dika, then they prefer Ina over Dika.
Now if you consider the entire group of electorate, two-thirds of them prefer Ina over Mina (i.e., group #1 and #3), two-thirds of them prefer Mina over Dika, and two-thirds of them prefer Dika over Ina! And that’s a paradox. Although individual voters have transitive preferences, the “society” has non-transitive preferences. A majority (two-thirds) of the voters prefers Ina over Mina, Mina over Dika and also Dika over Ina. Even if voters are rational, the society is not!
Can we ever derive societal preference from individual preferences? Perhaps not!
[Source: Innumeracy by John Allen Paulos]
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Earlier on this blog: The Monty Hall Paradox, A Mathematical Conundrum
