Sometimes known as the Arrow Paradox, after
Kenneth J. Arrow.
Three candidates are vying for the position of Governor:
Achilles,
Carroll, and
Zeno. In a
poll, 2/3 of the
voters polled prefer Achilles over Carrol, and 2/3 prefer Carroll over Zeno.
You would expect that most voters would therefore prefer Achilles over Zeno.
Not necessarily...
If the voters ranked their preferences as follows, most preferred to least preferred, each candidate can say he is preferred over another candidate:
1/3: A C Z
1/3: C Z A
1/3: Z A C
This seeming paradox occurs because we expect the problem to obey the
mathematical law of
transitivity. As a problem in
Game Theory, it is not bound by transitivity.
Any situation where at least 3 alternatives are ranked pairwise by at least 3 criteria can give rise to this sort of paradox. For instance, a man choosing between three possible brides may evaluate their respective strengths in sexual skill, economic advantage, and social grace. Assuming each criterion carries equal weight, he may find himself in the predicament of preferring Amy to Bev, Bev to Chrissy, and Chrissy to Amy.
A Blather of Paradoxes