Arrow's Impossibility Theorem
I seem to be posting a lot about paradoxes recently. I'll probably take a little break from paradoxes after this one.
Summary #
In 1951, Kenneth Arrow demonstrated that it is not possible to have a "fair" voting system that satisfied the following three criteria (imagine the group is voting on which fruit to eat: apples or pears):
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If every voter prefers apples to pears, then the group prefers apples to pears. (Sound familiar? It's called Pareto efficiency.)
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If every voter prefers apples to pears, then even if bananas are added to the set of options, the group will still prefer apples to pears.
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There is no dictator.
This is known as Arrow's Impossibility Theorem.
(Photo: Wikimedia)
Commentary #
The actual details of the theorem are interesting, and I refer you to Wikipedia (for those who are interested). There are situations, however where item 2 (where we added bananas) doesn't hold: imagine the game rocks-paper-scissors. In such a case, adding an alternative transforms the straightforward choice into a cyclic choice. I sometimes see this scenario when people compare different aspects of multiple candidates' platforms (or when they're choosing which car / laptop / soap / pants to purchase).
Sometimes, the trade-offs are hard; but sometimes they're impossible.