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From the Fun Fact files, here is a Fun Fact at the Medium level:

Arrow's Impossibility Theorem

Elections are democracy in action. People go to polls and express their preferences, and somehow we must aggregate the preferences of many individuals to make a joint decision. So the choice of voting method is very important. Is there an ideal voting method?

According a 1950 result by Kenneth Arrow, the answer is "no"---if by "ideal" you mean a preferential voting method that satisfies certain criteria that a "reasonable" voting method should have. For this work, Arrow received the Nobel Prize in Economics in 1972 for what was essentially a mathematical result!

To explain, he assumes a preferential voting method is a social welfare function: voters rank all candidates in order of preference, and based on these rankings, the method produces an outcome---another ranked list of all candidates that is supposed to represent the joint "will of the people".

One might ask a voting method to have these "reasonable" properties:

• No Dictators (ND): the outcome should not always be identical to the ranking of one particular person.
• Pareto Efficiency (PE): if every voter prefers candidate A to candidate B, then the outcome should rank candidate A above candidate B.
• Independence of Irrelevant Alternatives (IIA): the outcome's relative ranking of candidates A and B should not change if voters change the ranking of other candidates but do not change their relative rankings of A and B.
Then Arrow's Impossibility Theorem says: For elections with 3 or more candidates, there is no social welfare function that satisfies ND, PE, and IIA.

Presentation Suggestions:
One often hears people say that Arrow proved "there are no good/fair election methods". This is not true, since there are many election methods that are not covered by the hypotheses of Arrow's theorem. In particular, Arrow's result applies only to methods in which voters rank all candidates, a requirement not satisfied by many popular voting methods, e.g., approval voting or plurality voting. Furthermore, for any given context, one may question whether the "reasonable" criteria are truly reasonable in that context. And if there are only 2 candidates, then it is easy to see that plurality voting (which expresses preference for one candidate over the other) is a social welfare function that satisfies ND, PE, and (vacuously) IIA (since there are no other candidates).

Thus any discussion of Arrow's theorem should be qualified by clarifying the assumptions and conclusions of the result. Nonetheless, the result was a surprising and remarkable achievement.

The Math Behind the Fact:
Arrow's original work gave a larger set of five criteria for a "reasonable" preferential voting method. They include ND and IIA above, as well as these:

• Universality (U): The voting method ranks all candidates and the outcome is deterministic.
• Monotonicity (M): If a voter moves a candidate higher in her rankings, then that candidate should not have a lower ranking in the outcome.
• Citizen Sovereignty (CS): Every ranked outcome should be possible with a suitable set of voter rankings.

Essentially, U says that the voting method is a social welfare function. One can show also that IIA, CS, and M imply PE, so that the version of Arrow's Theorem stated above is stronger and slightly simpler to state. See the reference for a proof. You may also enjoy taking a course in game theory.

Su, Francis E., et al. "Arrow's Impossibility Theorem." Math Fun Facts. <http://www.math.hmc.edu/funfacts>.

References:

Subjects:    other
Level:    Medium
Fun Fact suggested by:   Francis Su
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