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Arrow's impossibility theorem

In voting systems, Arrow's impossibility theorem, or Arrow's paradox demonstrates the impossibility of designing rules for social decision making that obey a number of 'reasonable' criteria.

The theorem is due to the Bank of Sweden Prize ("Nobel prize in Economics") winning economist Kenneth Arrow, who proved it in his PhD thesis and popularized it in his 1951 book Social Choice and Individual Values.

The theorem's content, somewhat simplified, is as follows. A society needs to agree on a preference order among several different options. Each individual in the society has his or her own personal preference order. The problem is to find a general mechanism, called a social choice function, which transforms the set of preference orders, one for each individual, into a global societal preference order. This social choice function should have several desirable properties:

Arrow's theorem says that such a social choice function does not exist if the number of options is at least 3 and the society has at least 2 members.

Another version of Arrow's theorem can be obtained by replacing the monotonicity criterion with that of:

This statement is stronger, because assuming both monotonicity and independence of irrelevant alternatives implies Pareto efficiency.

With a narrower definition of "irrelevant alternatives" which excludes those candidates in the Smith set, some Condorcet methods meet all the criteria.

See also: Gibbard-Satterthwaite theorem, Voting paradox

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