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Arrow's impossibility theorem

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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 ('Nobel') Prize 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:

  • universality: the social choice function should create a complete societal preference order from every possible set of individual preference orders.
  • positive association of social and individual values (monotonicity): if an individual modifies his or her preference order by promoting a certain option, then the societal preference order should change only by (possibly) promoting that same option.
  • independence of irrelevant alternatives: if we restrict attention to a subset of options, and apply the social choice function only to those, then the result should be compatible with the outcome for the whole set of options.
  • citizen sovereignty: every possible societal preference order should be achievable by some set of individual prereference orders.
  • non-dictatorship: the social choice function should not simply follow the preference order of a single individual while ignoring all others.
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.

With a narrower definition of "irrelevant alternatives" which excludes those candidates in the Smith set, Condorcet's method meets all the criteria.

See also: Voting paradox

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