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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:

• unrestricted domain or universality: the social choice function should create a complete societal preference order from every possible set of individual preference orders. (The vote must have a result.)
• non-imposition or citizen sovereignty: every possible societal preference order should be achievable by some set of individual prereference orders. (Every result must be achievable somehow.)
• non-dictatorship: the social choice function should not simply follow the preference order of a single individual while ignoring all others.
• positive association of social and individual values or 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. (An individual should not be able to hurt a candidate by ranking it higher.)
• 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. (Removing some candidates should not have an effect on the relative ranking of the remaining candidates.)
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:

• unanimity or Pareto efficiency: if every individual prefers a certain option to another, then so must the resulting societal preference order.
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.