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The theorem is named after economist Kenneth Arrow , who demonstrated the theorem in his Ph.D. thesis and popularized it in his 1951 book '' Social Choice And Individual Values ''. The original paper was entitled "A Difficulty in the Concept of Social Welfare". Arrow, K.J., "A Difficulty in the Concept of Social Welfare", '' Journal Of Political Economy '' 58(4) (August, 1950), pp. 328–346. Arrow was a co-recipient of the 1972 Nobel Prize In Economics .


STATEMENT OF THE THEOREM


The need to aggregate , where one attempts to find an economic outcome which would be acceptable and stable; in decision making, where a person has to make a rational choice based on several criteria; and most naturally in Voting Systems , which are mechanisms for extracting a decision from a multitude of voters' preferences.

The framework for Arrow's theorem assumes that we need to extract a preference order on a given set of options (outcomes). Each individual in the society (or equivalently, each decision criterion) gives a particular order of preferences on the set of outcomes. We are searching for a Preferential Voting system, called a ''social welfare function'', which transforms the set of preferences into a single global societal preference order. The theorem considers the following properties, assumed to be reasonable requirements of a fair voting method:
  • non-dictatorship: the social welfare function should account for the wishes of multiple voters. It can't simply mimic the preferences of a single voter.

  • unrestricted domain or '''universality''': the social welfare function should account for all preferences among all voters to yield a unique and complete ranking of societal choices. Thus, the voting mechanism must account for all individual preferences, it must do so in a manner that results in a complete ranking of preferences for society, and it must Deterministically provide the same ranking each time voters' preferences are presented the same way.

  • Independence Of Irrelevant Alternatives (IIA): the social welfare function should provide the same ranking of preferences among a subset of options as it would for a complete set of options. Changes in individuals' rankings of ''irrelevant'' alternatives (ones outside the subset) should have no impact on the societal ranking of the ''relevant'' subset.

  • positive association of social and individual values or ''' Monotonicity ''': if any individual modifies his or her preference order by promoting a certain option, then the societal preference order should respond only by promoting that same option or not changing, never by placing it lower than before. An individual should not be able to hurt an option by ranking it ''higher''.

  • non-imposition or '''citizen sovereignty''': every possible societal preference order should be achievable by some set of individual preference orders. This means that the social welfare function is Onto : It has an unrestricted target space.


Arrow's theorem says that if the decision-making body has at least two members and at least three options to decide among, then it is impossible to design a social welfare function that satisfies all these conditions at once.

A later (1963) version of Arrow's theorem can be obtained by replacing the monotonicity and non-imposition criteria with:
  • Pareto Efficiency : if every individual prefers a certain option to another, then so must the resulting societal preference order. This, again, is a demand that the social welfare function will be minimally sensitive to the preference profile.


This version of the theorem is stronger—has weaker conditions—since monotonicity, non-imposition, and independence of irrelevant alternatives together imply Pareto efficiency, whereas Pareto efficiency, non-imposition, and independence of irrelevant alternatives together do not imply monotonicity.


FORMAL STATEMENT OF THE THEOREM