Impossibility Theorem

The first statement of the impossibility theorem is in Kenneth J. Arrow's Social Choice and Individual Values, 1951. This set theoretic theorem, based on the assumptions of rational choice theory, is a major social scientific finding that was partly responsible for Arrow's receipt of the Nobel Prize, shared with John R. Hicks, in economics in 1972, and the theorem served as the point of origin for the field of social choice. The theorem led to the dissolution of the early twentieth-century research paradigm of welfare economics, it led to a reevaluation of how democratic decision procedures arrive at representative expressions of individuals' preferences, and it proved that under most circumstances, it is impossible to have a valid statement of public interest or social welfare. The result has been used to challenge the cogency of the concept of “the public” as a meaningful social entity.

The impossibility theorem assumes that agents have complete and well-ordered preferences over all the outcomes under consideration in a collective choice situation. This requires that agents know whether they prefer, disprefer, or are indifferent between all two pairs of possible outcomes, and it requires that agents' preferences obey the logical relationship of transitivity requiring that if Adams is preferred to Madison, and Madison is preferred to Washington, that in turn Washington cannot be preferred to Adams. The impossibility theorem considers cases in which three or more agents make a collective choice from three or more alternatives in situations as diverse as democratic voting, establishing public policies that reflect social welfare, and the marketplace. The theorem is constructed to resolve the question of whether there is any mathematical procedure for amalgamating individual preferences that results in a collectively rational preference ordering of all the possible outcomes.

In addition to assuming that individuals' preferences are rational, the theorem stipulates that four minimal conditions must apply to the decision procedure so that its result is valid. The theorem requires that individuals be permitted to have any rational preference ordering over alternatives, that there not be a single dictator whose preference over a single pair of alternatives holds for the group decision, that the collective ranking over outcomes remains unchanged if one of the alternatives ceases to be considered, and that the collective choice uphold Vilfredo Pareto's condition holding that a unanimous preference over a pair of outcomes implies a collective preference over that pair. Most researchers agree that although these requirements are normative, they are sufficiently mild and beyond controversy.

The theorem proves that given these minimal assumptions, it is impossible to construct any procedure that results in a collectively rational expression of individual desires. Though highly technical in its statement, the theorem has important implications for philosophies of democracy and political economy. The theorem disregards the cogency of democratic will formation that rests on the assumptions of a “general will” suggested by Jean-Jacques Rousseau, or of a discursive public sphere advocated by John Dewey and Jürgen Habermas. The theorem rejects the possible formation of democratic will by relying on experts who paternalistically apply knowledge of what is best for a population. The theorem also denies that there could be either objective basic needs, or universal moral criteria, that any procedure for collective decision making should recognize, such as minimal nutrition standards or human rights.

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