Arrow's Impossibility Theorem

Kenneth J. Arrow (1921–) is one of the 20th century's leading economists. His most famous contribution to the social sciences is Arrow's impossibility theorem (Arrow, 1963). Arrow examined the logic of choice by a society based on the preferences of the individuals in that society and identified several reasonable conditions that a social choice rule used by a society should have. The impossibility theorem states that when there are three or more alternatives, it is impossible to find a social choice rule that always aggregates individual preferences into social preferences in a way that satisfies these desirable conditions. Thus, even if each individual's preferences are perfectly consistent and logical, the preferences of society as a whole might be incoherent.

Begin by assuming each individual in a society has preferences over a list of alternatives. A social choice rule is a rule that takes all individual preferences and combines them to determine the preferences of society as a whole. For example, individuals might have preferences over political candidates, and we seek a social choice rule (such as majority rule) to combine their preferences and determine who should take office. Arrow identified five reasonable conditions that a social choice rule should satisfy. The social choice rule should be the following:

• Complete. The social choice rule should provide a complete ranking of all alternatives. That is, for any alternatives A and B, the social choice rule should tell us if A is preferred to B, B is preferred to A, or if there is social indifference between A and B.
• Paretian. If every individual prefers A to B, then the social choice rule should rank A above B. Selecting alternative A over alternative B in this case would be a Pareto-optimal move (one that makes all individuals better off), and the social choice rule should rank A above B.
• Transitive. If the social choice rule ranks A above B and B above C, then A should be ranked higher than C.
• Independent of irrelevant alternatives. The ranking of A compared to B should not depend on preferences for other alternatives.
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• Nondictatorial. The social choice rule should not depend on the preferences of only one individual (a dictator).

The proof of Arrow's impossibility theorem demonstrates that it is impossible to find a social choice rule that always satisfies all five of these conditions when there are three or more alternatives. That is, no matter what rule a society adopts to determine social preferences, it will either have some undesirable “irrational” features or be determined entirely by one person.

The result of the impossibility theorem is troubling to those who study democratic choice. However, relaxing some of the conditions imposed by the impossibility theorem or adding additional restrictions does allow for making social decisions through the political process. An important example is when alternatives are assumed to be arranged along some one-dimensional spectrum, and individual preferences over these alternatives are single-peaked, meaning that strength of preference peaks at the most preferred alternative and decreases across other alternatives as they become further from the most preferred alternative on the spectrum. In this case, with an odd number of individuals, majority voting will lead to adopting the preferences of the median voter (e.g., Black, 1958; Downs, 1957). Societal norms and institutions can also offer a means to escape the implications of the impossibility theorem (e.g., Riker, 1980).

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