Arrow's Impossibility Theorem

Arrow's impossibility theorem is a proposition that it may be impossible to create a consistent collective decisionmaking rule. Consistent collective decision making treats individuals fairly and equitably and increases the likelihood that people will accept the outcome of a social decision. The theorem provides insights into the complexities of social decision making and the difficulties inherent in improving the voting process.

Difficulty in achieving consistent collective decisions is particularly evident when society is made up of more than one person and there are at least three choices being considered. Nobel Prize–winning economist Kenneth Arrow postulates that a voting scheme should satisfy six exhaustive axioms. First, the voting scheme should produce the same result regardless of the configuration of individual voter preferences. If pairwise voting (Choice A vs. Choice B, and the winner goes up against Choice C) is used, a different result may occur depending on which vote is held first (A vs. B or B vs. C).

This voting process may not lead to a single result if the pairwise voting process is allowed to go on indefinitely. This vote cycling is often referred to as the “paradox of voting.” If the voting process is not allowed to go on indefinitely, the one who sets the agenda (voting order) may be able to manipulate the outcome of the election. Neither scenario is desirable for group decision making.

Second, the voting rule should be able to rank all outcomes at the end of the voting process. Third, the ranking of outcomes should be responsive to the individuals in society. Social welfare must be a function of the welfare of the individual in society. Fourth, the outcome of the vote should not violate the law of transitivity. If Choice A is socially preferred to Choice B and B is preferred to C, then A should also be preferred to C. The social voting rule should produce a consistent outcome. Fifth, the outcome should be independent of irrelevant alternatives. If society is ranking Choices A, B, and C, only individual preferences of A, B, and C are relevant. Where individuals would rank Choice D is irrelevant because it is not included in the vote. Preferences concerning Choice D should not influence the vote on Choices A, B, and C. Last, the voting rule [Page 115]should not be solely based on the preferences of one individual in society. The social welfare function should not be determined by a dictator.

Each of these axioms is intuitively pleasing and seems very reasonable. A voting rule that satisfies all these criteria would most likely be accepted by society. However, when grouped together these axioms imply that it is unlikely that a society would be able to create a consistent decisionmaking rule. The results of collective decisions cannot be expected to be as consistent as the results of individual decisions. Social decisionmaking consistency is only possible for some patterns of individual preferences. For example, if all voters have identical preferences for Choices A, B, and C, a voting rule is likely to produce consistent results. According to the theorem, it is not completely impossible for consistent decisions to be made collectively, it is just highly unlikely.

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