Arrow's Impossibility Theorem

Background: the voting paradox

In eighteenth-century France, Borda (1781) proposed scoring rankings, and Condorcet (1785) proposed pairwise majority contests. Condorcet discovered that if three voters rank A>B>C, B>C>A and C>A>B, majority rule produces a cycle: A beats B, B beats C and C beats A. This is the Condorcet paradox, or voting paradox: individuals can each have consistent preferences while the group's preference is inconsistent. Duncan Black (1948) showed that when preferences are "single-peaked" along one dimension, cycles disappear and the median voter's preference wins.

What the theorem says

Kenneth Arrow generalized the problem in a 1950 paper and in Social Choice and Individual Values (1951). With three or more alternatives, he asked that a rule for forming social preferences meet these conditions. Unrestricted domain: it must produce a result for any profile of individual preferences. Pareto: if everyone prefers A to B, so must society. Independence of irrelevant alternatives: society's ranking of A and B must depend only on individuals' preferences between A and B. Non-dictatorship: no single person's preference may always become society's. Requiring social preferences to be transitive, Arrow proved no rule satisfies all four at once, so majority rule, the Borda count, runoffs and every other ranking-based method violates some condition. Geanakoplos (2005) gave three short proofs. Arrow received the Nobel Prize in economics in 1972.

Extensions and interpretive debates

Amartya Sen (1970) showed the "impossibility of a Paretian liberal," a conflict between minimal individual liberty and the Pareto condition, and argued that adding information such as interpersonal utility comparisons can escape impossibility; he received the 1998 Nobel Prize. Gibbard (1973) and Satterthwaite (1975) proved that with three or more alternatives no non-dictatorial rule is immune to strategic misrepresentation, adding game-theoretic implications. May (1952) showed that with two alternatives, majority rule is the unique rule meeting basic fairness conditions. William Riker (1982) drew on these results for a liberal reading of democracy in which the very idea of a "will of the people" is unstable, strongly influencing public choice theory. Others, such as Balinski and Laraki (2011), argue that evaluative voting with grades or scores instead of rankings can sidestep some of Arrow's conditions.

FAQ

Does Arrow's theorem mean democracy is impossible?

No. It shows there is no perfect way to aggregate rankings, so every voting system compromises by giving up some condition; which condition to relax is a choice of institutional design.

Does impossibility arise with only two alternatives?

No. With two alternatives, simple majority rule satisfies all the conditions; impossibility appears only with three or more.

Sources reviewed: 2026-10-26

References

1. de Condorcet, N. (1785). Essai sur l'application de l'analyse à la probabilité des décisions rendues à la pluralité des voix. Paris: Imprimerie Royale.
2. Borda, J.-C. de (1781). Mémoire sur les élections au scrutin. Histoire de l'Académie Royale des Sciences.
3. Black, D. (1948). On the rationale of group decision-making. Journal of Political Economy, 56(1), 23–34.
4. Arrow, K. J. (1950). A difficulty in the concept of social welfare. Journal of Political Economy, 58(4), 328–346.
5. Arrow, K. J. (1951). Social Choice and Individual Values. New York: Wiley.
6. May, K. O. (1952). A set of independent necessary and sufficient conditions for simple majority decision. Econometrica, 20(4), 680–684.
7. Sen, A. K. (1970). Collective Choice and Social Welfare. San Francisco: Holden-Day.
8. Sen, A. (1970). The impossibility of a Paretian liberal. Journal of Political Economy, 78(1), 152–157.
9. Gibbard, A. (1973). Manipulation of voting schemes: A general result. Econometrica, 41(4), 587–601.
10. Satterthwaite, M. A. (1975). Strategy-proofness and Arrow's conditions. Journal of Economic Theory, 10(2), 187–217.
11. Riker, W. H. (1982). Liberalism against Populism. San Francisco: W. H. Freeman.
12. Geanakoplos, J. (2005). Three brief proofs of Arrow's impossibility theorem. Economic Theory, 26(1), 211–215.
13. Balinski, M., & Laraki, R. (2011). Majority Judgment. Cambridge, MA: MIT Press.