In the collection Paradoxes and puzzles
What is Arrow's impossibility theorem?
Kenneth Arrow proved that no method of turning voters' ranked preferences into one group ranking can satisfy a short list of reasonable fairness conditions once there are three or more options.
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The short answer
Arrow's impossibility theorem is a result in social choice theory. It says that no procedure for combining individuals' rankings of alternatives into a single collective ranking can meet a set of conditions about fairness and rationality at the same time, as long as there are more than two alternatives. Kenneth Arrow himself called it the general possibility theorem. The options could be policies, candidates or anything else people can rank. [1] [2]
The question Arrow asked
Arrow asked which procedures can derive a social ordering of alternatives, from better to worse, from what is known about people's preferences. He listed properties such a procedure might be expected to have and then asked which procedures have them all. His answer was that none do. His framework is now widely used in welfare economics, and the theorem set the agenda for modern social choice theory. [1]
The result builds on the paradox of voting, found by the Marquis de Condorcet in 1785. Under pairwise majority decision, three voters ranking three options as ABC, BCA and CAB produce a cycle in the group's preferences rather than a clear ordering. [1]
The conditions that cannot all hold
The Stanford Encyclopedia of Philosophy lists five conditions. Unrestricted domain means the procedure must handle any combination of individual preferences. Social ordering means the result is always a proper ordering, never a cycle. Weak Pareto means that if everyone strictly prefers one option to another, so does the group. Non-dictatorship means no single person's strict preferences always become the group's. Independence of irrelevant alternatives means the group's comparison of two options depends only on how individuals rank those two. [1]
Arrow's theorem states that with more than two alternatives, no social welfare function satisfies all five. Other summaries group the conditions slightly differently, for example treating transitivity as its own axiom, but they describe the same result. A popular teaching version says that no ranked-ballot voting system with three or more candidates can satisfy all the listed criteria. [1] [2] [3]
What it does and does not mean
The theorem does not say voting is pointless or that every system gives bad results. It says every ranking-based method must give up at least one of the conditions. A university textbook puts it this way: any ranked-choice method for a single-winner election violates at least one of the majority, monotonicity, Condorcet or independence criteria. Some methods fail independence of irrelevant alternatives while still satisfying the Pareto condition. Borda counting is one example. Pairwise majority decision satisfies independence but fails the requirement of always producing an ordering. [4] [1]
Arrow's original proof appeared in 1951. Later authors have published variants and alternative proofs. [1]
Sources
- Arrow’s Theorem (Stanford Encyclopedia of Philosophy) (plato.stanford.edu) Found by source search. Not checked by a person.
- Arrow's impossibility theorem (oxfordreference.com) Found by source search. Not checked by a person.
- Maths in a minute: Arrow's theorem | plus.maths.org (plus.maths.org) Found by source search. Not checked by a person.
- 7.4 Arrow's Theorem, Conclusions and Exercises (mathbooks.unl.edu) Found by source search. Not checked by a person.
Questions to follow next
- What is the Condorcet paradox and how does it produce cycles?
- Which voting methods break independence of irrelevant alternatives?
- How have economists and philosophers responded to Arrow's theorem?