Arrow's impossibility theorem
Statement
For three or more alternatives, no social welfare function that aggregates individual complete, transitive preference rankings into a single collective ranking can simultaneously satisfy: unrestricted domain (works for any profile of individual preferences), Pareto efficiency (if every voter prefers to , so does society), independence of irrelevant alternatives (society's ranking of vs. depends only on individuals' rankings of vs. ), and non-dictatorship (no single voter's preferences always determine society's ranking).
Why is it true?
It feels like voting should be able to turn everyone's honest rankings into one fair collective ranking. Arrow's theorem shows this is a mirage once there are at least three options: any rule that never lets one voter dictate the outcome, and that judges versus using only opinions about versus , can be pushed by some combination of individually reasonable preferences into producing a collective ranking that cycles or otherwise fails to be consistent.
Proof sketch
A standard proof (the 'pivotal voter' argument) starts from a profile where everyone ranks last; by Pareto, society ranks last too. Moving to the top of each voter's ranking one at a time, some voter is 'pivotal': the first whose change flips society's ranking of from last to not-last. Using independence of irrelevant alternatives and further profile manipulations, one shows this pivotal voter is in fact a dictator over every pair of alternatives, contradicting non-dictatorship — unless one of the axioms fails.
Topics that use this theorem
Step-by-step proofs
No step-by-step proof yet for this theorem.
References
- Kenneth J. Arrow (1951). Social Choice and Individual Values
- Kenneth J. Arrow, Amartya K. Sen, Kotaro Suzumura (eds.) (2002). Handbook of Social Choice and Welfare, Vol. 1