Arrow's Impossibility

No ranked voting rule is fair, decisive, and non-dictatorial — pick at most two.

The idea

Arrow's impossibility theorem sets a limit on ranked voting. A voting rule takes one ranking of the alternatives from each voter and returns a single ranking for the group. Four conditions are asked of such a rule.

- Unrestricted domain: the rule returns a group ranking for every possible combination of individual rankings. - Pareto efficiency: if every voter ranks $A$ above $B$, so does the group. - Independence of irrelevant alternatives: the group's verdict on $A$ against $B$ depends only on how the voters rank $A$ against $B$, never on where anyone places a third alternative $C$. - Non-dictatorship: there is no single voter whose own ranking is always the group's ranking, whatever the others submit.

Theorem (Arrow's impossibility theorem).

When there are at least three alternatives, no voting rule satisfies all four of unrestricted domain, Pareto efficiency, independence of irrelevant alternatives, and non-dictatorship. Equivalently, every rule with unrestricted domain, Pareto efficiency, and independence of irrelevant alternatives is a dictatorship.

The second form is the sharper one: the first three conditions are consistent — rules satisfying them exist — but each such rule hands the entire decision to one voter. Every ranked voting system in use gives up one of the four conditions, and which one it gives up determines how it can misbehave.

Ways to work on it

• Walkthrough. Majority voting cycles on 3 voters / 3 candidates — and Arrow's theorem says why nothing better exists.
• Practice. Work through Arrow's theorem: no ranked voting rule satisfies all the fairness axioms at once.
• Hardest. Identify which of Arrow's axioms a real voting system gives up.

Not sure where to start? Take the ten-question placement test.