Kenneth Arrow proved no voting system is perfect

Person: Kenneth Arrow
Lived: 1921–2017
Field: Economics
From: American

The story

After wartime service as an Army Air Forces weather officer, Kenneth Arrow returned to graduate study at Columbia and in the late 1940s became a research associate at the Cowles Commission in Chicago, while spending summers at the RAND Corporation in California. He became interested in a simple-sounding question: how can the preferences of many individuals be combined into a single ranking for a whole society? The question matters for elections, committees and any group that must choose between options.

Arrow proved his answer in 1950, and it became the centrepiece of his Columbia PhD dissertation, published in 1951 as Social Choice and Individual Values. His impossibility theorem, which he himself called the General Possibility Theorem, showed that when there are three or more options, no method of turning individual rankings into a group ranking can satisfy a short list of reasonable conditions at the same time. These include respecting unanimous preferences, not letting a single person dictate the result, and keeping the choice between two options independent of irrelevant third options.

The result generalised a much older puzzle. In the eighteenth century the Marquis de Condorcet had shown that majority voting can produce cycles in which A beats B, B beats C and C beats A. Arrow showed that the difficulty is not peculiar to majority rule but affects every ranked method of collective choice. In voting terms, every ranked voting rule is open to spoiler effects, in which adding or removing a minor candidate changes the winner.

Far from being a counsel of despair, the theorem gave scholars a precise map of the trade-offs every voting rule must make. It inaugurated the modern field of social choice theory, became a touchstone in political theory and welfare economics, and inspired later work such as Amartya Sen's studies of liberty and welfare. Arrow himself later acknowledged that rating systems, in which voters grade each candidate separately, fall outside his theorem.

Today his framework is used to compare voting systems, analyse how often different rules produce spoilers, and think about fairness in committees, constitutions and ethical decision-making. In 1972 Arrow shared the Nobel Memorial Prize in Economic Sciences for his contributions to general equilibrium and welfare theory.

Why it matters

Arrow's theorem gave voters, lawmakers and designers of electoral systems a rigorous way to understand what any voting method can and cannot achieve.

About Kenneth Arrow

Economist whose impossibility theorem, general-equilibrium proofs and work on risk, information and health care reshaped economics; still the youngest winner of the Nobel economics prize.

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Kenneth Arrow

1921–2017 · Economics

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