The Impossibility

A society must often decide where its members disagree. One person puts the hospital first, another the school, another the road; the available resources do not permit all three. Suppose we know how every person ranks the alternatives but make no comparison of the intensity of their wants. Can a rule combine those rankings into one ordering for the group? Kenneth Arrow gave this question a form exact enough to prove something about it. The result concerns the conditions under which individual judgments can determine a collective judgment. It reaches beyond a particular voting procedure because the procedure itself is what must be chosen. Before asking which social outcome comes first, the theory asks what it would mean for a rule to produce a coherent answer from everyone’s preferences.

The difficulty can be seen with three people and three alternatives. The first ranks A above B above C; the second B above C above A; the third C above A above B. Each ordering is perfectly consistent. Yet two people prefer A to B, two prefer B to C and two prefer C to A. Pairwise majority preference runs in a circle. If sincere votes are taken in successive eliminations, the agenda can decide which alternative survives: first compare A with B, then their winner with C, and C wins; first compare B with C, then their winner with A, and A wins. The problem arises from combining different majorities, rather than from any voter contradicting himself. This is the Condorcet cycle. It shows a failure of majority rule to produce a transitive social ordering on this profile. By itself, it leaves open whether another rule could satisfy the requirements that majority rule misses.¹

Arrow’s theorem answers that further question by specifying a package of demands. There are finitely many people, at least two, and at least three alternatives. Each person has a complete and transitive ordering, and the rule must return such an ordering for society. It must work for every possible profile of the individual orderings: no inconvenient pattern may be excluded in advance. If everyone strictly prefers A to B, society must do so too; this is unanimity, or the weak Pareto condition. Society’s comparison of A with B must depend only on the individuals’ comparisons of those two alternatives, so that changing their rankings relative to C while leaving every A–B comparison intact cannot change the social A–B verdict. This is independence of irrelevant alternatives. Finally, no fixed individual may determine every strict social comparison regardless of the others. The theorem establishes that these demands cannot all be met. Under the remaining requirements, a rule defined on every profile must give that decisive position to one person.²

The proof explains why the conflict extends beyond the familiar cycle. Unanimity ensures that some group is decisive: the whole population can secure A over B by agreeing on it. Among the groups capable of determining some pairwise comparison, take one with the fewest members. Independence allows its authority over that pair to be followed while other rankings change; unrestricted domain permits the profiles needed for the comparison; transitivity connects the resulting social judgments. Arrow shows that a smallest decisive group with more than one member leads to a contradiction, so decisiveness must reach a single person. The same conditions then carry that person’s authority from one pair to every pair. Dictatorship here has an exact mathematical meaning: whenever that person strictly prefers one alternative, society does too, whatever anyone else thinks. It does not assert that the person controls an actual government, nor does it say what society must do when that person is indifferent. The force of the result comes from the role defined by the conditions.³

The conditions also show where the result stops. With only two alternatives and an odd number of at least three voters who rank them strictly, ordinary majority voting supplies a consistent social ranking. It respects unanimity and depends on those pairwise rankings alone; no fixed voter always controls it. There is no third alternative around which the comparisons can form a cycle. Likewise, seeing a coherent result in one election with many alternatives does not refute the theorem. The demand is for one rule that meets the requirements on every admissible profile, including profiles unlike the observed one. Restricting the allowable profiles changes that demand and may permit rules that the unrestricted theorem excludes. These limits matter because the collective decision before us is always particular. The theorem establishes an incompatibility among general requirements; it does not pronounce every actual disagreement incapable of resolution or every collective ranking incoherent.⁴

A points system makes one possible trade-off concrete. Give two points for first place, one for second and none for third, then rank alternatives by their totals. With two voters ranking A–C–B and B–A–C, A receives three points and B two. Change the rankings to A–B–C and B–C–A: now A receives two and B three. The first voter still prefers A to B, and the second still prefers B to A. Only their placements relative to C have changed, yet the social A–B ranking reverses. The points rule can return an orderly result by using information that independence forbids it to use for that comparison. Its violation is identifiable, and its merits can be argued on that basis. More generally, adding comparable measures of well-being would enlarge the information available beyond bare individual rankings. That move requires a justification for the comparisons and the way they are combined; the theorem does not supply either by ruling out the narrower package.⁵

The impossibility leaves the common decision with choices to make about its own foundation. A community can debate the information its rule should use, the range of disagreements it must accommodate and the consistency demanded of its conclusions. It can compare procedures by the consequences of the conditions they relax. Those are substantive judgments, and the theorem makes their cost more visible. The inability to derive every desired property from bare rankings does not remove the need to judge; it identifies what those rankings alone cannot deliver under the stated demands. The next plate, The Hedonimeter, returns to the hope of acquiring richer information: a direct reading of experience rather than an ordering inferred from choice. Such a reading would answer a different question. Whether it could be compared across people, and how that comparison should guide their common life, would still require an account of its own.