A person can want to choose his own reading and also want to choose somebody else’s. Preference records both desires without first deciding which carries a right. A society that means to respect personal liberty has to make that further distinction: some decisions are to belong to the person whose life they concern, even when others have views about them. Yet the same society may accept an apparently generous rule of common choice, that one outcome should rank above another whenever everybody strictly prefers it. Sen showed that these commitments can come into conflict. The trouble reaches beyond the case in which a majority overrides one dissenter. Even unanimity, combined with a very small sphere of individual decisiveness, can prevent a social ranking from yielding a best available alternative. The question is how much the satisfaction of preference can settle when the authority to choose is itself at issue.1
The theorem asks remarkably little of the liberty it protects, but something important of the rule’s reach. Every logically possible collection of individual rankings must be admitted. At least two people must each be decisive over at least one pair of alternatives: whichever member of that pair the person prefers must also be socially preferred to the other. These protected pairs are part of the rule, rather than favors reassigned whenever a difficult preference appears. Finally, the social relation must provide at least one best alternative in every nonempty available set, an alternative at least as good as every other member. Full transitivity of society’s ranking is unnecessary for the contradiction, and the proof imposes no independence-of-irrelevant-alternatives condition. Sen’s minimal liberty is weaker than giving every person command of a whole private domain. If even this smaller guarantee conflicts with the other demands, a larger guarantee cannot escape merely by being larger.2
Sen’s book example gives the conflict three possible outcomes. In x, the first person reads; in y, the second reads; in z, neither reads. The first would most like nobody to read the book, but would rather read it himself than have the second person exposed to it. His order is z, x, y. The second would most like the first to read it, enjoys the prospect of his encounter with the unwelcome book, and would next prefer to read it himself rather than have nobody read it. His order is x, y, z. Each ranking is perfectly coherent. Each also includes a judgment about the other person’s conduct. Give the first person the decision between his reading and nobody’s reading; give the second the analogous decision about his own reading. The protected choices are small enough to seem readily defensible.3
The first right requires society to prefer z to x, since the first reader chooses to abstain. The second requires y to rank above z, since the second reader chooses to read. But both people prefer x to y; unanimity therefore requires x above y. The resulting circuit is complete: y beats z, z beats x, and x beats y. Every alternative has another ranked above it. Society can physically select an outcome, but none is best under the relation that the three requirements have forced upon it. This failure does not come from adding up incomparable satisfactions or choosing the wrong voting threshold. There are only two people, and no measurement of how intensely either wants anything is needed. The rights and the unanimous comparison supply every link.
The contradiction identifies a limit on the universal rule, rather than an illness of every liberal decision. Many patterns of preference allow the rights and unanimity to coexist without a cycle. Sen’s unrestricted domain includes the difficult patterns too, including preferences about what another person should do. One response is to restrict which patterns the rule must handle; another is to limit the force of unanimity where it conflicts with a protected decision. Either changes an assumption, and its justification has to be supplied. The unanimous preference for the first person’s reading cannot by itself settle who has authority over that reading, because it is precisely the comparison that closes the circuit. Counting desires gives no instruction for resolving the collision with rights. A theory that treats every desire as decisive whenever everybody happens to share it has already made a substantive choice about the standing of personal liberty.4
The formal right in this proof is itself only part of freedom. In his later Nobel lecture, Sen distinguished the opportunity to achieve what one would choose in a private sphere from the process of being directly in charge of the choice. He acknowledged that the earlier formulation concentrated on the first and could not exhaust the second. Someone else’s arranging the outcome I happen to favor does not establish that I was allowed to decide it; equally, formal control can be inadequate if the opportunity to act is empty. This distinction deepens the question beyond the theorem’s machinery. Rights concern how decisions are made as well as which outcomes people prefer. Sen’s proposed direction is an explicit evaluation of the priorities between liberty and the fulfillment of desires, informed by people’s political values as well as their rankings of outcomes. More precise measurement of the same desires cannot do that work alone.5
A society can honor personal decisions and give weight to what people prefer. What it cannot have is Sen’s universal guarantee that every possible collection of preferences will satisfy all these requirements together. The distinction leaves room for institutions and judgment, while requiring them to say which claim has priority when the claims collide. Drawing a private boundary is one such judgment; deciding how far unanimous preference may reach across it is another. The liberty at stake is diminished if the right to choose survives only while everybody else likes the result. Yet rights also require an account of their extent, their exercise and their consequences. The table has reached a point at which a choice’s authority must be examined alongside its satisfaction. That question will remain when the next plate turns to the quieter power of arranging the options from which a person chooses.