Three people can each know exactly what they want, and their majority can want in a circle. Let the first rank the alternatives A, then B, then C; the second B, then C, then A; the third C, then A, then B. Two prefer A to B. Two prefer B to C. Two prefer C to A. Each person's order is coherent. The collective order is not. Nothing has gone wrong inside a mind, and nobody has lied about a preference. The difficulty appears in the addition: the majority that speaks for the whole changes its members as the question changes.
The circle is the voting paradox associated with Condorcet, but its importance extends beyond voting. Welfare economics had reason to hope that, if satisfactions could not be compared directly, individual rankings might still be combined into a social ordering. A community would not need to measure the intensity of every want; it would need a rule that took the members' preferences seriously and produced a coherent judgment among the available social states. Majority rule was an inviting candidate. The three-person example shows that it can fail. Kenneth Arrow's result asks a more disturbing question: whether another rule can avoid that failure while preserving a particular set of apparently reasonable requirements.
Arrow's theorem concerns a rule for converting a profile of individual rankings into a complete and transitive social ranking. Complete means that every pair can be compared; transitive means that the comparisons hang together, so that ranking A at least as high as B and B at least as high as C entails ranking A at least as high as C. Individuals are also assumed to have complete and transitive rankings. The alternatives are whole social states, not necessarily candidates, and an individual's preference can include concern for others as well as for herself. This is more demanding than selecting one winner from one ballot. A procedure that names a winner has done something necessary, but it has not thereby supplied the coherent ordering the theorem requires.
For a finite society of at least two people choosing among at least three alternatives, the crucial demands can be stated plainly. The rule must work for every admissible profile of individual rankings, not merely for profiles on which agreement happens to be easy. It must respect strict unanimity: if everyone prefers A to B, society must prefer A to B. Its comparison of A with B must depend only on how people rank A against B, not on their rankings involving other alternatives. And there must be no fixed individual whose strict preference always determines the social strict preference, whatever everyone else thinks. Together with a complete and transitive social ordering, these requirements cannot all be met. The conclusion is exact. So are the conditions that give it force.
Each condition protects a distinct claim about how individual judgments should count. Unrestricted domain refuses to dismiss disagreement merely because it makes aggregation inconvenient. Unanimity prevents the rule from overturning a judgment on which every person agrees. Independence prevents a comparison from changing when no one has altered her view of the pair being compared. Nondictatorship denies one person's order a permanent privilege over all the others. Collective transitivity asks the social judgment to have the consistency expected of an individual's judgment. Yet their conjunction is stronger than their separate plausibility suggests. Arrow's proof follows the power of a decisive set through different rankings: the other conditions make decisiveness spread between pairs, and a smallest decisive group cannot retain more than one member. The very safeguards meant to combine the rankings force, in the universal construction, the privilege the final safeguard forbids.
The majority cycle shows what one escape costs. Keep the pairwise votes and permit an intransitive collective relation. The preferences have been counted, but there is no stable top alternative in the example: whichever one is proposed, another beats it. A sequential agenda can still produce an outcome. If A first defeats B, C then defeats A, so C wins. If B first defeats C, A then defeats B, so A wins. If C first defeats A, B then defeats C, so B wins. The same sincere rankings can therefore yield any of the three outcomes under these agendas. The meeting has reached a decision, but the procedure organizing the comparisons has helped determine which decision it reaches. A single announced winner does not erase the cycle beneath it.
Another escape preserves a coherent ranking by giving points to positions. Award two points for first place, one for second, and none for third. In an invented two-voter example, rankings A–C–B and B–A–C give totals of three for A, two for B, and one for C. Now change the rankings to A–B–C and B–C–A. The first voter still prefers A to B; the second still prefers B to A. Yet B now scores three, A two, and C one. The social comparison of A with B reverses even though neither person's comparison of that pair has changed. The points have given the position of C a role in the verdict. That may be a trade a community is willing to make. It is a departure from independence, not a refutation of the theorem.
These point scores should not be mistaken for measured intensities. The interval between first and second is fixed by the voting convention, not discovered in the voter's experience. A rule can use that convention sensibly without calling it a unit of satisfaction. Equally, supplying cardinal numbers for each person does not automatically dissolve the problem: if each person's scale remains separately adjustable, the interpersonal calibration is still missing. The sum of wants requires a decision about what information belongs in the sum. Ordinal rankings, rank points, and comparable welfare measures are different kinds of information. Their differences matter precisely where an aggregation rule appears to turn separate judgments into one.
The theorem does not cover every question under the description of collective choice. With two alternatives and three voters, simple majority produces a complete, transitive ranking at every profile of strict preferences, respects unanimity, depends only on the pair, and gives no single voter permanent control. The third alternative is doing indispensable work in the impossibility result. A domain of preferences can also be restricted, though any restriction must be justified for the problem at hand rather than imposed merely to make an answer possible. Or a social judgment can leave some comparisons unresolved, giving up the demand for completeness. Each response identifies a requirement to revise. None promises that every difficulty encountered in actual public choice will disappear with it.
The more ambitious response is to admit information the original ordering problem excludes. Sen argues that useful social judgments can draw on partial interpersonal comparisons, deprivation, and the opportunities available to people, rather than on ordinal preference rankings alone. Such information has to be defended, and the principles for using it remain open to argument. But Arrow's theorem is not a prohibition on considering it. Nor does the theorem prove that democracy is futile, that every institution must be ruled by a tyrant, or that every actual choice is irrational. It shows the price of insisting on one particular combination of universal demands. The discovery is a constraint on the specification of a rule, not a verdict against the possibility of living together.
A collective decision can be made. The rule that makes it must disclose what it gives up.
Can the wants of millions be combined into a single will? They can be brought to bear on decisions through institutions whose compromises are intelligible and contestable. They cannot, on Arrow's stated domain, be converted into a universally coherent ordering while every one of his other conditions is preserved. The distinction matters. It replaces the dream of a procedure innocent of judgment with the work of choosing which judgments a procedure should embody. The sum does not speak before the rule has been chosen. And if we turn from that rule to a purported common measure of happiness, hoping to count the good more directly, the question follows us: what has been measured, whose experience does it represent, and what reason have we given it to speak for the whole?