Suppose you are designing a voting system for a group. You want it to be fair. You sit down and write out what "fair" means: the result should reflect everyone's preferences, a candidate who beats every other candidate head-to-head should win, irrelevant alternatives shouldn't affect the outcome, and no single person should be a dictator whose preferences automatically override everyone else's. These seem like minimal requirements — the floor of fairness, not the ceiling. Kenneth Arrow proved in 1951 that no voting system satisfies all of them simultaneously. The floor is too high. The requirements are mutually incompatible.
Arrow's theorem is a result in social choice theory, the mathematical study of how group preferences can be aggregated from individual ones. He formalized the requirements as four conditions. Unanimity: if every voter prefers A to B, the group ranking puts A above B. Independence of irrelevant alternatives (IIA): the group's ranking of A versus B depends only on how individuals rank A versus B, not on their rankings of C. Transitivity: the group ranking is consistent — if the group prefers A to B and B to C, it prefers A to C. Non-dictatorship: no single voter's preferences automatically determine the group ranking.
Arrow proved: any procedure that satisfies unanimity, IIA, and transitivity is a dictatorship. The only way to satisfy the first three conditions together is to designate one voter whose preferences become the group's preferences. The fourth condition — the one that says this is not allowed — cannot be added without breaking one of the others.
IIA is the key constraint. It says that to determine the group's ranking of any pair, you only need to look at how individuals rank that pair. This makes the aggregation function act pair-by-pair — like a system of pairwise votes. But pairwise voting is famously susceptible to cycles. The Condorcet paradox shows that three voters with preferences A>B>C, B>C>A, C>A>B produce a majority cycle: A beats B, B beats C, and C beats A. No winner emerges. Transitivity rules out these cycles. Arrow showed that ruling them out, under IIA, forces the procedure to become dictatorial.
The result matters practically because every real voting system violates one of the conditions. Plurality voting violates IIA massively — adding a third candidate can completely change who wins between the original two, which is why "spoiler effects" are real. Borda count violates IIA in a different way: changing where you rank an irrelevant candidate can alter your effective contribution to the ranking of other candidates. Ranked-choice voting violates transitivity: there are preference profiles where it fails to elect the Condorcet winner.
The impossibility theorem is sometimes interpreted as: elections don't work, democracy is incoherent, all voting systems are equally arbitrary. This goes too far. Arrow's theorem tells you that no aggregation procedure satisfies all four conditions simultaneously — it doesn't say the conditions are equally important, or that all violations are equally bad. Practical voting design is about choosing which condition to relax and by how much. These are genuine design choices with genuine tradeoffs.
What the theorem does reveal is something important about the nature of collective preference. Individual preferences are, by assumption, transitive and complete — each voter has a consistent ranking. But aggregating consistent individual preferences doesn't produce a consistent collective preference. The group is not an individual scaled up. It does not have preferences in the same sense. When we talk about "what the voters want," we are eliding a fundamental ambiguity: the answer depends on the aggregation method, and there is no method-neutral answer waiting to be discovered.
This has implications beyond voting. Any time you try to aggregate individual rankings — of search results, of policy priorities, of research proposals — Arrow's theorem lurks. You cannot build a ranking procedure from individual inputs that satisfies all four conditions. Something always gives. The social welfare function you choose is not revealing a pre-existing collective preference; it is constructing one, and the construction has fingerprints. Recognizing this is not a counsel of despair. It's a counsel of honesty about what aggregation can and cannot do.