Arrow's Theorem — There Is No Perfect Voting System

What just happened?

Three voters choose among three options (A, B, C) and each ranks them rationally. We want to build a single social ranking out of these three individual rankings. It seems enough to just “vote fairly” — and yet every method you try breaks something.

The preference profile (Condorcet’s paradox):

Check the pairwise majorities: A beats B (2:1), B beats C (2:1), but C beats A (2:1). A cycle arises:

Even though every voter has consistent, rational preferences, the group preferences form a cycle — there is no winner. This is the famous Condorcet paradox.

Reasonable fairness criteria (the invariant):

From a good voting system we expect natural conditions — among others:

In the panel beside it, each system satisfies some of them — but at least one criterion always glows red. Pairwise majority breaks the consistency of the ranking (a cycle); the Borda count breaks independence (IIA); and the only system satisfying Pareto, IIA, and consistency at once turns out to be… a dictatorship.

Arrow’s theorem (the impossibility theorem):

Kenneth Arrow proved in 1951 (in the book “Social Choice and Individual Values”) that when voters choose among at least three options, there exists no system aggregating individual rankings into a single social ranking that satisfies all of these “reasonable” conditions at once. They sound obvious and minimal — and yet they are mutually contradictory: every system must break at least one. This is exactly what Condorcet’s paradox illustrates, where the rational preferences of individuals add up to irrational, cyclic group preferences.

Kenneth Arrow and social choice theory

This is one of the most famous results of mathematics in the social sciences. For this and other work Arrow received the 1972 Nobel Prize in economics. The theorem has deep consequences for the theory of democracy, the design of electoral rules, and social choice theory — it reveals the mathematical limit of “perfect fairness” in collective decision-making.

Why this matters:

This is a topic beyond the core curriculum — a curiosity showing how mathematics describes democracy and elections. One conclusion is worth remembering: a “perfectly fair” voting system for ≥3 options does not exist, so every electoral rule is a deliberate compromise — we sacrifice one of the criteria.

Remember: with at least three options the conditions no dictator + unanimity + independence (IIA) are mutually contradictory. They cannot all hold at once — every voting system breaks at least one.