Suppose you and two friends are deciding what to eat. There are three options: pizza, sushi, and burgers. You all rank the three:
- You: pizza > sushi > burgers
- Friend A: sushi > burgers > pizza
- Friend B: burgers > pizza > sushi
What should the group choose?
Each option has exactly one first-place vote. Each option has exactly one second-place vote. Each option has exactly one third-place vote. There is no clear winner. Worse: any tie-breaking procedure you can invent will be susceptible to manipulation. If your friend changes their preference slightly, the “winner” changes. There is no fair way to aggregate these three rankings.
This isn’t a fluke. In 1951, the economist Kenneth Arrow proved that no voting system can simultaneously satisfy a small set of reasonable fairness criteria. The exact statement is now called Arrow’s impossibility theorem, and it won him the Nobel Prize in Economics in 1972. The theorem is one of the cleanest examples of pure mathematics intersecting with political theory, and its consequences shape how we think about elections, group decisions, and social choice.
This article is about what Arrow proved, why it matters, and what it implies for real-world voting systems.
The setup
A voting system takes the preferences of multiple voters and produces a collective decision. Each voter ranks the alternatives in some order; the system aggregates these rankings into a group ranking (or just a winner).
Arrow asked: what would a “fair” voting system look like? He proposed five reasonable criteria:
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Universal domain (Unrestricted): The system must accept any combination of voter preferences. You can’t refuse to handle some sets of rankings.
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Non-dictatorship: No single voter’s preferences should automatically determine the group outcome.
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Pareto efficiency (Unanimity): If everyone prefers A to B, the group should prefer A to B.
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Independence of irrelevant alternatives (IIA): Whether the group ranks A above B should depend only on how individual voters rank A vs. B — not on how they feel about some third option C. Adding or removing alternative C shouldn’t change the relative ranking of A and B.
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Transitivity: If the group prefers A to B, and B to C, the group should prefer A to C. The group’s ranking has to be a real ranking.
These five criteria seem mild. Each captures something obvious about what fairness should mean. Surely some voting system can satisfy all five.
What Arrow proved
Arrow’s theorem (1951): There is no voting system that satisfies all five criteria simultaneously when there are three or more alternatives.
For any voting system you propose, at least one of the criteria will fail. Either it’s a dictatorship, or it can produce intransitive group rankings, or it violates IIA, or it fails some other property.
Specifically, the most often-violated criterion in real-world systems is Independence of Irrelevant Alternatives. Almost every familiar voting system can have its outcomes changed by adding a “spoiler” candidate. The 2000 US presidential election, where Ralph Nader’s third-party candidacy is widely believed to have changed the outcome by drawing votes from Al Gore, is a classic example of IIA failing in plurality voting.
Arrow’s proof is technical but accessible to undergraduates. The core idea is that under the IIA assumption, you can examine pairs of alternatives separately. Each pair gets aggregated by some procedure depending only on how voters rank that pair. By analyzing how the procedure must behave on different subsets of voters, you can show that some specific voter ends up with dictatorial power over a particular pair — and that this dictatorial power must extend to all pairs.
Why this matters
Arrow’s theorem has practical and philosophical consequences.
Practical: every voting system you encounter — plurality, ranked-choice, approval voting, Condorcet methods, Borda count — fails at least one of Arrow’s criteria. Choosing a voting system is choosing which criterion to sacrifice. There’s no mathematically perfect system; there are only systems with different trade-offs.
Philosophical: democracy is, in some precise sense, impossible to make perfectly fair. No matter how cleverly you design your election system, some reasonable fairness condition will fail somewhere. This is sometimes mistaken to mean democracy is irrational — it isn’t. It means democracy involves value judgments about which trade-offs to accept, and those judgments are political, not mathematical.
Methodological: Arrow’s theorem is a foundational result in social choice theory — the mathematical study of how individual preferences combine into collective decisions. The field has grown enormously since 1951, with theorems characterizing when different voting systems work well, what manipulation strategies are possible, and what kinds of preferences allow which systems.
Comparison of real voting systems
Different systems make different trade-offs. Quick survey:
Plurality (first-past-the-post): Each voter picks one candidate; whoever gets the most wins. Simple, but vulnerable to spoiler effects (violates IIA dramatically) and tends to produce two-party systems via Duverger’s Law.
Two-round runoff: Top two from first round face off. Better than plain plurality at avoiding spoilers, but still vulnerable to specific failures (the candidate beaten in round 1 by a “compromise” candidate may have actually been preferred by more voters).
Ranked-choice voting (instant runoff): Voters rank candidates; lowest is eliminated and votes redistributed; repeat until someone has majority. Reduces spoiler effects substantially. But fails monotonicity in pathological cases — sometimes a candidate can lose by gaining more first-place votes.
Borda count: Voters rank all candidates; points awarded by rank; highest total wins. Good at finding consensus winners, but easy to manipulate strategically.
Condorcet methods: Find the candidate who would win pairwise against all others. Often works well but can fail when no Condorcet winner exists (the “Condorcet paradox” — exactly the pizza/sushi/burgers situation from the opening).
Approval voting: Voters approve as many candidates as they like; most approvals wins. Simple, but doesn’t capture preference intensity.
Score voting / Range voting: Voters give each candidate a score (e.g. 0-10); highest average wins. Captures intensity, but can be manipulated and isn’t ranked-based.
Each system fails Arrow’s criteria in different ways, and reasonable people disagree about which trade-offs are worst.
The Gibbard-Satterthwaite theorem
A second impossibility result, proved independently by Allan Gibbard (1973) and Mark Satterthwaite (1975), strengthens Arrow’s: every non-dictatorial voting system with three or more outcomes is manipulable. There is always some configuration of voter preferences in which a voter can get a better outcome by lying about their true preferences.
This is bleaker than Arrow. Not only is no voting system perfectly fair — every voting system gives voters incentives to misrepresent their preferences. Strategic voting is a permanent feature of democracy.
Real-world examples are everywhere:
- In plurality voting, voters often vote for “the lesser evil” rather than their true favorite to avoid wasting a vote.
- In ranked-choice voting, voters sometimes “bury” their true favorite below a strategic compromise.
- In approval voting, voters strategically choose how generously to approve.
There’s no system where honest voting is always optimal for every voter.
The Condorcet paradox
The simplest demonstration of voting paradoxes is the Condorcet paradox, named after the 18th-century French mathematician who first analyzed it. With three voters and three alternatives:
- Voter 1: A > B > C
- Voter 2: B > C > A
- Voter 3: C > A > B
Pairwise comparisons:
- A vs. B: A wins (voters 1 and 3 prefer A)
- B vs. C: B wins (voters 1 and 2 prefer B)
- C vs. A: C wins (voters 2 and 3 prefer C)
The collective preference is intransitive: A beats B, B beats C, C beats A. There’s no Condorcet winner. Any voting system has to break this cycle somehow, and how it breaks the cycle determines the outcome.
This is exactly the pizza/sushi/burger scenario from the opening. Real elections occasionally produce such cycles, especially in close three-way races. They’re not common, but they’re not vanishingly rare either.
Voting in practice
What can we conclude from all this for actual political elections?
No system is perfect. This is the basic conclusion. Every voting system has known failure modes. The choice between systems is a value judgment.
Some systems are clearly worse than others. Plurality is dominated by ranked-choice in most fairness criteria, and there’s strong empirical evidence that ranked-choice produces more representative outcomes in elections with more than two strong candidates. The political process for switching is hard, but the mathematical case is clear.
Strategic voting is unavoidable. No matter what system, some voters will benefit from misrepresenting their preferences. Designing a system to minimize manipulation incentives is a research area, but no perfectly resistant system exists.
Multiple winners are different. Most of these results apply to “elect one winner.” Proportional representation systems (e.g. STV, MMP) have their own analogous theorems but somewhat different trade-offs. They tend to produce more representative legislatures at the cost of being harder to explain.
Group size matters. Small committees can use methods (like consensus, deliberation) that don’t scale to mass elections. Large electorates need automated procedures that have predictable, manipulable-only-with-effort properties. Different scales suggest different systems.
What voting theory teaches
The deepest lesson of Arrow’s theorem is that collective decision-making is intrinsically harder than individual decision-making. An individual person can have transitive, consistent preferences. A group, given individuals with their own transitive preferences, cannot in general aggregate them into a transitive group preference without violating some other reasonable property.
This is not a defect of any particular system — it’s a structural feature of group decision-making itself. The mathematics of voting forces hard choices about what fairness means.
In a sense, Arrow’s theorem tells you that “the will of the people” is not a well-defined concept in any pure mathematical sense. It’s a useful approximation, an idealization, a political ideal — but the precise mathematical question of “what does the group want?” doesn’t have a unique answer.
This connects voting theory to deeper themes in mathematics. Gödel’s incompleteness theorem showed that no formal system can capture all of arithmetic. Turing’s halting problem showed that no algorithm can solve all algorithmic questions. Arrow’s theorem shows that no voting system can satisfy all reasonable fairness criteria. Each is a foundational impossibility result in its domain.
These limits are not failures of cleverness. They are inherent in the structure of the problems. Knowing them is part of understanding what mathematics — and political theory — can and cannot do. We can still vote, still argue about which systems are better, still improve democracy. We just can’t pretend there’s a perfect system waiting to be discovered.
That, more than anything, is what social choice theory has to tell us about how groups of humans should make decisions. Sometimes the deepest mathematical truth is that perfection isn’t on offer.
Frequently asked
Does Arrow's theorem mean democracy is impossible?
No. The theorem shows that no voting system can perfectly satisfy a particular set of fairness criteria simultaneously. Real democracies make trade-offs — different systems sacrifice different criteria, and which trade-off is best is a value judgment, not a mathematical one. Arrow's theorem clarifies what's possible, not what's good.
Is there a 'best' voting system?
There's no mathematically best system because there are different fairness criteria you might prioritize. Plurality voting is simple but vulnerable to spoiler effects. Ranked-choice voting handles this better but can fail other criteria. Approval voting and Condorcet methods have their own trade-offs. The right system depends on what fairness properties matter most for the specific decision.
Why don't more elections use ranked-choice voting?
Several reasons. Plurality is simpler to explain and count. Ranked-choice requires more voter education and more complex tabulation. Some systems satisfy mathematical criteria better but produce results voters find unintuitive. Political momentum matters too — established systems are hard to change. Adoption is growing in places like Maine, Alaska, New York City, Australia, and Ireland, but global rollout has been slow.