Home/Explainers/Arrow's Impossibility Theorem

A Fundamental Limit of Democracy

Why Perfect Voting
Is Mathematically Impossible

There is no voting system - none, ever, in principle - that can satisfy three basic fairness criteria. Kenneth Arrow proved it and won a Nobel Prize.

This isn't about specific flaws in specific voting systems. It's not about gerrymandering, voter fraud, or campaign finance.

It's a mathematical proof that fair aggregation of preferences is impossible.

Unanimity

If everyone prefers A to B, society should prefer A to B.

Independence (IIA)

Society's ranking of A vs B should only depend on how individuals rank A vs B.

No Dictator

No single voter should always determine the outcome.

With 3+ candidates, NO voting system satisfies all three.

Every voting system must violate at least one criterion.

PART I

Voting System Explorer

Let's start by seeing how different voting systems produce different winners from the same preferences. Drag to reorder each voter's preferences.

Create Your Electorate

Voter 1

1. A

2. B

3. C

Voter 2

1. A

2. B

3. C

Voter 3

1. B

2. C

3. A

Voter 4

1. B

2. C

3. A

Voter 5

1. C

2. A

3. B

PLURALITYTie / No Winner

First-place votes: A=2, B=2, C=1

Candidate A

Candidate B

Candidate C

What each method does:

Plurality: Whoever gets the most first-place votes wins. Simple, but ignores 2nd and 3rd preferences.

Borda Count: Points system: 2 for 1st, 1 for 2nd, 0 for 3rd. More nuanced, but vulnerable to strategic voting.

Instant Runoff: Eliminate the candidate with fewest first-place votes, redistribute their votes. Repeat until winner.

Condorcet: Find a candidate who beats everyone in head-to-head matchups. But what if there is none?

Try different voter preferences and watch how the same voters can produce different winners under different systems. This alone should make you suspicious that something fundamental is wrong.

PART II

The Condorcet Paradox

Before Arrow, the Marquis de Condorcet discovered something disturbing in 1785: majority preferences can cycle.

With three voters and three candidates, you can have: A beats B, B beats C, but C beats A. Rock-paper-scissors, but for democracy.

Preference Cycle Demo

CYCLE DETECTED

Voter 1

1. A

2. B

3. C

Voter 2

1. B

2. C

3. A

Voter 3

1. C

2. A

3. B

beats B

beats C

beats A

A beats B, B beats C, C beats A

There is no “best” candidate by majority rule!

This is why “majority rule” isn't a complete voting system. When preferences cycle, there is no majority winner. The social preference is intransitive even when every individual's preference is perfectly transitive.

Key insight: Individual rationality does not guarantee collective rationality. The group can be “irrational” even when every individual is rational.

PART III

Independence of Irrelevant Alternatives

This is the criterion that breaks most voting systems. IIA says: the social ranking between A and B should depend only on how voters rank A versus B - not on some third candidate C.

It sounds obvious. If I prefer pizza to sushi, learning that tacos exist shouldn't change my pizza-vs-sushi preference.

IIA Violation Demo (Borda Count)

Include Candidate C?

7 voters with fixed A vs B preferences:

With C in the race (Borda):

Winner: C

A: 6, B: 7, C: 8

Without C (First choice wins):

Winner: B

A: 2 first-place, B: 5 first-place

IIA Violation: B wins with C present, but A wins if C drops out.
The A vs B result changed because of an “irrelevant” alternative!

This is the “spoiler effect.” A third candidate who can't win themselves can still change who wins between the top two.

2000 US Presidential Election

Many believe Ralph Nader (Green Party) cost Al Gore the election in Florida. Nader received 97,421 votes in Florida; Gore lost to Bush by 537 votes. If Nader voters had split for Gore instead, the entire election result would have changed. This is IIA violation in the real world.

PART IV

The Proof Sketch

Arrow's proof is surprisingly elegant. It shows that satisfying Unanimity and IIA inevitably creates a dictator.

Proof Walkthrough

Step 1 of 6

Start with Unanimity + IIA

Assume we have a voting system that satisfies both Unanimity (if everyone prefers A to B, society prefers A) and IIA (the social ranking of A vs B depends only on individual rankings of A vs B).

The Deep Insight

The proof reveals that Unanimity + IIA together create a kind of “domino effect.” If one voter's preference is decisive in one situation (which Unanimity requires), IIA forces that same voter to be decisive in more situations. The conditions propagate until one voter controls everything.

PART V

What Dictatorship Looks Like

A “dictator” in Arrow's sense isn't necessarily a tyrant. It's simply someone whose preference always determines the outcome, regardless of what everyone else wants.

Dictatorship Demo

Dictator:

The DictatorDICTATOR

1. A

2. B

3. C

Voter 2

1. B

2. C

3. A

Voter 3

1. C

2. A

3. B

Voter 4

1. B

2. A

3. C

Voter 5

1. C

2. B

3. A

The Winner (always the dictator's top choice):

Change any non-dictator's preferences. The winner never changes.
Only the dictator's first choice matters.

This is what Arrow proved is inevitable if you want Unanimity and IIA. You can avoid dictatorship, but only by giving up one of the other two conditions.

PART VI

Living with Imperfection

Arrow's theorem doesn't say democracy is useless. It says we must choose which imperfection to accept.

Drop Unanimity?

Almost never done. A system that ignores unanimous preferences seems absurd. But technically, random selection (lottery) satisfies IIA and No Dictator.

Drop IIA?

This is what most voting systems do. Plurality, Borda, IRV all violate IIA. We accept that adding candidates can change outcomes. The spoiler effect is considered an acceptable cost of a “practical” system.

Accept a Dictator?

Monarchy, autocracy, or corporate hierarchy. One person decides. It's decisive and consistent, but few consider it fair for public decisions.

“The search for a perfect voting system is like the search for a perpetual motion machine - not just difficult, but mathematically impossible.”

The question isn't which system is perfect. It's which imperfection you can live with.

Practical Takeaways:

No voting reform can “fix” voting - there is no fix
Any voting system can be strategically manipulated (Gibbard-Satterthwaite theorem)
Ranked choice / IRV still has spoilers - just different ones
The best we can do is choose imperfections that cause least harm in our context

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Reference: Arrow (1951), “Social Choice and Individual Values”