Voting Theory: Why Democracy is Mathematically Impossible

In 1952, economist Kenneth Arrow proved something shocking:

There is no perfect voting system.

Any method of aggregating individual preferences into collective decisions must violate at least one desirable property.

Arrow’s Impossibility Theorem isn’t just an abstract mathematical curiosity — it explains:

• Why elections often feel unfair
• Why third-party candidates “spoil” elections
• Why strategic voting is rational
• Why democracies struggle with consistency

Welcome to voting theory — where mathematics reveals the fundamental limitations of democracy.

What Makes a Voting System “Good”?

Before diving into impossibility results, let’s define what properties we’d want in an ideal voting system.

1. Unrestricted Domain

The voting system should work for any possible set of individual preferences.

Translation: No matter what voters prefer, the system produces an outcome.

2. Non-Dictatorship

No single voter should always determine the outcome regardless of others’ preferences.

Translation: Everyone’s vote matters.

3. Pareto Efficiency (Unanimity)

If every voter prefers candidate A over candidate B, then the collective outcome should rank A above B.

Translation: If everyone agrees, the system should respect that.

4. Independence of Irrelevant Alternatives (IIA)

The ranking between A and B should depend only on voters’ preferences between A and B, not on their preferences for C.

Translation: Adding or removing a third candidate shouldn’t change the winner between the first two.

Arrow’s Impossibility Theorem: With 3 or more candidates, no voting system can satisfy all four properties simultaneously.

Arrow’s Impossibility Theorem

Statement:

For any social choice function (voting system) with at least 3 alternatives:

If it satisfies:

1. Unrestricted domain
2. Pareto efficiency
3. Independence of irrelevant alternatives

Then it must be a dictatorship (one person’s preferences always determine the outcome).

Translation: You can’t have a fair, rational voting system that respects everyone’s preferences.

This is a mathematical certainty, not a political opinion.

The Voting Paradox: Why Democracy Cycles

Even before Arrow, Condorcet (1785) discovered a paradox with majority voting.

Example: Three voters, three candidates

Voter preferences:

• Voter 1: A > B > C
• Voter 2: B > C > A
• Voter 3: C > A > B

Pairwise majority votes:

• 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)

Result: A > B > C > A — a cycle!

Collective preferences are intransitive even though each individual’s preferences are transitive.

Who should win?

• If we run A vs B, A wins
• If we run B vs C, B wins
• If we run C vs A, C wins

The winner depends on the order of voting!

Real-world implication: Whoever controls the agenda (order of votes) can manipulate the outcome, even with honest voting.

Common Voting Systems and Their Flaws

1. Plurality (First-Past-The-Post)

How it works: Each voter picks one candidate. Candidate with most votes wins.

Used in: U.S. presidential elections (within states), U.K. Parliament, Canada

Fatal flaw: Spoiler effect — third candidates can split the vote and elect the least preferred candidate.

Example: 2000 U.S. Presidential Election

• Bush: 48.4%
• Gore: 48.0%
• Nader: 2.7%

Result: Bush won. But polls suggested most Nader voters preferred Gore over Bush.

If Nader hadn’t run, Gore likely would have won Florida (and the presidency).

Violates IIA: Nader’s presence changed the outcome between Bush and Gore.

2. Runoff Voting (Two-Round System)

How it works: If no candidate gets >50%, hold a runoff between top two.

Used in: French presidential elections, many local elections

Flaw: Non-monotonicity — getting more votes can make you lose!

Example:

Round 1:

• A: 35%
• B: 33%
• C: 32%

Runoff: A vs B

Suppose in Round 2, A wins 51%-49%.

Now suppose some voters who ranked B>C>A switch to C>B>A (ranking C higher).

New Round 1:

• A: 35%
• C: 34%
• B: 31%

New Runoff: A vs C

A might lose to C 45%-55%!

Paradox: Voters ranking you higher can cause you to lose.

3. Instant Runoff Voting (Ranked Choice)

How it works: Voters rank candidates. Eliminate candidates with fewest first-choice votes, transferring their votes to next choice. Repeat until someone has majority.

Used in: Australia, Ireland, some U.S. cities (San Francisco, New York)

Flaws:

• Non-monotonicity: Ranking a candidate higher can hurt them
• Violates IIA: Eliminating a losing candidate can change the winner

Example (non-monotonicity):

Initial preferences:

• 35%: A > C > B
• 32%: B > C > A
• 33%: C > B > A

Round 1: B eliminated (fewest first-choice votes) B’s votes transfer to C Result: C wins over A

Now suppose 5% of B>C>A voters change to C>B>A (ranking C higher):

• 35%: A > C > B
• 27%: B > C > A
• 38%: C > B > A

Round 1: B still eliminated But now A wins because C’s supporters didn’t need B’s transfers

Paradox: More support for C (38% vs 33% first choice) led to C losing!

4. Borda Count

How it works: Voters rank candidates. Points awarded (3 points for 1st, 2 for 2nd, 1 for 3rd). Highest total wins.

Used in: Eurovision Song Contest, some awards (Heisman Trophy)

Flaw: Vulnerable to strategic manipulation

Example:

Honest preferences:

• 51%: A > B > C
• 49%: C > B > A

Honest Borda scores:

• A: 51×3 + 49×1 = 202
• B: 100×2 = 200
• C: 49×3 + 51×1 = 198

A wins.

But if C supporters vote strategically (rank B last instead of middle):

• 51%: A > B > C
• 49%: C > A > B

New Borda scores:

• A: 51×3 + 49×2 = 251
• B: 51×2 + 49×1 = 151
• C: 49×3 + 51×1 = 198

Now A wins by even more? Actually, let’s recalculate strategically for C voters:

If C voters rank C > A > B instead of C > B > A:

• A: 51×3 + 49×2 = 251
• B: 100×1 = 100
• C: 49×3 + 51×1 = 198

Strategic voting helps C, but not enough to win.

Better strategy: Bury the main rival (A):

• 51%: A > B > C
• 49%: C > B > A (rank main rival last)

The system rewards dishonest voting.

5. Approval Voting

How it works: Vote for as many candidates as you “approve of.” Most approvals wins.

Advantages:

• Simple
• Reduces spoiler effect
• Encourages voting for third parties

Flaw: No resistance to strategic approval

You must decide: Do I approve of my second choice to prevent my last choice from winning? But that might cause my second choice to beat my first choice!

6. Condorcet Method

How it works: The winner is the candidate who would beat every other candidate in pairwise majority votes.

Problem: A Condorcet winner might not exist (due to Condorcet’s paradox/cycle).

When cycles occur, different tie-breaking rules give different winners:

• Copeland method
• Kemeny-Young
• Schulze method
• Ranked pairs

Each has different properties and flaws.

Strategic Voting: The Gibbard-Satterthwaite Theorem

Arrow’s theorem shows no perfect voting system exists.

But can we at least have a system where honest voting is optimal?

Answer: No.

Gibbard-Satterthwaite Theorem (1973, 1975):

For any voting system with at least 3 candidates:

If it’s:

1. Non-dictatorial
2. Deterministic (same votes always produce same outcome)
3. Surjective (every candidate can win under some votes)

Then it’s manipulable — strategic voting can improve your outcome compared to honest voting.

Translation: In any reasonable voting system, you should sometimes vote dishonestly!

Example: Plurality voting

Your true preferences: C > B > A

Polls show: A at 40%, B at 35%, C at 25%

If you vote honestly for C: A wins

If you vote strategically for B: B might win (which you prefer to A)

Honest voting is suboptimal!

Median Voter Theorem: Why Politics Is Boring

Theorem (Black, 1948; Downs, 1957):

In single-dimensional political competition (e.g., left-right spectrum) with majority voting, candidates converge to the median voter’s position.

Why?

Setup:

• Voters arranged on left-right spectrum
• Candidates choose positions
• Voters vote for closest candidate

Result: Both candidates move toward the median to capture more voters.

Example:

Voters distributed:

• 20% far left
• 30% center-left
• 25% center (median)
• 15% center-right
• 10% far right

If Candidate 1 is at center-left, Candidate 2 should move to center to capture center + right voters (50%) and win.

If Candidate 1 moves to center in response, they split the vote.

Equilibrium: Both candidates at the median.

Implications:

• Explains why major parties seem similar
• Extreme candidates lose (they’re too far from median)
• Third parties can’t win (they pull away from median)

Real-world:

• U.S. presidential candidates both moderate toward center after primaries
• “Flip-flopping” is rational repositioning toward median

The Problem of Cycles and Agenda Control

When Condorcet cycles exist, the order of voting determines the winner.

Example:

Preferences:

• 1/3: A > B > C
• 1/3: B > C > A
• 1/3: C > A > B

Cycle: A > B, B > C, C > A

If we vote on:

Agenda 1: First vote A vs B (A wins), then winner vs C (C wins). C is the outcome.

Agenda 2: First vote B vs C (B wins), then winner vs A (A wins). A is the outcome.

Agenda 3: First vote A vs C (C wins), then winner vs B (B wins). B is the outcome.

Different agendas, different winners!

Real-world: Parliamentary procedure, committee voting, legislation

Whoever controls the agenda (order of votes) effectively controls the outcome.

This is why parliamentary rules matter so much!

Solutions and Trade-offs

Since perfect voting doesn’t exist, what can we do?

1. Accept trade-offs

Choose which properties to sacrifice:

• Plurality: Sacrifice IIA (spoiler effect)
• IRV: Sacrifice monotonicity
• Borda: Sacrifice resistance to manipulation
• Approval: Sacrifice revealing preference intensity

2. Use different systems for different purposes

Small elections: Condorcet methods with tie-breaking Large elections: Approval or Score voting (rate candidates 0-10) Proportional representation: Allocate seats by vote share (party-list systems)

3. Change the game

Proportional representation: Instead of winner-take-all, allocate seats proportionally

• Used in most European parliaments
• Allows third parties to win seats
• Reduces spoiler effect

Quadratic voting: Voters have budget of “credits” to allocate

• Can express intensity of preferences
• Proposed by Glen Weyl (2018)
• Used in some corporate governance

Real-World Implications

1. Electoral Reform

Understanding impossibility theorems helps evaluate proposed reforms:

• IRV proponents: Reduces spoiler effect (true), but has non-monotonicity (often ignored)
• Approval voting proponents: Simple and reduces spoilers (true), but incentivizes strategy
• Proportional representation: Reduces wasted votes, but may increase fragmentation

No reform is perfect — it’s about choosing acceptable trade-offs.

2. Why Third Parties Fail

In plurality systems, third parties face structural disadvantages:

• Spoiler effect deters supporters from voting for them
• Strategic voters choose “lesser of two evils”
• Median voter theorem predicts convergence to two major parties

Not a conspiracy — it’s mathematics.

3. Why Democracies Seem Unstable

Condorcet cycles explain:

• Policy flip-flops when control changes
• Different coalitions producing different outcomes from same preferences
• Agenda manipulation in legislatures

Instability isn’t always dysfunction — sometimes it’s inherent to preference aggregation.

Key Takeaways

1. Arrow’s Impossibility Theorem — no voting system satisfies all desirable properties with 3+ candidates
2. Condorcet paradox — majority preferences can cycle (A>B>C>A), making collective choice irrational
3. Gibbard-Satterthwaite Theorem — all reasonable voting systems are manipulable (strategic voting helps)
4. Median voter theorem — candidates converge to the median voter position
5. Spoiler effect — third candidates can change winner between top two (violates IIA)
6. Agenda control — order of voting determines outcome when cycles exist
7. No perfect system — every voting method has flaws; choose trade-offs wisely
8. Electoral systems matter — different rules produce dramatically different outcomes

Voting theory reveals that democracy’s problems aren’t just political — they’re mathematical. Understanding these limitations helps us design better (though never perfect) systems.

Practice Problem

Three candidates (A, B, C) compete for office. Voters have the following preferences:

• 35% of voters: A > B > C
• 33% of voters: B > C > A
• 32% of voters: C > B > A

Questions:

1. Who wins under plurality voting (vote for one candidate)?
2. Who wins under instant runoff voting (IRV)?
3. Is there a Condorcet winner (candidate who beats all others in pairwise votes)?
4. What does this reveal about different voting systems?

This post is part of the Game Theory Series, where we explore the mathematics of strategic decision-making. Voting theory reveals the fundamental mathematical limits of democracy and shows why all voting systems involve trade-offs.