The Mathematics of Elections and Voting

Most people assume that once the votes are counted, the winner is obvious. Mathematics tells a more interesting story: the very same ballots can produce different winners depending on the counting rule you use. In this article we explore the main voting methods, work through a full example, and see how mathematicians divide parliamentary seats fairly.

Preference Ballots and Preference Schedules

Instead of asking voters for just one name, imagine asking them to rank all the candidates. A ranking such as A > B > C means "I like A most, then B, then C". When many voters submit rankings, we group identical ballots together into a preference schedule.

Here is the schedule we will use throughout. One hundred students are electing a class representative from three candidates: Ahmed (A), Bushra (B) and Chaudhry (C).

Nothing about the voters changes from here on. Only the counting rule does.

Worked Example 1: Four Methods, Three Winners

Method 1: Plurality (First Past the Post)

Count only first choices:

• A: 40 votes
• C: 35 votes
• B: 25 votes

Winner: Ahmed, with 40%. Notice that 60% of voters ranked Ahmed last. Plurality is the system used for National Assembly seats in Pakistan, as well as in the UK, India and Canada.

Method 2: Plurality with Run-off

If nobody has a majority (more than 50), eliminate everyone except the top two. Here the top two are A (40) and C (35), so B is eliminated.

Step 1: B's 25 voters move to their next choice, which is C.

Step 2: New totals: A = 40, C = 35 + 25 = 60.

Winner: Chaudhry, with 60%. France uses a two-round version of this method to elect its president.

Method 3: Borda Count

Give points for each position. With three candidates, a 1st place earns 2 points, a 2nd place 1 point and a 3rd place 0 points.

Ahmed:

$$ 40 \times 2 + 35 \times 0 + 25 \times 0 = 80 $$

Bushra:

$$ 40 \times 1 + 35 \times 1 + 25 \times 2 = 40 + 35 + 50 = 125 $$

Chaudhry:

$$ 40 \times 0 + 35 \times 2 + 25 \times 1 = 70 + 25 = 95 $$

Check: each voter hands out \(2 + 1 + 0 = 3\) points, so the total should be \(100 \times 3 = 300\). Indeed \(80 + 125 + 95 = 300\).

Winner: Bushra, with 125 points. The method is named after the French naval officer Jean-Charles de Borda, who proposed it in 1770. Variants are used to elect members of parliament in Nauru, to choose two minority representatives in Slovenia, and in many sports awards.

Method 4: Condorcet's Head-to-Head Method

The Marquis de Condorcet argued in 1785 that a candidate who beats every other candidate in a one-on-one contest deserves to win.

• B vs A: voters ranking B above A are \(35 + 25 = 60\); voters ranking A above B are 40. B wins 60–40.
• B vs C: voters ranking B above C are \(40 + 25 = 65\); C above B are 35. B wins 65–35.
• A vs C: A above C is 40; C above A is \(35 + 25 = 60\). C wins 60–40.

Bushra beats both rivals, so she is the Condorcet winner. Plurality, which chose Ahmed, has picked the candidate who loses every head-to-head contest.

Our final scoreboard: plurality elects A, run-off elects C, Borda and Condorcet elect B. Same voters, three different winners.

The Condorcet Paradox

Condorcet himself discovered a troubling fact. Suppose three equal groups of voters hold these rankings:

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

In head-to-head contests, A beats B (groups 1 and 3), B beats C (groups 1 and 2), and C beats A (groups 2 and 3). The group preference goes round in a circle, just like rock–paper–scissors, so there is no Condorcet winner at all. Individual voters are perfectly logical, yet the group as a whole is not.

Arrow's Impossibility Theorem

In 1951 the American economist Kenneth Arrow asked whether any ranked voting method could satisfy a short list of reasonable fairness conditions at once, such as:

• If every voter prefers X to Y, the group should too (unanimity).
• The group ranking of X versus Y should depend only on how voters rank X versus Y, not on some third candidate (independence of irrelevant alternatives).
• No single voter is a dictator whose preference always decides the result.

Arrow proved that with three or more candidates, no ranked voting system can satisfy all these conditions simultaneously. The result helped earn him the 1972 Nobel Prize in Economics. It does not mean elections are pointless; it means every system involves trade-offs that citizens should understand.

Worked Example 2: Apportioning Seats Fairly

After an election, or when dividing an assembly among regions by population, we must turn fractions into whole seats. Consider a council of 7 seats shared by three districts:

• District P: 5,240,000 people
• District Q: 3,170,000 people
• District R: 1,590,000 people

Step 1: Total population.

$$ 5{,}240{,}000 + 3{,}170{,}000 + 1{,}590{,}000 = 10{,}000{,}000 $$

Step 2: Standard divisor (people per seat).

$$ d = \frac{10{,}000{,}000}{7} \approx 1{,}428{,}571 $$

Step 3: Standard quotas (population ÷ divisor).

• P: \(5{,}240{,}000 \div 1{,}428{,}571 \approx 3.668\)
• Q: \(3{,}170{,}000 \div 1{,}428{,}571 \approx 2.219\)
• R: \(1{,}590{,}000 \div 1{,}428{,}571 \approx 1.113\)

Check: \(3.668 + 2.219 + 1.113 = 7.000\).

Step 4: Give each district the whole-number part. P gets 3, Q gets 2, R gets 1, a total of 6 seats.

Step 5: Hamilton's method. One seat remains, and it goes to the district with the largest fractional remainder. The remainders are 0.668, 0.219 and 0.113, so P receives it.

Final allocation: P = 4, Q = 2, R = 1, totalling 7 seats.

This largest-remainder method was proposed by Alexander Hamilton in 1792. It seems natural, but it can produce the Alabama paradox: in the 1880 US census calculations, increasing the House from 299 to 300 seats would have reduced Alabama's seats from 8 to 7. To avoid such quirks, other methods named after Jefferson, Webster and Huntington–Hill use adjusted divisors instead.

Did You Know?

• Australia has used ranked-choice "instant run-off" voting for its House of Representatives since 1918.
• The Eurovision Song Contest uses a points system that is essentially a modified Borda count.
• In the 2000 US presidential election, the result in Florida, which decided the whole election, was separated by just 537 votes out of nearly 6 million.

Key Takeaways

• The same preference ballots can produce different winners under plurality, run-off, Borda count and Condorcet methods.
• A Condorcet winner beats every rival head-to-head, but group preferences can form a cycle with no such winner.
• Arrow's theorem shows no ranked voting system with three or more candidates can satisfy every basic fairness condition.
• Apportionment methods such as Hamilton's convert population quotas into whole seats, but can create paradoxes.

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