The Winner Is… Determined by Voting Rules

I want to explore a truth that feels both mathematical and political:

In an election, the winner is not determined only by the voters. The winner is also determined by the voting rule.

That sounds controversial at first. Surely democracy is just “the person with the most votes wins”? But mathematics tells a more complicated story. Different voting systems can produce different winners from the exact same set of voters and preferences.

The Mathematics of Choice

Imagine an election with three candidates: Caesar, Brutus, and Cicero.

Suppose 100 citizens rank them as follows:

Voters	1st choice	2nd choice	3rd choice
40	Caesar	Brutus	Cicero
35	Brutus	Cicero	Caesar
25	Cicero	Brutus	Caesar

At first glance, Caesar seems strongest: he has the largest number of first-choice votes (40). Under first-past-the-post, Caesar wins immediately.

But now apply a different rule.

Instant Runoff Voting

Under instant runoff voting, the candidate with the fewest first-choice votes is eliminated first.

1. Cicero has the fewest first-choice votes (25), so he is eliminated.

2. Those 25 voters preferred Brutus next, so their votes transfer to Brutus.

3. Brutus now has 35 + 25 = 60 votes, while Caesar remains at 40.

Under this rule, Brutus wins 60–40.

Same voters. Same preferences. Different winner.

Condorcet’s Insight

Now compare candidates head-to-head:

1. Caesar vs Brutus

   40 voters prefer Caesar over Brutus, but 35 + 25 = 60 voters prefer Brutus over Caesar.

   Brutus beats Caesar 60–40.

2. Brutus vs Cicero

   40 + 35 = 75 voters prefer Brutus over Cicero, while 25 prefer Cicero.

   Brutus beats Cicero 75–25.

3. Caesar vs Cicero

   40 voters prefer Caesar over Cicero, while 35 + 25 = 60 prefer Cicero over Caesar.

   Cicero beats Caesar 60–40.

Brutus defeats every other candidate in a one-on-one contest. In voting theory, that makes him the Condorcet winner.

Yet first-past-the-post elected Caesar instead.

Why This Matters

This is not a trick. It is a theorem-level reality of social choice mathematics.

The 18th-century mathematician Marquis de Condorcet discovered that collective preferences can behave strangely. Later, Kenneth Arrow proved something even more unsettling: no voting system can perfectly convert individual preferences into a collective decision while satisfying a set of reasonable fairness conditions.

This is known as Arrow’s Impossibility Theorem.

In plain English: there is no flawless voting system.

Every rule makes trade-offs:

• First-past-the-post is simple and decisive, but it can elect a candidate opposed by a majority.

• Instant runoff captures broader support, but it can behave strategically and sometimes eliminate a candidate who might win head-to-head contests.

• Condorcet methods honour pairwise majorities, but they can become complex and occasionally produce cycles with no clear winner.

Mathematics does not tell us which system is morally best. But it does destroy the illusion that voting rules are neutral.

It means that before we ask “Who should win?”, we must ask “How are we deciding?”.

A republic that uses first-past-the-post may reward passionate minorities and clear factions. A republic that uses ranked voting may reward coalition-builders and compromise candidates. The structure of the system shapes political behaviour long before a single ballot is cast.

In Rome, Caesar did not rise in a vacuum. Institutions mattered. Rules mattered. Incentives mattered. Mathematics reminds us that political outcomes are often products of systems, not just personalities.

A Final Reflection

There is something almost Roman about this conclusion. We like to imagine history turning on the greatness of individuals: Caesar, Brutus, Cicero. But mathematics whispers a quieter truth:

sometimes the constitution matters more than the candidate.

Change the rule, and you may change the republic.

So when we say, “The winner is…”, the mathematician answers:

“…determined by voting rules.”