Who Should Win When Everyone Disagrees?

A Classroom Vote That Ended in Confusion

Imagine your class is picking a single field trip. Four destinations are on the ballot: Adventure Park (A), Beach Day (B), Castle Museum (C), and Dragon Boat (D). Everyone writes down their ranking from best to worst. You collect 21 votes and the results look like this:

• 3 students rank: A first, B second, C third, D last.
• 5 students rank: A first, C second, B third, D last.
• 7 students rank: B first, D second, C third, A last.
• 6 students rank: C first, B second, D third, A last.

Who should go? The simplest rule—pick the candidate with the most first‑place votes—makes A the winner with 8 votes. But 13 students put A in last place. D gets zero first‑place votes, yet every single student ranks B above D. That leaves two contenders: B and C. Which one better reflects the group’s real opinion? Two mathematicians from the 1700s gave opposite answers.

Condorcet’s Idea: The Undisputed Champion

The Marquis de Condorcet (1743–1794) said: let each candidate fight every other in a one‑on‑one vote. The candidate who beats everyone else is the Condorcet winner—the truest choice of the group.

Using the field‑trip rankings, we can run these head‑to‑head matches:

• C vs A: In the first and second groups, A is ranked above C, so 3 + 5 = 8 students prefer A. In the third and fourth groups, C is above A, so 7 + 6 = 13 prefer C. C beats A 13–8.
• C vs B: The second and fourth groups put C above B (5 + 6 = 11); the first and third put B above C (3 + 7 = 10). C beats B 11–10.
• C vs D: Every group except the fourth ranks D below C, giving C a 14–7 win.

C defeats every opponent. According to Condorcet, C should be the winner—even though C got only 6 first‑place votes. The thought is simple: if you can beat anyone else in a direct contest, you’re the strongest candidate.

Borda’s Revenge: The Point Champion

Jean‑Charles de Borda (1733–1799) thought Condorcet’s method missed something important. Beating someone by a narrow margin isn’t the same as beating them by a landslide, and losing by a few votes is different from being crushed. Borda’s fix was to assign points: with four candidates, a first‑place ranking gives 3 points, second place 2, third place 1, and last place 0. The candidate with the highest total score wins. That total is called the Borda count.

Apply that to our trip ballot:

• A appears first on 8 ballots (3 + 5) → 8 × 3 = 24 points. A never ranks second or third on any ballot, so the final score is 24.
• B appears first on 7 ballots → 7 × 3 = 21 points. B lands second on 3 + 6 = 9 ballots → 9 × 2 = 18 points. B is third on 5 ballots → 5 × 1 = 5 points. Total: 21 + 18 + 5 = 44.
• C first on 6 ballots → 6 × 3 = 18. Second on 5 ballots → 5 × 2 = 10. Third on 3 + 7 = 10 ballots → 10 × 1 = 10. Total: 38.
• D never first; second on 7 ballots → 7 × 2 = 14; third on 6 ballots → 6 × 1 = 6. Total: 20.

The Borda count says B is the winner, and by a comfortable margin. Borda’s method records not just who wins each duel but by how much. This seems more complete. Yet Condorcet’s supporter would object: in a one‑on‑one fight, B loses to C. Can B really be the group’s best choice?

The Vicious Cycle: When No One Beats Everyone

Condorcet’s method sounds clean until we meet its biggest problem: sometimes there is no Condorcet winner. Imagine just three voters ranking three candidates:

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

In a one‑on‑one vote, A beats B (voters 1 and 3 prefer A over B). B beats C (voters 1 and 2 prefer B over C). C beats A (voters 2 and 3 prefer C over A). The majority preference goes round in a circle: A > B > C > A. This is the Condorcet paradox (or a majority cycle). No candidate beats everyone else. So Condorcet’s own rule would declare a three‑way tie, which isn’t helpful.

Borda’s method always spits out a winner, but it too can stumble in ways that feel unfair. In our original 21‑voter example, A is a Condorcet loser—she loses to every other candidate in a head‑to‑head. Yet Borda’s winner, B, is not that loser. In other scenarios, though, a scoring rule like Borda can crown a Condorcet loser. So both approaches have cracks. Which crack bothers you more?

Arrow’s Big Discovery: No Voting System Is Perfect

In the 20th century, the economist Kenneth Arrow looked at the whole mess and proved something startling. He asked: is there any voting method that uses rankings and satisfies four common‑sense fairness rules?

1. Unanimity: If every voter ranks A above B, then B should not win.
2. No Dictator: No single person’s preference should always decide the winner.
3. Universal Domain: The rule must work for any set of rankings, no matter how strange.
4. Independence of Irrelevant Alternatives: The final ranking of A versus B should depend only on how voters rank A and B—not on where they place other candidates like C or D.

Arrow showed that if you have three or more candidates, no ranking‑based method can satisfy all four rules at once. Every method will violate at least one. That result is called Arrow’s impossibility theorem.

To see the idea, watch Borda count break rule 4. Start with three voters and candidates A, B, C, plus an extra X that everyone ranks last:

• 3 voters: A > B > C > X
• 2 voters: B > C > A > X
• 2 voters: C > A > B > X

The Borda scores give a ranking of A > B > C. Now keep the relative order of A, B, C exactly the same, but move X into different spots:

• 3 voters: A > B > C > X
• 2 voters: B > C > X > A
• 2 voters: C > X > A > B

The Borda scores flip the order of the top three: now C > B > A. Adding an irrelevant candidate nobody cares about scrambles the result. So Borda fails rule 4. Plurality rule, Condorcet‑consistent methods, and all the rest fail some condition or another. Arrow’s theorem tells us the dream of a perfectly fair voting system is mathematically impossible.

Why the Fight Over Voting Still Matters Today

You might never have heard of Condorcet or Borda, but their argument is alive in every election. Different countries and cities use different voting methods—plurality, ranked‑choice voting (also called the Hare rule), approval voting, and more—precisely because they come to different conclusions about what “the group wants.” Some methods aim to find a Condorcet winner; others favor the Borda idea of weighing strength of support. Arrow’s theorem shows that none of them can tick every box, so political scientists and mathematicians keep debating which trade‑offs are most acceptable.

Next time your class votes on a movie, a field trip, or a class president, watch how the rules shape the outcome. Would a head‑to‑head champion feel fairer than a points champion? Could you accidentally create a cycle? The puzzle that stumped two brilliant 18th‑century thinkers now sits in your hands.

Think about it

1. If your classroom used a rule where everyone votes only for their top choice, could a candidate win even though most people didn’t really want them? What kind of unfairness might that cause?
2. You’re designing a method to pick a class president. Would you rather reward the person who could beat every rival one‑on‑one, or the person who gets the highest average rating from everyone? Why?
3. Arrow proved that no voting system is flawless. Does that mean it’s pointless to search for better rules, or does it make the search even more important?