How an election can have no winner
Tonight's idea
Ava, Ben and Cleo rank three dinners.
| 1st | 2nd | 3rd | |
|---|---|---|---|
| Ava | Pizza | Sushi | Tacos |
| Ben | Sushi | Tacos | Pizza |
| Cleo | Tacos | Pizza | Sushi |
Pizza or sushi? Ava and Cleo say pizza: 2 to 1. Sushi or tacos? Ava and Ben say sushi: 2 to 1. Tacos or pizza? Ben and Cleo say tacos: 2 to 1. The majority prefers pizza to sushi to tacos to pizza. Draw the three as dots with an arrow from each winner to the loser and you get a cycle. There is no dinner that beats every other, so "what the majority wants" has no answer. Condorcet spotted this in 1785.
Real elections avoid the question by picking a system, and the surprise is that reasonable systems disagree with each other. Take 100 voters and three candidates:
| Voters | 1st | 2nd | 3rd |
|---|---|---|---|
| 40 | A | B | C |
| 35 | C | B | A |
| 25 | B | C | A |
- Plurality: most first-place votes wins. A wins with 40.
- Runoff: drop the candidate with fewest first places, then count again between the other two. B is dropped; in A versus C the 25 B-voters switch to C, so C wins 60 to 40.
- Borda count: 2 points for a first place, 1 for a second, 0 for a third. A gets 80, B gets \(40 + 35 + 50 = 125\), C gets \(70 + 25 = 95\). B wins.
Same ballots, three systems, three different winners. And notice that A, the plurality winner, is ranked last by 60 of the 100 voters.
In 1951 Kenneth Arrow proved there is no way out: no ranking system with three or more options can satisfy all of a short list of obviously fair rules at the same time.
Watch one: the pairwise table
For the 100-voter election, compare each pair. A vs B: only the 40 A-first voters prefer A, so B wins 60–40. A vs C: again only 40 prefer A, so C wins 60–40. B vs C: the 40 A-voters and the 25 B-voters prefer B, so B wins 65–35. B beats everyone head to head. A candidate like that is called a Condorcet winner, and B does not win under plurality or runoff.
Your turn
- Twenty students rank three class trips: 9 rank Zoo > Museum > Beach, 7 rank Beach > Museum > Zoo, 4 rank Museum > Beach > Zoo. Find the plurality winner, the runoff winner (drop the trip with fewest first places, then count again between the other two), and the Borda winner (2, 1, 0 points).
- For the same twenty students, make the pairwise table: Zoo vs Museum, Zoo vs Beach, Museum vs Beach. Is there a Condorcet winner?
- Invent your own three-voter, three-option cycle like the dinner one. Write the rankings and check all three votes come out 2–1 in a circle.
- In one sentence, explain how "most first-place votes" can elect someone that most voters rank last.
Stretch: voting with a straight face
Back to the plurality election, where A wins. The 25 voters who honestly rank B first know B cannot win. Suppose they all vote for C instead, their second choice. Who wins now, and is that a better result for them than voting honestly?
Now the Borda count, where B wins. The 40 A-first voters would prefer A. Suppose they rank C second instead of B, to drag B's score down. Recompute the Borda totals. Who wins, and how do the A-voters feel about it?
Answers
- Plurality: Zoo 9, Beach 7, Museum 4, so Zoo. Runoff: Museum is dropped; Zoo vs Beach, the 4 Museum-voters prefer Beach, so Beach 11, Zoo 9, Beach. Borda: Zoo \(18\), Museum \(9 + 7 + 8 = 24\), Beach \(14 + 4 = 18\), so Museum. Three winners again.
- Zoo vs Museum: 9 to 11, Museum. Zoo vs Beach: 9 to 11, Beach. Museum vs Beach: 13 to 7, Museum. Museum beats both, so it is the Condorcet winner, and it finished last under plurality.
- Any rotation works, for example X > Y > Z, Y > Z > X, Z > X > Y. Check: X beats Y (voters 1 and 3), Y beats Z (1 and 2), Z beats X (2 and 3).
- If the other candidates split the remaining votes, a candidate can have the biggest single pile of first choices while being the last choice of everyone who did not put them first.
Stretch. Plurality: if the 25 switch to C, the totals are A 40, C 60, so C wins. The B-voters prefer C to A, so lying got them a better result. This is called tactical voting and it is why plurality elections tend to collapse to two big candidates. Borda with the A-voters lying (A > C > B): A \(80\), B \(35 + 50 = 85\), C \(70 + 25 + 40 = 135\). C wins. The A-voters rank C last, so their lie backfired: they knocked out B, their second choice, and handed the election to their least favourite. Strategic voting can help or hurt, and working out which requires guessing what everyone else will do.