One person can have perfectly consistent preferences. But when many people vote together, the majority can behave cyclically. This puzzle reveals an elegant crack in the idea of “what the majority wants.”
The Impossible Vote
Riddle statement
Three candidates are running in an election: A, B, and C.
Each voter ranks the three candidates from best to worst.
Can it happen that, by majority:
A is preferred to B;
B is preferred to C;
and C is preferred to A?
Show solution
Solution
Yes, it can happen.
It is enough to have three voters, or three equal-sized groups of voters, with these preferences:
First group:
Second group:
Third group:
Now compare the candidates in pairs.
Between A and B:
the first group prefers A to B; the third group also prefers A to B; only the second group prefers B to A.
So A beats B by majority.
Between B and C:
the first group prefers B to C; the second group also prefers B to C; only the third group prefers C to B.
So B beats C by majority.
Between C and A:
the second group prefers C to A; the third group also prefers C to A; only the first group prefers A to C.
So C beats A by majority.
We therefore have:
Each individual voter is consistent, but the collective majority forms a cycle.
Answer: yes. The majority can prefer A to B, B to C, and C to A at the same time.