voting · paradox
Condorcet paradox
Every option loses to another, one pair at a time.
Three friends are choosing a restaurant. Each person has a clear favorite order. When they vote in pairs, they get a circle instead of a winner.
Call the restaurants A, B, and C. Friend 1 ranks them A best, then B, then C. Friend 2 ranks them B best, then C, then A. Friend 3 ranks them C best, then A, then B.
Now they vote head-to-head. A versus B: Friend 1 and Friend 3 prefer A, so A wins 2–1. B versus C: Friend 1 and Friend 2 prefer B, so B wins 2–1. C versus A: Friend 2 and Friend 3 prefer C, so C wins 2–1.
A beats B, B beats C, and C beats A. There is no restaurant that beats the other two.
Every friend has a normal, consistent ranking. Nothing is random or contradictory inside one person's list. The contradiction appears only when the three rankings are combined with majority vote.
This is the Condorcet paradox: with three or more options, majority rule can produce a loop even when every voter is perfectly logical. Different ways of resolving the loop can pick different winners from the same votes.