Drag voters' ideal points along a policy axis. The theorem predicts the group will always converge on the median — when preferences are single-peaked.
Voter ideal points — drag to move
Utility curves — each voter prefers policies closer to their ideal point
Single-peaked preference: A voter's utility has exactly one peak (their ideal point) and declines monotonically in both directions. It never dips and then rises again. A preference with a "valley" in the middle is not single-peaked.
Four examples — click each to see its curve
Voter A
Ideal: Left (20) Ranking: Left ≻ Centre ≻ Right
✓ Single-peaked
Voter B
Ideal: Centre (50) Ranking: Centre ≻ Left ≻ Right
✓ Single-peaked
Voter C
Ideal: Far Left (10) Ranking: Far Left ≻ Far Right ≻ Centre