In a one-dimensional policy space with single-peaked preferences and majority rule, outcomes converge on the median voter.
Hotelling's 1929 ice cream vendors on a beach both move to the centre; Black proved the median-voter result for single-peaked preferences in 1948, and Downs developed spatial party competition in 1957. The convergence version also assumes two vote-maximising candidates, and it fails where turnout, primaries or multiple parties change the incentive.
In practice
- Two supermarket chains open branches on the same corner and sell nearly identical ranges.
- Two candidates for a leadership post arrive at almost identical platforms by the final hustings, having started far apart.