When voters choose by majority rule on one issue dimension and have single-peaked preferences, the option preferred by the median voter beats every alternative in a head-to-head vote.
In more detail
Duncan Black proved the result in 1948, and Anthony Downs (1957) applied it to party competition, predicting that two vote-seeking parties converge on the median voter's position. The theorem depends on one dimension, single-peaked preferences and full turnout. With several issue dimensions, abstention, activists or more than two parties, convergence is not guaranteed and empirical support is mixed.