In political science and social choice, Black's median voter theorem says that if voters and candidates are distributed along a political spectrum, any Condorcet consistent voting method will elect the candidate preferred by the median voter. The median voter theorem thus shows that under a realistic model of voter behavior, Arrow's theorem does not apply, and rational choice is possible for societies. The theorem was first derived by Duncan Black in 1948, and independently by Kenneth Arrow.

Similar median voter theorems exist for rules like score voting and approval voting when voters are either strategic and informed or if voters' ratings of candidates fall linearly with ideological distance.

An immediate consequence of Black's theorem, sometimes called the Hotelling-Downs median voter theorem, is that if the conditions for Black's theorem hold, politicians who only care about winning the election will adopt the same position as the median voter. However, this strategic convergence only occurs in voting systems that actually satisfy the median voter property (see below).

Say there is an election where candidates and voters have opinions distributed along a one-dimensional political spectrum. Voters rank candidates by proximity, i.e. the closest candidate is their first preference, the second-closest is their second preference, and so on. Then, the median voter theorem says that the candidate closest to the median voter is a majority-preferred (or Condorcet) candidate. In other words, this candidate preferred to any one of their opponents by a majority of voters. When there are only two candidates, a simple majority vote satisfies this condition, while for multi-candidate votes any majority-rule (Condorcet) method will satisfy it.

Proof sketch: Let the median voter be Marlene. The candidate who is closest to her will receive her first preference vote. Suppose that this candidate is Charles and that he lies to her left. Marlene and all voters to her left (by definition a majority of the electorate) will prefer Charles to all candidates to his right, and Marlene and all voters to her right (also a majority) will prefer Charles to all candidates to his left. ∎

We will say that a voting method has the "median voter property in one dimension" if it always elects the candidate closest to the median voter under a one-dimensional spatial model. We may summarize the median voter theorem as saying that all Condorcet methods possess the median voter property in one dimension.

It turns out that Condorcet methods are not unique in this: Coombs' method is not Condorcet-consistent but nonetheless satisfies the median voter property in one dimension. Approval voting satisfies the same property under several models of strategic voting.

It is impossible to fully generalize the median voter theorem to spatial models in more than one dimension, as there is no longer a single unique "median" for all possible distributions of voters. However, it is still possible to demonstrate similar theorems under some limited conditions.