In political science and social choice, Black's median voter theorem says that if voters and candidates are distributed along a one-dimensional 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), which would exclude all rules currently in use for national elections (party primaries, two-round systems, first-preference plurality, and instant-runoff voting). The median mandate views the median voter preferences grant political legitimacy.
Statement and proof of the theorem
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 is 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. ∎
The assumption that preferences are cast in order of proximity can be relaxed to say merely that they are single-peaked. The assumption that opinions lie along a real line can be relaxed to allow more general topologies. Spatial / valence models: Suppose that each candidate has a valence (attractiveness) in addition to his or her position in space, and suppose that voter i ranks candidates j in decreasing order of vj – dij where vj is j 's valence and dij is the distance from i to j. Then the median voter theorem still applies: Condorcet methods will elect the candidate voted for by the median voter.
The median voter property 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.
Extensions to higher dimensions In higher dimensional space the median can be generalized to the geometric median. For spatial models the McKelvey–Schofield chaos theorem shows that there might be no Condorcet winner. However, it is still possible to demonstrate similar theorems under some limited conditions.
The table shows an example of an election given by the Marquis de Condorcet, who concluded it showed a problem with the Borda count. The Condorcet winner on the left is A, who is preferred to B by 41:40 and to C by 60:21. The Borda winner is instead B. However, Donald Saari constructs an example in two dimensions where the Borda count (but not the Condorcet winner) correctly identifies the candidate closest to the center (as determined by the geometric median). The diagram shows a possible configuration of the voters and candidates consistent with the ballots, with the voters positioned on the circumference of a unit circle. In this case, A's mean absolute deviation is 1.15, whereas B's is 1.09 (and C's is 1.70), making B the spatial winner. Thus the election is ambiguous in that two different spatial representations imply two different optimal winners. This is the ambiguity we sought to avoid earlier by adopting a median metric for spatial models; but although the median metric achieves its aim in a single dimension, the property does not fully generalize to higher dimensions.
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