prefer Charles to all candidates to his right, and Marlene and all voters to her right will prefer Charles to all candidates to his left. Assumptions The theorem also applies when the number of voters is even, but the details depend on how ties are resolved. 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. History The theorem was first set out by Duncan Black in 1948. He wrote that he saw a large gap in economic theory concerning how voting determines the outcome of decisions, including political decisions. Black's paper triggered research on how economics can explain voting systems. In 1957 Anthony Downs expounded upon the median voter theorem in his book An Economic Theory of Democracy. 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 summarise 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. Extension to distributions in more than one dimension The median voter theorem applies in a restricted form to distributions of voter opinions in spaces of any dimension. A distribution in more than one dimension does not necessarily have a median in all directions (which might be termed an 'omnidirectional median'); however a broad class of rotationally symmetric distributions, including the Gaussian, does have a median of this sort. Whenever the distribution of voters has a unique median in all directions, and voters rank candidates in order of proximity, the median voter theorem applies: the candidate closest to the median will have a