Statement and proof of the theorem
Assume that there is an odd number of voters and at least two candidates, and assume that opinions are distributed along a spectrum. Assume that each voter ranks the candidates in an order of proximity such that the candidate closest to the voter receives their first preference, the next closest receives their second preference, and so forth. Then there is a median voter and we will show that the election will be won by the candidate who is closest to him or her. Proof — 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. Then Marlene and all voters to her left (comprising a majority of the electorate) will 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. ∎ The Condorcet criterion is defined as being satisfied by any voting method which ensures that a candidate who is preferred to every other candidate by a majority of the electorate will be the winner, and this is precisely the case with Charles here; so it follows that Charles will win any election conducted using a method satisfying the Condorcet criterion. Hence under any voting method which satisfies the Condorcet criterion the winner will be the candidate preferred by the median voter. For binary decisions the majority vote satisfies the criterion; for multiway votes several methods satisfy it (seeAssumptions
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''.Anthony Downs, " An Economic Theory of Democracy" (1957).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 majority preference over all his or her rivals, and will be elected by any voting method satisfying the median voter property in one dimension.See Valerio Dotti's thesiRelation between the median in all directions and the geometric median
Whenever a unique omnidirectional median exists, it determines the result of Condorcet voting methods. At the same time theHotelling's law
The more informal assertion – the median voter ''model'' – is related to Harold Hotelling's 'principle of minimum differentiation', also known as ' Hotelling's law'. It states that politicians gravitate toward the position occupied by the median voter, or more generally toward the position favored by the electoral system. It was first put forward (as an observation, without any claim to rigor) by Hotelling in 1929. Hotelling saw the behavior of politicians through the eyes of an economist. He was struck by the fact that shops selling a particular good often congregate in the same part of a town, and saw this as analogous the convergence of political parties. In both cases it may be a rational policy for maximizing market share. As with any characterization of human motivation it depends on psychological factors which are not easily predictable, and is subject to many exceptions. It is also contingent on the voting system: politicians will not converge to the median voter unless the electoral process does so. If an electoral process gives more weight to rural than to urban voters, then parties are likely to converge to policies which favor rural areas rather than to the true median.Uses of the median voter theorem
The theorem is valuable for the light it sheds on the optimality (and the limits to the optimality) of certain voting systems. Valerio Dotti points out broader areas of application:The ''Median Voter Theorem'' proved extremely popular in the Political Economy literature. The main reason is that it can be adopted to derive testable implications about the relationship between some characteristics of the voting population and the policy outcome, abstracting from other features of the political process.He adds that...
The median voter result has been applied to an incredible variety of questions. Examples are the analysis of the relationship between income inequality and size of governmental intervention in redistributive policies (Meltzer and Richard, 1981), the study of the determinants of immigration policies (Razin and Sadka, 1999), of the extent of taxation on different tyes of income (Bassetto and Benhabib, 2006),M. Bassetto and J. Benhabib, "Redistribution, Taxes, and the Median Voter" (2006). and many more.
See also
*References
Further reading
* * * * Dasgupta, Partha and Eric Maskin, "On the Robustness of Majority Rule", Journal of the European Economic Association, 2008. * * * * * * * * * * * {{cite journal , last=Waldfogel , first=Joel , title=The Median Voter and the Median Consumer: Local ''Private'' Goods and Population Composition , journal= Journal of Urban Economics , year=2008 , volume=63 , issue=2 , pages=567–582 , doi=10.1016/j.jue.2007.04.002 , ssrn=878059External link