# Median voter theorem

The median voter theorem is a proposition relating to ranked preference voting put forward by Duncan Black in 1948.[1] It states that if voters and policies are distributed along a one-dimensional spectrum, with voters ranking alternatives in order of proximity, then any voting method which satisfies the Condorcet criterion will elect the candidate closest to the median voter. In particular, a majority vote between two options will do so.

The theorem is associated with public choice economics and statistical political science. Partha Dasgupta and Eric Maskin have argued that it provides a powerful justification for voting methods based on the Condorcet criterion.[2]

A loosely related assertion had been made earlier (in 1929) by Harold Hotelling.[3] It is not a true theorem and is more properly known as the median voter theory or median voter model. It says that in a representative democracy, politicians will converge to the viewpoint of the median voter.[4]

## 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 his or her first preference, the next closest receives his or her 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.

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. The Condorcet criterion can be considered as a method in its own right (the Condorcet method), and has a natural extension due to Ramon Llull (1299), sometimes known as Copeland's method.

### 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.[5]

The assumption that opinions lie along a real line can be relaxed to allow more general topologies.[6]

### 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 with his paper titled An Economic Theory of Political Action in Democracy, Anthony Downs expounded upon the median voter theorem.[7]

### Extension to distributions in more than one dimension

The median voter theorem in two dimensions

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 an omnidirectional median, i.e. a point which coincides with the one-dimensional median for every projection of the distribution onto a single dimension. However a broad class of rotationally symmetric distributions, including the Gaussian, does have a median of this sort. Whenever the distribution of voters has an omnidirectional median, 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 Condorcet criterion.

Proof. See the diagram, in which the grey shading represents the density of the voter distribution and M is the omnidirectional median. Let A and B be two candidates, of whom A is the closer to the median. Then the voters who rank A above B are precisely the ones to the left (i.e. the 'A' side) of the solid red line; and since A is closer than B to M, the median is also to the left of this line.

A distribution with no omnidirectional median

Now, since M is an omnidirectional median, it coincides with the one-dimensional median in the particular case of the direction shown by the blue arrow, which is perpendicular to the solid red line. Thus if we draw a broken red line through M, perpendicular to the blue arrow, then we can say that half the voters lie to the left of this line. But since this line is itself to the left of the solid red line, it follows that more than half of the voters will rank A above B. ∎

It is easy to construct voter distributions with no omnidirectional median. The simplest example consists of a distribution limited to 3 points not lying in a straight line, such as 1, 2 and 3 in the second diagram. Each voter location coincides with the median under a certain set of one-dimensional projections. If A, B and C are the candidates, then 1 will vote A-B-C, 2 will vote B-C-A, and 3 will vote C-A-B, giving a Condorcet cycle.

## Hotelling'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.[3]

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.

## References

1. ^ Duncan Black, 'On the Rationale of Group Decision-making' (1948).
2. ^ P. Dasgupta and E. Maskin, "The fairest vote of all" (2004); "On the Robustness of Majority Rule" (2008).
3. ^ a b Hotelling, Harold (1929). "Stability in Competition". The Economic Journal. 39 (153): 41–57. doi:10.2307/2224214. JSTOR 2224214.
4. ^ Holcombe, Randall G. (2006). Public Sector Economics: The Role of Government in the American Economy. p. 155. ISBN 9780131450424.
5. ^ See Black's paper.
6. ^ Berno Buechel, 'Condorcet winners on median spaces' (2014).
7. ^ Downs, Anthony (1957). An Economic Theory of Democracy. Harper Collins.