Median voter theorem

In political science and social choice theory, Black's median voter theorem states that if voters and candidates are distributed along a one-dimensional spectrum and voters have single peaked preferences, any voting method satisfying the Condorcet criterion will elect the candidate preferred by the median voter.

The median voter theorem thus serves two important purposes:


 * 1) It shows that under a somewhat-realistic model of voter behavior, most voting systems will produce similar results.
 * 2) It justifies the median voter property, a voting system criterion generalizing the median voter theorem, which says election systems should choose the candidate most well-liked by the median voter, when the conditions of the median voter theorem apply.

Instant-runoff voting and plurality fail the criterion, while approval voting, Coombs' method, and all Condorcet methods satisfy it. Score voting satisfies the property under strategic and informed voting (where it is equivalent to approval voting), or if voters’ ratings of candidates fall linearly with ideological distance. Systems that fail the median voter criterion exhibit a center-squeeze phenomenon, encouraging extremism rather than moderation.

A related assertion was made earlier (in 1929) by Harold Hotelling, who argued politicians in a representative democracy would converge to the viewpoint of the median voter, basing this on his model of economic competition. However, this assertion relies on a deeply simplified voting model, and is only partly applicable to systems satisfying the median voter property. It cannot be applied to systems like instant-runoff voting or plurality at all, even in two-party systems.

Statement and proof of the theorem
Consider a group of voters who have to elect one from a set of two or more candidates. For simplicity, assume the number of voters is odd.

The elections are one-dimensional. This means that the opinions of both candidates and voters are distributed along a one-dimensional spectrum, and 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.

A Condorcet winner is a candidate who is preferred over every other candidate by a majority of voters. In general, a Condorcet winner might not exist. The median voter theorem says that:


 * 1) In one-dimensional elections, a Condorcet winner always exists.
 * 2) The Condorcet winner is the candidate closest to the median voter.

In the above example, the median voter is denoted by M, and the candidate closest to him is C, so the median voter theorem says that C is the Condorcet winner. It follows that Charles will win any election conducted using a method satisfying the Condorcet criterion. In particular, when there are only two candidates, the majority rule satisfies the Condorcet criterion; for multiway votes, several methods satisfy it (see Condorcet method).

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. 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.

Extensions

 * 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 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 general, it is impossible to generalize the median voter theorem to spatial models in more than one dimension, though several attempts have been made to do so.

Nongeneralization
The table on the left 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 it is the Borda count that 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.

Omnidirectional medians
Despite this result, the median voter theorem can be applied to distributions that are rotationally symmetric, e.g. Gaussians, which have a single median that is the same in all directions. 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.

It follows that all Condorcet methods – and also Coombs' method – satisfy the median voter property in spaces of any dimension for voter distributions with omnidirectional medians.

It is easy to construct voter distributions which do not have a median in all directions. 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. This is the subject of the McKelvey–Schofield theorem.

Proof. See the diagram, in which the grey disc represents the voter distribution as uniform over a circle and M is the median in all directions. 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.

Now, since M is a median in all directions, 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.

Relation 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 the geometric median can arguably be identified as the ideal winner of a ranked preference election (see comparison of electoral systems). It is therefore important to know the relationship between the two. In fact whenever a median in all directions exists (at least for the case of discrete distributions), it coincides with the geometric median.

Lemma. Whenever a discrete distribution has a median M in all directions, the data points not located at M  must come in balanced pairs (A,A ' ) on either side of M  with the property that A – M – A ' is a straight line (ie. not like A0– M – A2 in the diagram).

Proof. This result was proved algebraically by Charles Plott in 1967. Here we give a simple geometric proof by contradiction in two dimensions.

Suppose, on the contrary, that there is a set of points Ai which have M as median in all directions, but for which the points not coincident with M  do not come in balanced pairs. Then we may remove from this set any points at M, and any balanced pairs about M, without M ceasing to be a median in any direction; so M  remains an omnidirectional median.

If the number of remaining points is odd, then we can easily draw a line through M such that the majority of points lie on one side of it, contradicting the median property of M.

If the number is even, say 2n, then we can label the points A0, A1,... in clockwise order about M starting at any point (see the diagram). Let θ be the angle subtended by the arc from M –A0 to M –An. Then if θ &lt; 180° as shown, we can draw a line similar to the broken red line through M which has the majority of data points on one side of it, again contradicting the median property of M ; whereas if θ &gt; 180° the same applies with the majority of points on the other side. And if θ = 180°, then A0 and An form a balanced pair, contradicting another assumption.

Theorem. Whenever a discrete distribution has a median M in all directions, it coincides with its geometric median.

Proof. The sum of distances from any point P to a set of data points in balanced pairs (A,A ' ) is the sum of the lengths A – P – A '. Each individual length of this form is minimized over P when the line is straight, as happens when P coincides with M. The sum of distances from P to any data points located at M is likewise minimized when P and M  coincide. Thus the sum of distances from the data points to P is minimized when P coincides with M.

Hotelling-Downs median voter theorem
A related theorem was discussed by Harold Hotelling as his 'principle of minimum differentiation', also known as 'Hotelling's law'. It states that if the voting system satisfies the median voter property, both candidates care only about maximizing their margin of victory, and candidates can choose their views "freely" (with voters supporting the candidate most like them), all candidates will take the same set of positions as the median voter.

As a special case, this law applies to the situation where there are exactly two candidates in the race.


 * 1) There are exactly two candidates in the race, with the winner selected by a simple majority;
 * 2) It is impossible for more candidates to enter or exit the race;
 * 3) Voters and candidates differ only along a one-dimensional ideological spectrum;
 * 4) Candidates can choose any set of positions on policy issues they wish;
 * 5) Both candidates only care about maximizing their expected vote share.

This theorem was first described by Hotelling in 1929. In practice, none of these conditions hold for modern American elections, though they may have held in Hotelling's time (when nominees were publicly-unknown candidates chosen by closed party caucuses in ideologically diverse parties). Most importantly, politicians must win primary elections, which often include challengers or competitors, to be chosen as major-party nominees, meaning politicians must compromise between appealing to the primary and general electorates. Similar effects imply candidates do not converge to the median voter under electoral systems that do not satisfy the median voter theorem, including primary elections, plurality, the two-round system, or instant-runoff voting.

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 types of income (Bassetto and Benhabib, 2006), and many more."