# Median voter theorem

In political science and social choice, the 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 that is compatible with majority-rule will elect the candidate preferred by the median voter.
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.
A different argument due to Anthony Downs and Harold Hotelling is only loosely-related to Black's median voter theorem, but is often confused with it. This model argues that politicians in a representative democracy will converge to the viewpoint of the median voter, because the median voter theorem implies that a candidate who wishes to win will adopt the positions of the median voter. However, this argument only applies to systems satisfying the median voter property, and cannot be applied to systems like ranked choice voting (RCV) or plurality voting outside of limited conditions (see § Hotelling–Downs model).

## 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 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 v$_j$ – d$_{ij}$ where v$_j$ is j 's valence and d$_{ij}$ 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

It is impossible to fully generalize the median voter theorem to spatial models in more than one dimension, as there is no longer a single unique "median" for all possible distributions of voters. However, it is still possible to demonstrate similar theorems under some limited conditions.
| Ranking | Votes |
| A-B-C | 30 |
| B-A-C | 29 |
| C-A-B | 10 |
| B-C-A | 10 |
| A-C-B | 1 |
| C-B-A | 1 |
| | Number of voters |
| A > B | 41:40 |
| A > C | 60:21 |
| B > C | 69:12 |
| Total | 81 |
The table shows an example of an election given by the Marquis de Condorcet, who concluded it showed a problem with the Borda count.$^{:90}$ 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.

### 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 median voter methods satisfy the same 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. 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 A$_0$– M – A$_2$ 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 A$_i$ 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 A$_0$, A$_1$,... in clockwise order about M starting at any point (see the diagram). Let θ be the angle subtended by the arc from M –A$_0$ to M –A$_n$. Then if θ < 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 > 180° the same applies with the majority of points on the other side. And if θ = 180°, then A$_0$ and A$_n$ 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 model

A related observation was discussed by Harold Hotelling as his 'principle of minimum differentiation', also known as 'Hotelling's law'. It states that if:
* Candidates can choose ideological positions without consequence,
* Candidates only care about winning the election (not their actual beliefs),
* All other criteria of the median voter theorem are met (i.e. voters rank candidates by ideological distance),
* The voting system satisfies the median voter criterion,
Then all politicians will converge to the median voter. As a special case, this law applies to the situation where there are exactly two candidates in the race, if it is impossible or implausible that any more candidates will join the race, because a simple majority vote between two alternatives satisfies the Condorcet criterion.
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 often previously-unknown and 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. As a result, politicians must compromise between appealing to the median voter in 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 plurality voting, plurality-with-primaries, plurality-with-runoff, or ranked-choice runoff (RCV).

## 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.

## Empirical evidence and contradictions

In the United States Senate, each state is allocated two seats. Levitt (1996) examined the voting patterns of pairs of senators from the same state when one belonged to the Democratic Party and the other to the Republican Party. According to the Median Voter Theorem, the voting patterns of two senators representing the same state should be identical, regardless of party affiliation. However, reality differs. Moreover, Levitt found that the similarity in their voting patterns was only slightly higher than that of randomly paired senators. This finding suggests that senators' ideological leanings have a stronger influence on their decisions than voters' preferences, contradicting the prediction of the Median Voter Theorem.
Pande (2003) studied political changes in India between 1960 and 1992 that increased political representation for marginalized groups. The data she collected showed that as a result of these changes, transfer payments to these populations increased even though the overall electorate (which had already included these groups) remained unchanged. This finding contradicts the Median Voter Theorem, as the model predicts that such a political shift should not alter the political equilibrium.
Chattopadhyay and Duflo (2004) examined another political change in India, which mandated that women lead one-third of village councils. These councils are responsible for providing various public goods to rural communities. According to the Median Voter Theorem, this policy should not have affected the composition of public goods supplied by local governments, as a female candidate still needs to be elected by a majority vote. As long as the median voter's preferences remain unchanged, the allocation of public goods should remain stable. However, empirical data showed that in villages where a woman was elected, the distribution of public goods shifted toward those preferred by women. Furthermore, in districts where women were elected for a second term, the allocation of public goods continued to reflect women's preferences. It is important to note, however, that while the composition of public goods changed when a woman led the village council, this does not necessarily imply an improvement or decline in overall social welfare.
Similar findings were reported by Miller (2008), who analyzed the impact of granting women the right to vote across the United States in 1920. Miller built on previous research indicating that women prioritize child welfare more than men and demonstrated that extending voting rights to women led to an immediate shift in federal policy. This change resulted in a significant increase in healthcare spending and a consequent reduction in child mortality rates by 8%–15%. However, unlike previous cases, Miller's findings actually support the Median Voter Theorem. This is because granting women suffrage altered the composition of the electorate, shifting the median voter’s position toward the preferences of the new female voters.
Lee, Moretti, and Butler (2004) investigated whether voters influence politicians' positions or merely choose from existing policy stances. They found that an exogenous shift in the voter base does not alter candidates' positions. For instance, an increase in Democratic voters in a given area does not push a Republican candidate’s stance further to the left, and vice versa. This finding suggests that the electorate selects from the positions that politicians already hold, rather than shaping those positions, contradicting the prediction of the Median Voter Theorem, which assumes candidates are ideologically neutral.
Gerber and Lewis (2015) analyzed voting data from a series of referendums in California to estimate the preferences of the median voter. They found that elected officials are constrained by the preferences of the median voter in homogeneous regions but less so in heterogeneous ones.
In contrast, Brunner and Ross (2010), who also studied voter data from two referendums in California, found that the decisive voter in votes concerning public expenditure was not the median voter, but rather a voter from the fourth income decile. This finding aligns with other studies suggesting that low-income voters often form coalitions with high-income voters to oppose increases in public spending.
Referendum data from Switzerland was used by Stadelmann, Portmann, and Eichenberger (2012) to examine the degree to which legislators' votes align with the preferences of the median voter in their districts. Their research showed that the Median Voter Model explains legislative voting behavior better than an alternative random voting hypothesis, but only by a modest margin of 17.6%. Additionally, they found that support from the median voter in a senator’s district increases the likelihood of the senator supporting a given proposal by 8.4% in parliament.
Milanovic (2000), using data from 79 countries, concluded that the greater the inequality in a country's pre-tax income distribution, the more aggressive the redistributive policies of the winning government. This finding supports the Median Voter Theorem.