Median voter theorem

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The median voter theorem in political science and social choice theory, developed by Duncan Black, 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.^[1] 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.^[2]

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,^[3] basing this on his model of economic competition.^[3][4] 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 ranked choice voting (RCV) or first-past-the-post at all, even in two-party systems.^{[5][6][note 1]}

The median voter theorem says that in dimensional elections, the candidate closest to the median voter is the majority-preferred (Condorcet) candidate, i.e. a candidate who a majority of voters prefer to every other candidate.

In a "one-dimensional" election, where the opinions of candidates and voters are distributed along a one-dimensional spectrum and a voter ranks candidates by proximity, such that the candidate closest to the voter receives their first preference, the next closest receives their second preference, and so forth, the median voter theorem says that "C," the candidate closest in views to the median voter "M," will be the Condorcet winner of any election conducted using a method satisfying the Condorcet criterion. In particular, when there are only two candidates, a majority rule system satisfies the Condorcet criterion; for multi-candidate votes, several methods 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.^[7]
• The assumption that opinions lie along a real line can be relaxed to allow more general topologies.^[8]
• 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.

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.^[9] Approval voting satisfies the same property under several models of strategic voting.

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.

The table shows an example of an election given by the Marquis de Condorcet, who concluded it showed a problem with the Borda count.^[10]: 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).^[11]

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.

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.^[12]

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.

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.^[13] 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, A₁,... 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.

A related observation was discussed by Harold Hotelling as his 'principle of minimum differentiation', also known as 'Hotelling's law'. It states that if:

1. Candidates choose ideological positions solely with the intention of winning the election (not based on their actual beliefs),
2. All other criteria of the median voter theorem are met (i.e. voters rank candidates by ideological distance),
3. 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.^[4] 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. 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).^[5][14]

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.^[12]

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),^[15] the study of the determinants of immigration policies (Razin and Sadka, 1999),^[16] of the extent of taxation on different types of income (Bassetto and Benhabib, 2006),^[17] and many more.

• Arrow's impossibility theorem
• McKelvey–Schofield chaos theorem
• Median mechanism
• Ranked voting
• Median voting rule

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• The Median Voter Model