In political science and social choice, Black's median voter theorem says that if voters and candidates are distributed along a one-dimensional political spectrum, any Condorcet consistent voting method will elect the candidate preferred by the median voter. The median voter theorem thus shows that under a realistic model of voter behavior, Arrow's theorem does not apply, and rational choice is possible for societies. The theorem was first derived by Duncan Black in 1948, and independently by Kenneth Arrow.
Similar median voter theorems exist for rules like score voting and approval voting when voters are either strategic and informed or if voters' ratings of candidates fall linearly with ideological distance.
An immediate consequence of Black's theorem, sometimes called the Hotelling-Downs median voter theorem, is that if the conditions for Black's theorem hold, politicians who only care about winning the election will adopt the same position as the median voter.Anthony Downs, "An Economic Theory of Democracy" (1957). However, this strategic convergence only occurs in voting systems that actually satisfy the median voter property (see below), which would exclude all rules currently in use for national elections (party primaries, two-round systems, first-preference plurality, and instant-runoff voting).
The median mandate perspective views the median voter preferences as granting political legitimacy.
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 is 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.See Black's paper. The assumption that opinions lie along a real line can be relaxed to allow more general topologies.Berno Buechel, "Condorcet winners on median spaces" (2014). 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.
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.B. Grofman and S. L. Feld, "If you like the alternative vote (a.k.a. the instant runoff), then you ought to know about the Coombs rule" (2004). Approval voting satisfies the same property under several models of strategic voting.
Extensions to higher dimensions In higher dimensional space the median can be generalized to the geometric median. For spatial models the McKelvey–Schofield chaos theorem shows that there might be no Condorcet winner. However, it is still possible to demonstrate similar theorems under some limited conditions.
RankingVotesA-B-C30B-A-C29C-A-B10B-C-A10A-C-B1C-B-A1 Number of votersA > B41:40A > C60:21B > C69:12Total81 The table shows an example of an election given by the Marquis de Condorcet, who concluded it showed a problem with the Borda count.George G. Szpiro, "Numbers Rule" (2010). 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).Eric Pacuit, "Voting Methods", The Stanford Encyclopedia of Philosophy (Fall 2019 Edition), Edward N. Zalta (ed.).
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.See Valerio Dotti's thesis "Multidimensional Voting Models" (2016).
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.C. R. Plott, "A Notion of Equilibrium and its Possibility Under Majority Rule" (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 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.
Downsian model
The Downsian model (also called the Hotelling–Downs model) builds on Harold Hotelling's principle of minimum differentiation, also known as Hotelling's law. Anthony Downs adapted Hotelling's spatial competition framework to politics in 1957, creating a model that predicts politicians will converge to the median voter's position under four conditions:
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.
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.
Hotelling's original principle was first described in 1929 for business competition, while Downs later applied this framework to electoral politics. In case of the Duverger's law with a two-party system the Downsian model predicts the two parties are incentivized to move towards the median voter.