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Inpolitical science andsocial choice,Black'smedian voter theorem says that if voters and candidates are distributed along apolitical spectrum, anyCondorcet consistent voting method will elect the candidate preferred by themedian voter.[1][2] The median voter theorem thus shows that under a realistic model of voter behavior,Arrow's theorem does not apply, andrational choice is possible for societies. The theorem was first derived byDuncan Black in 1948,[3] and independently byKenneth Arrow.
Similar median voter theorems exist for rules likescore voting andapproval voting[4][5] when voters are eitherstrategic and informed or if voters' ratings of candidatesfall linearly with ideological distance.
Animmediate consequence of Black's theorem, sometimes called theHotelling-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.[6][7][8] However, this strategic convergence only occurs in voting systems that actually satisfy the median voter property (see below).[9][5][10]
Say there is an election where candidates and voters have opinions distributed along a one-dimensionalpolitical 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 amajority-preferred (orCondorcet) 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 simplemajority vote satisfies this condition, while for multi-candidate votes any majority-rule (Condorcet) method will satisfy it.
Proof sketch: Let themedian 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. ∎
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.[13] Approval voting satisfies the same property under several models of strategic voting.
It is impossible to fully generalize the median voter theorem tospatial 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 theMarquis de Condorcet, who concluded it showed a problem with theBorda count.[14]: 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 thegeometric median).[15]
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'smean 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.[16]
It follows that allmedian 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 theMcKelvey–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 thegeometric 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 medianM in all directions, the data points not located atM must come in balanced pairs (A,A ' ) on either side ofM with the property thatA – M – A ' is a straight line (ie.not likeA 0 – M – A 2 in the diagram).
Proof. This result was proved algebraically by Charles Plott in 1967.[17] Here we give a simple geometric proof by contradiction in two dimensions.
Suppose, on the contrary, that there is a set of pointsAi which haveM as median in all directions, but for which the points not coincident withM do not come in balanced pairs. Then we may remove from this set any points atM, and any balanced pairs aboutM, withoutM ceasing to be a median in any direction; soM remains an omnidirectional median.
If the number of remaining points is odd, then we can easily draw a line throughM such that the majority of points lie on one side of it, contradicting the median property ofM.
If the number is even, say 2n, then we can label the pointsA 0,A1,... in clockwise order aboutM starting at any point (see the diagram). Let θ be the angle subtended by the arc fromM –A 0 toM –A n . Then if θ < 180° as shown, we can draw a line similar to the broken red line throughM which has the majority of data points on one side of it, again contradicting the median property ofM ; whereas if θ > 180° the same applies with the majority of points on the other side. And if θ = 180°, thenA 0 andA n form a balanced pair, contradicting another assumption.
Theorem. Whenever a discrete distribution has a medianM in all directions, it coincides with its geometric median.
Proof. The sum of distances from any pointP to a set of data points in balanced pairs (A,A ' ) is the sum of the lengthsA – P – A '. Each individual length of this form is minimized overP when the line is straight, as happens whenP coincides withM. The sum of distances fromP to any data points located atM is likewise minimized whenP andM coincide. Thus the sum of distances from the data points toP is minimized whenP coincides withM.
TheDownsian model[18][19][20] (also called theHotelling–Downs model) builds onHarold Hotelling'sprinciple of minimum differentiation, also known asHotelling'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:
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 theCondorcet criterion.
Hotelling's original principle was first described in 1929 for business competition,[7] while Downs later applied this framework to electoral politics. In practice, none of these conditions hold for modern American elections, though they may have held in Hotelling's time (when nominees were oftenpreviously-unknown and chosen by closedparty caucuses in ideologically diverse parties). Most importantly, politicians must winprimary 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 underelectoral systems that do not satisfy the median voter theorem, includingplurality voting,plurality-with-primaries,plurality-with-runoff, orranked-choice runoff (RCV).[5][21]
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:
TheMedian 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.[16]
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),[22] the study of the determinants of immigration policies (Razin and Sadka, 1999),[23] of the extent of taxation on different types of income (Bassetto and Benhabib, 2006),[24] and many more.
In theUnited StatesSenate, eachstate 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.[25]
Pande (2003) studied political changes inIndia 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.[26]
Chattopadhyay andDuflo (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 ofpublic 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.[27]
Similar findings were reported by Miller (2008), who analyzed the impact ofgranting 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.[28]
Lee, Moretti, and Butler (2004) investigated whether voters influence politicians' positions or merely choose from existing policy stances. They found that anexogenous shift in the voter base does not alter candidates' positions. For instance, an increase inDemocratic voters in a given area does not push aRepublican 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.[29]
Gerber and Lewis (2015) analyzed voting data from a series of referendums inCalifornia 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.[30]
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 incomedecile. This finding aligns with other studies suggesting that low-income voters often form coalitions with high-income voters to oppose increases in public spending.[31]
Referendum data fromSwitzerland 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.[32]
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.[33]
