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Electoral systems 


In political science and social choice theory, Black's median voter theorem states that if voters and candidates are distributed along a onedimensional 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:
 It shows that under a realistic model of voter behavior, Arrow's theorem does not apply, and rational social choice is in fact possible (with Condorcet methods).
 It justifies the median voter property, a voting system criterion generalizing the median voter theorem, which says election systems should choose the candidate most wellliked by the median voter, when the conditions of the median voter theorem apply.
Instantrunoff voting and plurality fail the criterion, while approval voting,^{[1]}^{[2]} Coombs' method, and all Condorcet methods^{[3]} 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 centersqueeze 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,^{[4]} basing this on his model of economic competition.^{[4]}^{[5]} 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 instantrunoff voting or plurality at all, even in twoparty systems.^{[2]}^{[6]}^{[note 1]}
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Median Voter Theorem Animation

The Theory of the Median Voter

Game Theory 101 (#40): Hotelling's Game and the Median Voter Theorem

Explainer: The Median Voter Theorem, Turnout, and the Democratic Primary

MEDIAN VOTER THEOREM (PUBLIC FINANCE/ECONOMICS)
Transcription
The Median Voter Theorem has its origins in Harold Hotelling's spatial model of firm strategies in a competitive marketplace. To help us understand the nuts and bolts of Hotelling's model, we will take the scenario where two parties, in our case two ice cream vendors, compete to make the most profit from sale of ice creams on a hypothetical neighborhood block. Let us assume for simplicity that all the houses on this hypothetical block are equally spaced and are of equal size with equal number of residents living in each house. We will assume that these two ice cream vendors sell the same product at the same low price. We will also assume that it is a hot sunny summer day and everyone on this hypothetical block will buy ice cream from one of these two vendors. Since both vendors sell the same product at the same price, the residents' decision to buy ice cream from one of these two vendors will depend on how close the ice cream truck is to their home. The customers' decision rule will be to buy ice cream from the vendor who is closest to their home. We will also assume that these ice cream vendors are mobile. They can push the truck from one end of the street to the other. They are also able to move their truck around the other as well. The goal of both vendor is to make the most profit from the sale of their ice cream. Let us assume that Joe will start his business by placing himself at the west end of the block. Having no competitors, he has the advantage of having all the customers on that block coming to buy ice cream from his truck. Now let us introduce Joe's competition, Ted, who places his ice cream business on the east side of the block. The two vendors now split the customers down the middle. Joe takes every customer from the center to the west end of the block while Ted takes every customer from the center to the east. What is the winning strategy for one of these two vendors? Holding all else constant, if Ted pushes his truck towards Joe, he can have some of Joe's customers on the west side of the street to come and buy ice cream from Ted's truck. Having said this, is there an optimal strategy for Ted if all else is held constant? Clearly, the best strategy is for Ted to situate his truck right next to Joe so that all the customers to the east side of Joe's truck will come to buy ice cream from Ted. Now, let us loosen one of our conditions about Joe staying put. If we allow Joe to react to Ted, is there something that Joe can do to better his chances of winning this competition of ice cream sales against Ted? The answer is “yesâ€? he can. Holding Ted's place constant, Joe can simply situate his cart to the right side of Ted. Ted, on the other hand, can move his cart to the right side of Joe; and Joe can do the same. They will continue. But will they ever stop? And when they do, where do they stop? The answer is yes they will stop. They stop in the middle of this block. The reason for this is that it serves neither Ted nor Joe any better to move away from the center of the block. That is, their marginal gain in sale of ice cream from an incremental move away from the center of the block is less than zero. They actually lose sales if either vendor moves away from the center of the block. The actual theoretical basis for the convergence of these two players to the middle of the street is based on a game theoretic concept of "iterative deletion of dominated strategies". That is, for each successive deletion of the dominated strategies from either end of the street, the strategy that leads the players to the middle of the street allows the players to be better off no matter where the other player is positioned. Now let's take a look at how the spatial model of strategic firm behavior translates into party strategy in a competitive electoral democracy. Assume now that instead of ice cream vendors, we have parties. And instead of selling ice creams, each party espouses a particular ideology, which can be mapped on a unidimensional ideological space running from extreme liberal to extreme conservative. We will assume also that the voting population is equally distributed along this ideological plane. We will also assume that each voter's decision to vote for either one of these two parties will depend on how close each party's ideological stance is relative to the voter's ideology. If the electoral rule is such that the party with the most number of votes is the winner and we assume that voter participation rate is 100%, what is the party's winning strategy? Similar to the ice cream vendors, each party's winning strategy will be to situate themselves right next to each other at the middle of the ideological spectrum. The middle will be referred to as the median. We will also change one of the assumptions about the voting population without losing generality of our model. Instead of assuming that the voting population is uniformly distributed across the ideological spectrum, we will assume a normal distribution. That is, the distribution of the voting population is single peaked and symmetric around the median. This is not an unrealistic assumption to make since we are suggesting that there are less ideological extremists than there are moderates. The winning strategy still remains the same for each party. Each party will seek to seize the median voter.
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 onedimensional. This means that the opinions of both candidates and voters are distributed along a onedimensional 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:
 In onedimensional elections, a Condorcet winner always exists.
 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 singlepeaked.^{[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.
History
The theorem was first set out by Duncan Black in 1948.^{[9]} 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.^{[10]}
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 onedimensional 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 Condorcetconsistent but nonetheless satisfies the median voter property in one dimension.^{[11]} 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 it is possible under some limited conditions.
Nongeneralization
Ranking  Votes 

ABC  30 
BAC  29 
CAB  10 
BCA  10 
ACB  1 
CBA  1 
Number of voters  

A > B  41:40 
A > C  60:21 
B > C  69:12 
Total  81 
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.^{[12]}^{: 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 it is the Borda count that correctly identifies the candidate closest to the center (as determined by the geometric median).^{[13]}
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.^{[14]}
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 onedimensional projections. If A, B and C are the candidates, then '1' will vote ABC, '2' will vote BCA, and '3' will vote CAB, 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 onedimensional 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 A_{ 0 }– M – A_{ 2} in the diagram).
Proof. This result was proved algebraically by Charles Plott in 1967.^{[15]} 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.
HotellingDowns median voter 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 choose ideological positions solely with the intention of winning the election (not based on 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.^{[5]} In practice, none of these conditions hold for modern American elections, though they may have held in Hotelling's time (when nominees were publiclyunknown 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 majorparty 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, pluralitywithprimaries, pluralitywithrunoff, or rankedchoice runoff (RCV).^{[2]}^{[16]}
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.^{[14]}
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),^{[17]} the study of the determinants of immigration policies (Razin and Sadka, 1999),^{[18]} of the extent of taxation on different types of income (Bassetto and Benhabib, 2006),^{[19]} and many more.
See also
 Arrow's impossibility theorem
 McKelvey–Schofield chaos theorem
 Median mechanism
 Ranked voting
 Median voting rule
Notes
 ^ An exception exists if third parties and independent candidacies are not just "unimportant", but completely incapable of winning any votes, even in theory (e.g. being prohibited from appearing on the ballot by sore loser laws). In other words, there must be no possibility that voters will defect to a thirdparty campaign; it is not just enough for voters not to actually defect, as it is possible the voters would switch to a thirdparty candidate if the majorparty candidate moved to the center. When such conditions hold, the election is decided by a simple majority between the two candidates, which is a Condorcet method.
References
 ^ Cox, Gary W. (1985). "Electoral Equilibrium under Approval Voting". American Journal of Political Science. 29 (1): 112–118. doi:10.2307/2111214. ISSN 00925853. JSTOR 2111214.
 ^ ^{a} ^{b} ^{c} Myerson, Roger B.; Weber, Robert J. (March 1993). "A Theory of Voting Equilibria". American Political Science Review. 87 (1): 102–114. doi:10.2307/2938959. hdl:10419/221141. ISSN 15375943. JSTOR 2938959.
 ^ P. Dasgupta and E. Maskin, "The fairest vote of all" (2004); "On the Robustness of Majority Rule" (2008).
 ^ ^{a} ^{b} Holcombe, Randall G. (2006). Public Sector Economics: The Role of Government in the American Economy. Pearson Education. p. 155. ISBN 9780131450424.
 ^ ^{a} ^{b} Hotelling, Harold (1929). "Stability in Competition". The Economic Journal. 39 (153): 41–57. doi:10.2307/2224214. JSTOR 2224214.
 ^ Mussel, Johanan D.; Schlechta, Henry (20230721). "Australia: No party convergence where we would most expect it". Party Politics. doi:10.1177/13540688231189363. ISSN 13540688.
 ^ See Black's paper.
 ^ Berno Buechel, "Condorcet winners on median spaces" (2014).
 ^ Black, Duncan (19480201). "On the Rationale of Group Decisionmaking". Journal of Political Economy. 56 (1): 23–34. doi:10.1086/256633. ISSN 00223808. S2CID 153953456.
 ^ Anthony Downs, "An Economic Theory of Democracy" (1957).
 ^ 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).
 ^ George G. Szpiro, "Numbers Rule" (2010).
 ^ Eric Pacuit, "Voting Methods", The Stanford Encyclopedia of Philosophy (Fall 2019 Edition), Edward N. Zalta (ed.).
 ^ ^{a} ^{b} See Valerio Dotti's thesis "Multidimensional Voting Models" (2016).
 ^ C. R. Plott, "A Notion of Equilibrium and its Possibility Under Majority Rule" (1967).
 ^ Robinette, Robbie (20230901). "Implications of strategic position choices by candidates". Constitutional Political Economy. 34 (3): 445–457. doi:10.1007/s10602022093786. ISSN 15729966.
 ^ A. H. Meltzer and S. F. Richard, "A Rational Theory of the Size of Government" (1981).
 ^ A. Razin and E. Sadka "Migration and Pension with International Capital Mobility" (1999).
 ^ M. Bassetto and J. Benhabib, "Redistribution, Taxes, and the Median Voter" (2006).
Further reading
 Buchanan, James M.; Tollison, Robert D. (1984). The Theory of Public Choice. Vol. II. Ann Arbor: University of Michigan Press. ISBN 0472080415.
 Clinton, Joshua D. (2006). "Representation in Congress: Constituents and the Roll Calls in the 106th House". Journal of Politics. 68 (2): 397–409. doi:10.1111/j.14682508.2006.00415.x.
 Congleton, Roger (2003). "The Median Voter Model" (PDF). In Rowley, C. K.; Schneider, F. (eds.). The Encyclopedia of Public Choice. Kluwer Academic Press. ISBN 9780792386070.
 Dasgupta, Partha and Eric Maskin, "On the Robustness of Majority Rule", Journal of the European Economic Association, 2008.
 Downs, Anthony (1957). "An Economic Theory of Political Action in a Democracy". Journal of Political Economy. 65 (2): 135–150. doi:10.1086/257897. S2CID 154363730.
 Holcombe, Randall G. (1980). "An Empirical Test of the Median Voter Model". Economic Inquiry. 18 (2): 260–275. doi:10.1111/j.14657295.1980.tb00574.x.
 Holcombe, Randall G.; Sobel, Russell S. (1995). "Empirical Evidence on the Publicness of State Legislative Activities". Public Choice. 83 (1–2): 47–58. doi:10.1007/BF01047682. S2CID 44831293.
 Husted, Thomas A.; Kenny, Lawrence W. (1997). "The Effect of the Expansion of the Voting Franchise on the Size of Government". Journal of Political Economy. 105 (1): 54–82. doi:10.1086/262065. S2CID 41897793.
 Krehbiel, Keith (2004). "Legislative Organization". Journal of Economic Perspectives. 18 (1): 113–128. doi:10.1257/089533004773563467. S2CID 249607866.
 McKelvey, Richard D. (1976). "Intransitives in Multidimensional Voting Models and Some Implications for Agenda Control". Journal of Economic Theory. 12 (3): 472–482. doi:10.1016/00220531(76)900405.
 Schummer, James; Vohra, Rakesh V. (2013). "Mechanism Design Without Money". In Nisan, Noam; Roughgarden, Tim; Tardos, Eva; Vazirani, Vijay (eds.). Algorithmic Game Theory. New York: Cambridge University Press. pp. 246–252. ISBN 9780521872829.
 Rice, Tom W. (1985). "An Examination of the Median Voter Hypothesis". Western Political Quarterly. 38 (2): 211–223. doi:10.2307/448625. JSTOR 448625.
 Romer, Thomas; Rosenthal, Howard (1979). "The Elusive Median Voter". Journal of Public Economics. 12 (2): 143–170. doi:10.1016/00472727(79)900100.
 Sobel, Russell S.; Holcombe, Randall G. (2001). "The Unanimous Voting Rule is not the Political Equivalent to Market Exchange". Public Choice. 106 (3–4): 233–242. doi:10.1023/A:1005298607876. S2CID 16736216.
 Waldfogel, Joel (2008). "The Median Voter and the Median Consumer: Local Private Goods and Population Composition". Journal of Urban Economics. 63 (2): 567–582. doi:10.1016/j.jue.2007.04.002. S2CID 152378898. SSRN 878059.