Editing Median voter theorem (section)

==Statement and proof of the theorem==
[[File:Median voter.png|left|thumb|443x443px|A [[proof without words]] of the median voter 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 [[Condorcet winner|''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. {{Tombstone}}

* The assumption that preferences are cast in order of proximity can be relaxed to say merely that they are [[single peaked preferences|single-peaked]].<ref>See Black's paper.</ref>
* The assumption that opinions lie along a real line can be relaxed to allow more general topologies.<ref>Berno Buechel, "Condorcet winners on median spaces" (2014).</ref>
* '''''Spatial&nbsp;/&nbsp;valence models:''''' Suppose that each candidate has a ''[[valence politics|valence]]'' (attractiveness) in addition to his or her position in space, and suppose that voter ''i'' ranks candidates ''j'' in decreasing order of ''v<sub>j</sub>''&nbsp;–&nbsp;''d<sub>ij</sub>'' where ''v<sub>j</sub>'' is ''j''&nbsp;'s valence and ''d<sub>ij</sub>'' 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.<ref name="b-2004">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).</ref> Approval voting satisfies the same property under several models of strategic voting.