Editing Median voter theorem (section)

=== Omnidirectional medians ===
[[File:Median Voter Theorem.svg|thumb|The median voter theorem in two dimensions]]Despite this result, the median voter theorem can be applied to distributions that are rotationally symmetric, e.g. [[Multivariate normal distribution|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.<ref name="dotti-2016">See Valerio Dotti's thesis [https://discovery.ucl.ac.uk/id/eprint/1516004/1/thesis_Valerio_Dotti_final.pdf "Multidimensional Voting Models"] (2016).</ref>

It follows that all [[Median voter criterion|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 chaos theorem|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.

[[File:Voting Paradox example.png|thumb|left|A distribution with no median in all directions]]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.