Open main menu
Home
Random
Recent changes
Special pages
Community portal
Preferences
About Wikipedia
Disclaimers
Incubator escapee wiki
Search
User menu
Talk
Dark mode
Contributions
Create account
Log in
Editing
Median voter theorem
(section)
Warning:
You are not logged in. Your IP address will be publicly visible if you make any edits. If you
log in
or
create an account
, your edits will be attributed to your username, along with other benefits.
Anti-spam check. Do
not
fill this in!
== Extensions to higher dimensions == It is impossible to fully generalize the median voter theorem to [[Spatial model of voting|spatial 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. [[File:SaariExample.png|Saari's example of a domain where the Condorcet winner is not the socially-optimal candidate.|alt=Saari's example|thumb]] {| class="wikitable sortable floatleft" !Ranking !Votes |- | style="background:white" |A-B-C | style="background:white" |30 |- | style="background:white" |B-A-C | style="background:white" |29 |- | style="background:#fadadd" |C-A-B | style="background:#fadadd" |10 |- | style="background:#fadadd" |B-C-A | style="background:#fadadd" |10 |- | style="background:#cfeeee" |A-C-B | style="background:#cfeeee" |1 |- | style="background:#cfeeee" |C-B-A | style="background:#cfeeee" |1 |} {| class="wikitable floatleft" ! !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 the [[Marquis de Condorcet]], who concluded it showed a problem with the [[Borda count]].<ref name="george g-2010">George G. Szpiro, "Numbers Rule" (2010).</ref>{{rp|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 the [[geometric median]]).<ref>Eric Pacuit, [https://plato.stanford.edu/archives/fall2019/entries/voting-methods/ "Voting Methods"], The Stanford Encyclopedia of Philosophy (Fall 2019 Edition), Edward N. Zalta (ed.).</ref> 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 === [[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. ===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 [[Median#Multivariate median|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. [[File:Median Plott.svg|thumb|Diagram for the lemma]]'''''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''<sub> 0 </sub>– ''M'' – ''A''<sub> 2</sub> in the diagram). '''''Proof'''''. This result was proved algebraically by Charles Plott in 1967.<ref>C. R. Plott, "A Notion of Equilibrium and its Possibility Under Majority Rule" (1967).</ref> Here we give a simple geometric proof by contradiction in two dimensions. Suppose, on the contrary, that there is a set of points ''A<sub>i</sub>'' 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 2''n'', then we can label the points ''A''<sub> 0</sub>, ''A''<sub>1</sub>,... in clockwise order about ''M'' starting at any point (see the diagram). Let θ be the angle subtended by the arc from ''M'' –''A''<sub> 0</sub> to ''M'' –''A''<sub> ''n'' </sub>. 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''<sub> 0</sub> and ''A''<sub> ''n''</sub> 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''.
Edit summary
(Briefly describe your changes)
By publishing changes, you agree to the
Terms of Use
, and you irrevocably agree to release your contribution under the
CC BY-SA 4.0 License
and the
GFDL
. You agree that a hyperlink or URL is sufficient attribution under the Creative Commons license.
Cancel
Editing help
(opens in new window)