Kemeny–Young method: Difference between revisions
Add statistical interpretation and do a bit of cleanup.
m (changed "the the" to "the") |
(Add statistical interpretation and do a bit of cleanup.) |
||
Line 1:
{{Wikipedia|Kemeny–Young method}}
Each possible complete ranking of the candidates is given a "distance" score. For each pair of candidates, find the number of ballots that order them the opposite way as the given ranking. The distance is the sum across all such pairs. The ranking with the least distance wins.
The winning candidate is the top candidate in the winning ranking.
== Statistical interpretation ==
==Strategic Vulnerability==▼
The Kemeny-Young method produces the maximum likelihood estimate for a voting model where the voters know a best order of the candidates, and for each pair of candidates, ranks that pair correct with some probability <math>p > \frac 1 2 </math>, or reversed with probability <math>1-p</math>.<ref name="Young pp. 51–64">{{cite journal | last=Young | first=Peyton | title=Optimal Voting Rules | journal=Journal of Economic Perspectives | publisher=American Economic Association | volume=9 | issue=1 | date=1995-02-01 | issn=0895-3309 | doi=10.1257/jep.9.1.51 | pages=51–64}}</ref> This is called a Mallows model.<ref name="Tang 2018">{{cite arXiv | last=Tang | first=Wenpin | title=Mallows Ranking Models: Maximum Likelihood Estimate and Regeneration | date=2018-08-26 | eprint=1808.08507 | class=math.ST}}</ref>
Kemeny-Young is vulnerable to [[Tactical voting|compromising]], [[Tactical voting|burying]], and [[Strategic nomination|crowding]]. It fails [[cloneproofness]] because adding a clone can cause a non-clone to be elected, and this effect increases as the number of clones increases. ▼
▲Kemeny-Young is vulnerable to [[Tactical voting|compromising]], [[Tactical voting|burying]], and [[Strategic nomination|crowding]]. It fails [[
==Example==
Line 29 ⟶ 30:
It can be shown that this ranking is the one with the lowest distance score (this is because this is the [[Condorcet ranking]], and therefore switching any pair of candidates would require overturning the majority of voters in that pairing rather than the minority). Therefore, the winning ranking is Nashville>Chattanooga>Knoxville>Memphis, and so the winning candidate is Nashville.
===Example with a Condorcet cycle
{{ballots|
25 A>B>C
40 B>C>A
35 C>A>B}}
A>B: 60>40, B>C: 65>35, C>A:75>25. There are 6 main rankings to consider here:
* A>B>C: A>B opposed by 40, A>C by 75, and B>C by 35. Score is 150. So the minimum score so far is 150.
* A>C>B: A>C by 75, A>B by 40, C>B by 65. Score is 180. Since this is greater than the minimum (150) this is disqualified.
* B>A>C: B>A by 60, B>C by 35, A>C by 75. Score is 170. Disqualified by 150.
* B>C>A: B>C by 35, B>A by 60, C>A by 25. Score is 120. This is the new minimum, so A>B>C is now disqualified.
* C>A>B: C>A by 25, C>B by 65, A>B by 40. Score is 130. Disqualified by 120.
* C>B>A: C>B by 65, C>A by 25, B>A by 60. Score is 150. Disqualified by 120.
So the final ranking is B>C>A, with B winning.
Line 54 ⟶ 56:
Some text of this article is derived with permission from [http://condorcet.org/emr/methods.shtml#Kemeny-Young Electoral Methods: Single Winner].
==References==
<references/>
[[Category:Single-winner voting methods]]
| |||