Editing Condorcet method (section)

===Kemeny–Young method===
{{Unreferenced section|date=March 2021}}
{{Main|Kemeny–Young method}}
The Kemeny–Young method considers every possible sequence of choices in terms of which choice might be most popular, which choice might be second-most popular, and so on down to which choice might be least popular. Each such sequence is associated with a Kemeny score that is equal to the sum of the [[#Pairwise counting and matrices|pairwise counts]] that apply to the specified sequence. The sequence with the highest score is identified as the overall ranking, from most popular to least popular.

When the pairwise counts are arranged in a matrix in which the choices appear in sequence from most popular (top and left) to least popular (bottom and right), the winning Kemeny score equals the sum of the counts in the upper-right, triangular half of the matrix (shown here in bold on a green background).

{| class="wikitable" style="text-align:center"
!
! ...over '''Nashville'''
! ...over '''Chattanooga'''
! ...over '''Knoxville'''
! ...over '''Memphis'''
|-
! Prefer '''Nashville'''...
| —
| style="background:#cfc;"| '''68'''
| style="background:#cfc;"| '''68'''
| style="background:#cfc;"| '''58'''
|-
! Prefer '''Chattanooga'''...
| 32
| —
| style="background:#cfc;"| '''83'''
| style="background:#cfc;"| '''58'''
|-
! Prefer '''Knoxville'''...
| 32
| 17
| —
| style="background:#cfc;"| '''58'''
|-
! Prefer '''Memphis'''...
| 42
| 42
| 42
| —
|}

In this example, the Kemeny Score of the sequence Nashville > Chattanooga > Knoxville > Memphis would be 393.

Calculating every Kemeny score requires considerable computation time in cases that involve more than a few choices. However, fast calculation methods based on [[integer programming]] allow a computation time in seconds for some cases with as many as 40 choices.