Editing Schulze method (section)

== Example ==
In the following example 45 voters rank 5 candidates.
{| class="wikitable"
!Number of voters
!Order of preference
|-
|5
|ACBED
|-
|5
|ADECB
|-
|8
|BEDAC
|-
|3
|CABED
|-
|7
|CAEBD
|-
|2
|CBADE
|-
|7
|DCEBA
|-
|8
|EBADC
|}
The pairwise preferences have to be computed first. For example, when comparing ''{{mvar|A}}'' and ''{{mvar|B}}'' pairwise, there are {{math|1=5+5+3+7=20}} voters who prefer ''{{mvar|A}}'' to ''{{mvar|B}}'', and {{math|1=8+2+7+8=25}} voters who prefer ''{{mvar|B}}'' to ''{{mvar|A}}''. So <math>d[A, B] = 20</math> and <math>d[B, A] = 25</math>. The full set of pairwise preferences is:
[[File:Schulze_method_example1.svg|right|thumb|300x300px|[[Directed graph]] labeled with pairwise preferences d[*, *] ]]
{| class="wikitable" style="text-align:center"
|+ Matrix of pairwise preferences
|-
! !! <math>d[*,A]</math> !! <math>d[*,B]</math> !! <math>d[*,C]</math> !! <math>d[*,D]</math> !! <math>d[*,E]</math>
|-
! <math>d[A,*]</math>
| || style="background:#fdd;"|20 || style="background:#dfd;"|26 || style="background:#dfd;"|30 || style="background:#fdd;"|22
|-
! <math>d[B,*]</math>
| style="background:#dfd;"|25 || || style="background:#fdd;"|16 || style="background:#dfd;"|33 || style="background:#fdd;"|18
|-
! <math>d[C,*]</math>
| style="background:#fdd;"|19 || style="background:#dfd;"|29 || || style="background:#fdd;"|17 || style="background:#dfd;"|24
|-
! <math>d[D,*]</math>
| style="background:#fdd;"|15 || style="background:#fdd;"|12 || style="background:#dfd;"|28 || || style="background:#fdd;"|14
|-
! <math>d[E,*]</math>
| style="background:#dfd;"|23 || style="background:#dfd;"|27 || style="background:#fdd;"|21 || style="background:#dfd;"|31 || 
|}
The cells for d[X, Y] have a light green background if d[X, Y] > d[Y, X], otherwise the background is light red. There is no undisputed winner by only looking at the pairwise differences here.

Now the strongest paths have to be identified. To help visualize the strongest paths, the set of pairwise preferences is depicted in the diagram on the right in the form of a [[directed graph]].  An arrow from the node representing a candidate X to the one representing a candidate Y is labelled with d[X, Y].  To avoid cluttering the diagram, an arrow has only been drawn from X to Y when d[X, Y] > d[Y, X] (i.e. the table cells with light green background), omitting the one in the opposite direction (the table cells with light red background).

One example of computing the strongest path strength is p[B, D] = 33: the strongest path from B to D is the direct path (B, D) which has strength 33. But when computing p[A, C], the strongest path from A to C is not the direct path (A, C) of strength 26, rather the strongest path is the indirect path (A, D, C) which has strength min(30, 28) = 28. The ''strength'' of a path is the strength of its weakest link.

For each pair of candidates X and Y, the following table shows the strongest path from candidate X to candidate Y in red, with the weakest link underlined.{{clear}}
{| class="wikitable" style="text-align:center"
|+ Strongest paths
|-
! {{diagonal split header|From|To}}
! A !! B !! C !! D !! E
!
|-
! A
| {{n/a}} || [[Image:Schulze method example1 AB.svg|border|none|150px]] A-(30)-D-<u>(28)</u>-C-(29)-B || [[Image:Schulze method example1 AC.svg|border|none|150px]] A-(30)-D-<u>(28)</u>-C || [[Image:Schulze method example1 AD.svg|border|none|150px]] A-<u>(30)</u>-D || [[Image:Schulze method example1 AE.svg|border|none|150px]] A-(30)-D-(28)-C-<u>(24)</u>-E
! A
|-
! B
| [[Image:Schulze method example1 BA.svg|border|none|150px]] B-<u>(25)</u>-A || {{n/a}} || [[Image:Schulze method example1 BC.svg|border|none|150px]] B-(33)-D-<u>(28)</u>-C || [[Image:Schulze method example1 BD.svg|border|none|150px]] B-<u>(33)</u>-D || [[Image:Schulze method example1 BE.svg|border|none|150px]] B-(33)-D-(28)-C-<u>(24)</u>-E
! B
|-
! C
| [[Image:Schulze method example1 CA.svg|border|none|150px]] C-(29)-B-<u>(25)</u>-A || [[Image:Schulze method example1 CB.svg|border|none|150px]] C-<u>(29)</u>-B || {{n/a}} || [[Image:Schulze method example1 CD.svg|border|none|150px]] C-<u>(29)</u>-B-(33)-D || [[Image:Schulze method example1 CE.svg|border|none|150px]] C-<u>(24)</u>-E
! C
|-
! D
| [[Image:Schulze method example1 DA.svg|border|none|150px]] D-(28)-C-(29)-B-<u>(25)</u>-A || [[Image:Schulze method example1 DB.svg|border|none|150px]] D-<u>(28)</u>-C-(29)-B || [[Image:Schulze method example1 DC.svg|border|none|150px]] D-<u>(28)</u>-C || {{n/a}} || [[Image:Schulze method example1 DE.svg|border|none|150px]] D-(28)-C-<u>(24)</u>-E
! D
|-
! E
| [[Image:Schulze method example1 EA.svg|border|none|150px]] E-(31)-D-(28)-C-(29)-B-<u>(25)</u>-A || [[Image:Schulze method example1 EB.svg|border|none|150px]] E-(31)-D-<u>(28)</u>-C-(29)-B || [[Image:Schulze method example1 EC.svg|border|none|150px]] E-(31)-D-<u>(28)</u>-C || [[Image:Schulze method example1 ED.svg|border|none|150px]] E-<u>(31)</u>-D || {{n/a}}
! E
|-
!
! A !! B !! C !! D !! E
! {{diagonal split header|To|From}}
|}

{| class="wikitable" style="text-align:center"
|+Strengths of the strongest paths
|-
! !! <math>p[*,A]</math> !! <math>p[*,B]</math> !! <math>p[*,C]</math> !! <math>p[*,D]</math> !! <math>p[*,E]</math>
|-
! <math>p[A,*]</math>
| || style="background:#dfd;"|28 || style="background:#dfd;"|28 || style="background:#dfd;"|30 || style="background:#fdd;"|24
|-
! <math>p[B,*]</math>
| style="background:#fdd;"|25 || || style="background:#fdd;"|28 || style="background:#dfd;"|33 || style="background:#fdd;"|24
|-
! <math>p[C,*]</math>
| style="background:#fdd;"|25 || style="background:#dfd;"|29 || || style="background:#dfd;"|29 || style="background:#fdd;"|24
|-
! <math>p[D,*]</math>
| style="background:#fdd;"|25 || style="background:#fdd;"|28 || style="background:#fdd;"|28 || || style="background:#fdd;"|24
|-
! <math>p[E,*]</math>
| style="background:#dfd;"|25 || style="background:#dfd;"|28 || style="background:#dfd;"|28 || style="background:#dfd;"|31 || 
|}
Now the output of the Schulze method can be determined. For example, when comparing ''{{mvar|A}}'' and ''{{mvar|B}}'', since <math>(28 =) p[A,B] > p[B,A] (= 25)</math>, for the Schulze method candidate ''{{mvar|A}}'' is ''better'' than candidate ''{{mvar|B}}''. Another example is that <math>(31 =) p[E,D] > p[D,E] (= 24)</math>, so candidate E is ''better'' than candidate D. Continuing in this way, the result is that the Schulze ranking is <math>E > A > C > B > D</math>, and ''{{mvar|E}}'' wins. In other words, ''{{mvar|E}}'' wins since <math>p[E,X] \ge p[X,E]</math> for every other candidate X.