Ranked Robin: Difference between revisions
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</math>, and <math display="inline">C</math> are tied after '''Level 2''', then <math>A_w-A_l=B_w-B_l=C_w-C_l</math> (where wins and loses are calculated across the entire field of candidates), but it's likely that <math>A_w\neq B_w\neq C_w</math> (and by proxy that <math>A_l\neq B_l\neq C_l</math>). The tied candidate with the greatest loss margin will also have the greatest win margin, and the tied candidate with the least loss margin will have the least win margin. Elect the tied candidate with the least loss and win margins as that is the least polarizing tied candidate.
'''Level 4:''' If there is still a tie after '''Level 3''', it is unlikely that '''Level 4''' will break that tie, as it will only work if the tied candidates have matchup losses against other candidates. Find the shortest [[beatpath]] from each tied candidate to each other tied candidate. For each tied candidate, for each shortest beatpath to another tied candidate, for each pairwise victory in the beatpath, subtract the number of ballots on which the losing candidate was preferred over the winning candidate from the number of ballots on which the winning candidate was preferred over the losing candidate. Sum these differences within each selected beatpath to get the total strength of each selected beatpath. Sum each tied candidate's total beatpath strengths over other tied candidates. Elect the tied candidate with the greatest sum of beatpath strengths. If there are multiple shortest beatpaths from one tied candidate to another, select the one with the lowest total strength.
'''Example of a ballot set that requires all 4 tie-breaking levels:'''<blockquote>10: Eli>Deegan>Ava=Cedric>Fabio
9: Bianca=Deegan>Eli>Cedric
8: Deegan>Eli>Ava=Bianca=Cedric
8: Bianca>Ava>Fabio>Cedric
8: Fabio>Cedric>Ava>Deegan>Bianca
7: Ava>Eli>Bianca>Fabio
6: Fabio>Bianca=Cedric>Ava
6: Cedric>Deegan=Eli>Ava=Bianca>Fabio
5: Deegan>Ava=Bianca>Eli>Cedric
4: Cedric>Bianca>Ava
4: Ava>Bianca=Fabio
2: Bianca=Fabio>Ava=Eli</blockquote>Here's the preference matrix:
{| class="wikitable"
|+Bold indicates wins, ''Italics indicates losses'', <u>Underline indicates ties</u>
|'''# of voters who prefer'''
|''Ava''
|''Bianca''
|''Cedric''
|''Deegan''
|''Eli''
|''Fabio''
|'''Row total (wins)'''
|-
|'''Ava over'''
|—
|<u>29</u>
|''26''
|'''39'''
|'''42'''
|'''52'''
|'''188'''
|-
|'''Bianca over'''
|<u>29</u>
|—
|'''35'''
|''31''
|'''46'''
|'''47'''
|'''188'''
|-
|'''Cedric over'''
|'''33'''
|''28''
|—
|<u>32</u>
|''32''
|'''42'''
|'''167'''
|-
|'''Deegan over'''
|''38''
|'''37'''
|<u>32</u>
|—
|'''30'''
|'''38'''
|'''175'''
|-
|'''Eli over'''
|''33''
|''31''
|'''41'''
|''19''
|—
|'''45'''
|'''169'''
|-
|'''Fabio over'''
|''16''
|''24''
|''35''
|''35''
|''28''
|—
|'''138'''
|-
|''Column total (losses)''
|''149''
|''149''
|''169''
|''156''
|''178''
|''224''
|'''1025''' - ''1025'' = <u>0</u>
|}
'''Ranked Robin:''' Ava and Bianca tie for pairwise beating the greatest number of other candidates, '''3'''.
'''Level 1:''' Ava and Bianca tie for the greatest total difference in votes against other tied finalists (both <math>29-29=0</math>).
'''Level 2:''' Ava and Bianca tie for the greatest total difference in votes against all other candidates (both <math>188-149=39</math>).
'''Level 3:''' Ava and Bianca tie for the least ''losing'' (and '''winning''') votes between them, ''149'' (and '''188''').
'''Level 4:''' The shortest beatpath from Ava to Bianca is Ava→Deegan→Bianca and the shortest beatpath from Bianca to Ava is Bianca→Cedric→Ava. The difference between the number of voters who prefer Ava over Deegan and the number of voters who prefer Deegan over Ava is <math>39-38=1</math>. From Deegan to Bianca, the difference is <math>37-31=6</math>. The sum of the differences in the beatpath from Ava to Bianca (the total beatpath strength) is <math>1+6=7</math>. From Bianca to Cedric, the difference is <math>35-28=7</math>. From Cedric to Ava, the difference is <math>33-26=7</math>. The total beatpath strength from Bianca to Ava is <math>7+7=14</math>. Bianca has the greatest (sum of) total beatpath strength(s) among tied candidates, so Bianca is elected.<references />
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