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=== Tie-breaking methods ===
If there is a tie (including [[Condorcet paradox|Condorcet cycles]]), use the "Level '''1"<sup>st</sup> Degree''' tie-breaking method to resolve it. If there is still a tie, use "Levelthe '''2"<sup>nd</sup> Degree''' tiebreaker, and so on.
'''Level 1<sup>st</sup> Degree:''' Declare the tied candidates finalists and eliminate all other candidates. For each finalist, subtract the number of ballots on which they lost to each other finalist from the number of ballots on which they beat each other finalist. The finalist with the greatest total difference is elected. For example, let <math>A_w</math> be the number of ballots on which finalist <math display="inline">A</math> beats each other finalist (equivalent to the sum of the values in <math display="inline">A</math>'s row in a preference matrix consisting only of finalists) and let <math>A_l</math> be the number of ballots on which <math display="inline">A</math> loses to each other finalist (equivalent to the sum of the values in '<math display="inline">A</math>s column in preference matrix consisting only of finalists); <math display="inline">A</math>'s total difference is <math>A_w-A_l</math>. This is mathematically equivalent to the [https://en.wikipedia.org/wiki/Borda_count#Tournament-style_counting_of_ties tournament-style of the Borda count] (among only the finalists), where candidates get, per ballot, 1 point for each candidate they beat and ½ point for each candidate they tie.
'''Level 2<sup>nd</sup> Degree:''' For each tied finalist, subtract the number of ballots on which they lost to each other candidate (including eliminated candidates) from the number of ballots on which they beat each other candidate (including eliminated candidates). The tied finalist with the greatest total difference is elected.
'''Level 3<sup>rd</sup> Degree:''' It is highly unlikely that there will still be a tie after the '''Level 2<sup>nd</sup> Degree''' tiebreaker, but if there is, it is not recommended to use tie-breaking methods beyond the '''Level 2<sup>nd</sup> Degree''' tiebreaker for government elections as voter trust may be shaken more by using the '''Level 3<sup>rd</sup> Degree''' tiebreaker and beyond than drawing lots or hosting another election. In the event that there is a tie after '''Level 2''', the differences for the tied candidates will be the same, but the values used to calculate them will likely be different. Elect the tied candidate whose values are closest to the tied differences. For example, if <math display="inline">A</math>, <math display="inline">B
</math>, and <math display="inline">C</math> are tied after the '''Level 2<sup>nd</sup> Degree''' tiebreaker, 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<sup>th</sup> Degree:''' If there is still a tie after the '''Level 3<sup>rd</sup> Degree''' tiebreaker, it is unlikely that the '''Level 4<sup>th</sup> Degree''' tiebreaker 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 levelsdegrees:'''<blockquote>10: Eli>Deegan>Ava=Cedric>Fabio===
<blockquote>10: Eli>Deegan>Ava=Cedric>Fabio
9: Bianca=Deegan>Eli>Cedric
|''Eli''
|''Fabio''
|'''Row total (winsvotes for)'''
|-
|'''Ava over'''
|'''138'''
|-
|''Column total (lossesvotes against)''
|''149''
|''149''
'''Ranked Robin:''' Ava and Bianca tie for pairwise beating the greatest number of other candidates, '''3'''.
'''Level 1<sup>st</sup> Degree:''' Ava and Bianca tie for the greatest total difference in votes for and against other tied finalists (both <math>29-29=0</math>).
'''Level 2<sup>nd</sup> Degree:''' Ava and Bianca tie for the greatest total difference in votes for and against all other candidates (both <math>188-149=39</math>).
'''Level 3<sup>rd</sup> Degree:''' Ava and Bianca tie for the least ''losing'' (and '''winning''') votes between them, ''149'' (and '''188''').
'''Level 4<sup>th</sup> Degree:''' 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 />
== Presentation of results ==
If there is a [[Condorcet winner criterion|Condorcet Winner]], then simply show each of the winner's pairwise matchups against other candidates. This can either be shown as percentage of total votes for each candidate in a given pairwise matchup, or as the percentage point difference in favor of the winner if there's a desire to show less information. If there is no Condorcet Winner but a single candidate wins without any tie-breaker, show how many matchups each candidate won in addition to each of the winner's pairwise matchups. These two scenarios will cover the vast majority of real-world elections.
==== '''Two different ways to present the results of the same election with Condorcet Winner Ava:''' ====
<blockquote>Ava: 54%《》Bianca: 46%
Ava: 59%《》Cedric: 39%
Ava: 63%《》Deegan: 31%
Ava: 64%《》Eli: 22%
Ava: 64%《》Fabio: 13%</blockquote><blockquote>Ava vs. Bianca: +8% points
Ava vs. Cedric: +20% points
Ava vs. Deegan: +32% points
Ava vs. Eli: +42% points
Ava vs. Fabio: +51% points</blockquote>
==== Example of how to present the results of an election where the winner Ava is not a Condorcet Winner: ====
<blockquote>Ava won 4 matchups (against Cedric, Deegan, Eli, and Fabio)
Bianca won 3 matchups (against Ava, Eli, and Fabio)
Cedric won 3 matchups (against Bianca, Eli, and Fabio)
Deegan won 3 matchups (against Bianca, Cedric, and Fabio)
Eli won 2 matchups (against Deegan and Fabio)
Fabio lost all matchups</blockquote><blockquote>Ava vs. Bianca: -8% points
Ava vs. Cedric: +20% points
Ava vs. Deegan: +32% points
Ava vs. Eli: +42% points
Ava vs. Fabio: +51% points</blockquote>
=== If there's a tie ===
When there is a '''1<sup>st</sup> Degree''' tie, it's often a [[Condorcet paradox|Condorcet cycle]] (rock-paper-scissors-style tie) with more than two candidates (now finalists). In the case that there are only two finalists, present results as the above example without a Condorcet Winner, but highlight the matchup between the two finalists, which alone breaks the tie. Otherwise, present results according to the level desired as described below.
'''Level 1:''' Simply state who the winner is.
'''Level 2:''' Show which candidates are finalists (and optionally which candidates were eliminated).
'''Level 3:''' Show how many matchups each candidate won as shown above.
'''Level 4:''' Show each finalist's total difference in votes for and against other finalists. These values can be given a name like "Total Advantage".
'''Level 5:''' Show the breakdown of each finalist's total difference in votes for and against other finalists by showing the difference within each finalist matchup. These values can be given a name like "Matchup Advantage"
'''Level 6:''' Show a preference matrix that's just wins and losses (and ties).
'''Level 7:''' Show a preference matrix using percentages.
'''Level 8:''' Show the full preference matrix.
==== Example of showing Level 4 with 3 finalists in a Condorcet cycle: ====
<blockquote>Ava, Bianca, and Cedric are finalists.
Ava's Total Advantage: -23.5% points
Bianca's Total Advantage: -3.3% points
Cedric's Total Advantage: +33.8% points
Cedric is elected!</blockquote>Note that all of the Total Advantages sum to 0. This can be used to check the math performed.
==== Example of showing Level 5 with 3 finalists in a Condorcet cycle: ====
<blockquote>Ava, Bianca, and Cedric are finalists.
Ava vs. Bianca: +22.1% points
Ava vs. Cedric: -45.6% points
Ava's Total Advantage: -23.5% points
Bianca vs. Cedric: +18.8% points
Bianca vs. Ava: -22.1% points
Bianca's Total Advantage: -3.3% points
Cedric vs. Ava: +45.6% points
Cedric vs. Bianca: -18.8% points
Cedric's Total Advantage: +33.8% points
Cedric is elected!</blockquote><references />
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