Ranked Robin
Ranked Robin is a Condorcet voting method focused on the presentation of the results such that everyday voters can understand them without extensive education. Ranked Robin uses a ranked ballot. Voters are free to rank multiple candidates equally on their ballots. The candidate who beats the most other candidates head-to-head is elected, much like a round-robin tournament.
History
Ranked Robin was invented by Sass on 30 September 2021 and named by Sara Wolk on 7 November 2021. As an enthusiast of cardinal voting methods and a strong advocate for voter empowerment, Sass saw a timely need for a sufficiently-accurate ranked voting method that was on par with the simplicity of voting methods like STAR Voting and even Approval Voting, particularly in the United States. Ranked Robin is nearly identical to the earliest known Condorcet method, invented by Ramon Llull in his 1299 treatise Ars Electionis^{[1]}, which was similarly replicated by Marquis de Condorcet centuries later, and then again by Arthur Herbert Copeland. A mathematically identical method to Ranked Robin including the first tie-breaking mechanic was described by Partha Dasgupta and Eric Maskin in 2004^{[2]}. The primary innovation of Ranked Robin is the reduction and formatting of results in such a way that they are palatable to a general audience, as a full preference matrix can be overwhelming for most voters. This innovation can likely be adapted to simplify the results of other voting methods that use pairwise counting, particularly those that first restrict the set of winners such as Smith-efficient voting methods.
Balloting
Voters may rank as many candidates as they would like. Voters are free to rank multiple candidates equally. Skipped ranks are ignored and will neither hurt nor help a voter's vote. All candidates left unranked are considered tied for the last rank, below the lowest rank marked on a voter's ballot.
Local counting
Ranked Robin is summable through the use of preference matrices. Full preference matrices can be created simply by hand if needed and then reported directly to the media and the public, allowing ballots and ballot data to remain local for recounts and risk-limiting audits without risking the threat of vote selling and voter coercion^{[3]}. This decentralization of tallying allows elections to remain robust against scaled election attacks, which is vital in jurisdictions that run geographically-spread or high-profile elections. In contrast, voting methods that are not precinct summable, like Ranked Choice (Instant Runoff) Voting and many expressive proportional voting methods, lose these benefits and can lead to distrust in election outcomes if fraud, attacks, or even simple mistakes happen under a centralized counting authority.
Tabulation
Elect the candidate who pairwise beats the greatest number of candidates.
Example election
8:Ava>Cedric>Deegan>Bianca>Eli 6:Ava=Bianca=Cedric>Eli>Deegan 6:Eli>Ava>Bianca=Cedric=Deegan 6:Deegan>Bianca=Cedric>Eli>Ava 4:Bianca>Ava>Eli>Deegan>Cedric 3:Eli>Deegan>Bianca=Cedric>Ava 2:Deegan=Eli>Bianca=Cedric>Ava
Create a preference matrix from the ballots.
# of voters who prefer | Ava | Bianca | Cedric | Deegan | Eli |
Ava over | — | 14 | 18 | 24 | 18 |
Bianca over | 15 | — | 4 | 10 | 24 |
Cedric over | 11 | 8 | — | 14 | 20 |
Deegan over | 11 | 19 | 15 | — | 14 |
Eli over | 17 | 11 | 15 | 19 | — |
Ava pairwise beats the greatest number of candidates, 3, so she is elected as the winner.
Tie-breaking mechanics
Frequency of ties
Almost all public elections using Ranked Robin will not have any ties for the winning candidate. However, ties under Ranked Robin may potentially be more common than ties under Choose-one Voting. While there are 4 degrees of tiebreakers defined, ties after the 1^{st} Degree tiebreaker are about as rare as ties under Choose-one Voting, and ties after the 2^{nd} Degree tiebreaker are much rarer than that.
Degrees of ties
If there is a tie (including Condorcet cycles), use the 1^{st} Degree tie-breaking method to resolve it. If there is still a tie, use the 2^{nd} Degree tiebreaker, and so on.
1^{st} Degree: Declare the tied candidates finalists. Elect the finalist with the greatest sum of win margins over the other finalists.
To find a win margin for a finalist in a matchup, subtract the number of votes ranking the opponent finalist higher from the number of votes ranking the finalist in question higher. Add these margins up for each finalist to find their sum of win margins. Note that any of these values can be positive or negative.
Since those two steps are both addition, they can also be combined into one step. For example, let be the number of votes preferring finalist over each other finalist (equivalent to the sum of the values in 's row in a preference matrix that consists only of finalists) and let be the number of votes preferring each other finalist over (equivalent to the sum of the values in 's column in a preference matrix that consists only of finalists); 's sum of win margins is .
2^{nd} Degree: Elect the tied finalist with the greatest sum of win margins over all candidates.
3^{rd} Degree: It is highly unlikely that there will still be a tie after the 2^{nd} Degree tiebreaker, but if there is, it is not recommended to use tie-breaking methods beyond the 2^{nd} Degree tiebreaker for public elections as voter trust may be shaken more by using the 3^{rd} Degree tiebreaker and beyond than drawing lots or hosting another election.
In the event that there is a tie after the 2^{nd} Degree tiebreaker, the sum of win margins for the tied candidates will be the same, but the values used to calculate them will likely be different. Elect the tied finalist whose values are closest to the tied margins.
For example, if , , and are tied after the 2^{nd} Degree tiebreaker, then (where win margins are calculated across the entire field of candidates), but it's likely that (and by proxy that ). The tied candidate with the greatest number of votes against them will also have the greatest number of votes in favor of them, and the tied candidate with the fewest number of votes against them will have the fewest number of votes for them. Elect the tied candidate with the fewest number of votes on both metrics (for the motivation that they are the least polarizing tied candidate).
4^{th} Degree: If there is still a tie after the 3^{rd} Degree tiebreaker, it is unlikely that the 4^{th} 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, find the win margin. Sum these win margins 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. (Every number in this calculation will be positive.) 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 degrees
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 4: Ava=Bianca>Fabio 2: Bianca=Fabio>Ava=Eli
Here's the preference matrix:
# of voters who prefer | Ava | Bianca | Cedric | Deegan | Eli | Fabio | Row total (votes for) |
Ava over | — | 29 | 26 | 39 | 42 | 56 | 204 |
Bianca over | 29 | — | 35 | 31 | 46 | 51 | 204 |
Cedric over | 33 | 28 | — | 32 | 32 | 42 | 167 |
Deegan over | 38 | 37 | 32 | — | 30 | 38 | 175 |
Eli over | 33 | 31 | 41 | 19 | — | 45 | 169 |
Fabio over | 16 | 24 | 39 | 39 | 32 | — | 150 |
Column total (votes against) | 149 | 149 | 181 | 168 | 190 | 232 | 1069 - 1069 = 0 |
Ranked Robin: Ava and Bianca tie for pairwise beating the greatest number of other candidates, 3.
1^{st} Degree: Ava and Bianca tie for the greatest sum of win margins over other finalists (both ).
2^{nd} Degree: Ava and Bianca tie for the greatest sum of win margins over all candidates (both ).
3^{rd} Degree: Ava and Bianca tie for the least losing (and winning) votes between them, 149 (and 204).
4^{th} 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 win margin of Ava over Deegan is . From Deegan to Bianca, the win margin is . The sum of the win margins in the beatpath from Ava to Bianca (the total beatpath strength) is . From Bianca to Cedric, the win margin is . From Cedric to Ava, the win margin is . The total beatpath strength from Bianca to Ava is . Bianca has the greatest sum of beatpath strengths among tied candidates, so Bianca is elected.
Presentation of results
Ranked Robin results can be broken down differently depending on whether there are any ties and how much detail is desired.
If there is no tie
Show how many matchups each candidate won and highlight the overall winner. Optionally, show the overall winner's margins over other candidates.
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
Ava is elected!
Ava vs. Bianca: -6% points
Ava vs. Cedric: +20% points
Ava vs. Deegan: +32% points
Ava vs. Eli: +42% points
Ava vs. Fabio: +51% points
If there's a tie
First show how many matchups each candidate won and highlight the finalists. Then, show each finalist's sum of win margins over other finalists and highlight the winner. Optionally, show each matchup among finalists including their win margins.
Ava won 4 matchups (against Bianca, Deegan, Eli, and Fabio)
Bianca won 4 matchups (against Cedric, Deegan, Eli, and Fabio)
Cedric won 4 matchups (against Ava, Deegan, Eli, and Fabio)
Deegan won 2 matchups (against Eli and Fabio)
Eli won 1 matchups (against Fabio)
Fabio lost all matchups
Ava, Bianca, and Cedric are finalists.
Ava's total win margin over other finalists: -23.5% pointsBianca's total win margin over other finalists: -3.3% points
Cedric's total win margin over other finalists: +26.8% points
Cedric is elected!
Ava, Bianca, and Cedric are finalists.
Ava vs. Bianca: +22.1% pointsAva vs. Cedric: -45.6% points
Ava's total win margin over other finalists: -23.5% points
Bianca vs. Cedric: +18.8% points
Bianca vs. Ava: -22.1% points
Bianca's total win margin over other finalists: -3.3% points
Cedric vs. Ava: +45.6% points
Cedric vs. Bianca: -18.8% points
Cedric's total win margin over other finalists: +26.8% points
Cedric is elected!
In the rare case of a 2^{nd} Degree tie, if there are many candidates, it is recommended to focus on who the tied finalists are and their sum of win margins over all candidates (which likely will not sum to 0).
Legal and economic viability
When legally defined as always reducing to a finalist set first and then electing the finalist with the greatest sum of win margins over other finalist (as described in the 1^{st} Degree tiebreaker), Ranked Robin always elects a majority preferred winner, arguably including in cases of 2^{nd} Degree ties. This legal definition does not change the outcomes of Ranked Robin. Many municipalities in the United States are subject to a majority clause in their respective state's election code, often requiring those jurisdictions to run two or more elections for certain races. Ranked Robin can satisfy many of these majority clauses in a single election, allowing municipalities to eliminate an election if so desired, helping to offset the costs of implementing Ranked Robin, typically entirely within one election cycle.
If there is only 1 finalist, then they are voted for by a majority of voters who had a preference among finalists.
If there are multiple finalists, at least 1 finalist will have a positive sum of win margins and at least 1 finalist will have a negative sum of win margins because the sum of all win margins will always equal 0. Because the finalist with the greatest sum of win margins is elected, that winner is guaranteed to have a positive sum of win margins, demonstrating that, among finalists, they are a majority preferred winner.
If there is a 2^{nd} Degree tie, all of the finalists could potentially (but rarely) have a negative sum of win margins when compared to all candidates, but it could be argued that because the finalist with the greatest sum of win margins is elected, the winner was voted for by a majority of voters who had a preference among finalists. This argument is further strengthened in the case that exactly 2 finalists experience a 2^{nd} Degree tie, which covers almost all cases of 2^{nd} Degree ties. If this argument is found not to satisfy a particular majority clause, it may be desirable to leave the 2^{nd} Degree tiebreaker out of the legislation and legally declare a tie in the equivalent case of a 2^{nd} Degree tie, which is about as rare as a tie under Choose-one Voting.
Furthermore, in most cases with only 1 finalist, including all elections with a Condorcet Winner, the winner will be majority preferred over all other candidates because the winner’s sum of win margins is positive; however, there are rare theoretical cases in which the only finalist has a negative sum of win margins over all other candidates. If the “majority preferred among finalists” argument doesn’t legally hold when there’s only 1 finalist, then this rare case could either explicitly be denoted as not electing a majority winner (thus requiring an extra election to be run), or an alternative winner could be calculated by selecting the candidate with the greatest sum of win margins among all candidates (completely ignoring any reduction to a set of finalists).
Criteria
Passed
- Unrestricted domain
- Non-imposition (a.k.a. citizen sovereignty)
- Non-dictatorship
- Homogeneity
- Condorcet criterion
- Majority criterion
- Pareto criterion (a.k.a. unanimity)
- Monotonicity criterion
- Majority loser criterion
- Condorcet loser criterion
- Reversal symmetry
- Resolvability criterion
- Polynomial time
- Mutual majority criterion
- Smith criterion
- Independence of Smith-dominated alternatives
- Independence of Irrelevant Ballots
Failed
- Participation
- Independence of irrelevant alternatives
- Consistency
- Independence of clones
- Sincere Favorite
A note on cloneproofness
Ranked Robin can fail clone independence only in elections with no Condorcet Winner, through crowding and teaming. It can be argued that a party stands nothing to gain (or lose) by running clones as far as the crowding vulnerability is concerned because all a losing candidate A can achieve by triggering a clone failure is to change the winner from some B to some other C, which doesn't help A since A lost anyway (unless C happens to be politically closer to A than B does). However, in elections expected to lack a Condorcet Winner, the teaming incentive may be exploitable since it directly benefits a candidate who runs clones. If the there's expected to be a tie leading to more than one finalist, then this could allow teaming to succeed. For instance, consider this pre-cloning election:
12: A>B>C>D>E>F 11: B>C>A>D>E>F 10: C>A>B>D>E>F
The finalists are A, B, and C. A and B tie for sum of win margins, but this can be shifted in favor of A by teaming:
12: A1>A2>B>C>D>E>F 11: B>C>A1>A2>D>E>F 10: C>A1>A2>B>D>E>F
By "sacrificing" A2 to prevent B from becoming a finalist, A1 wins after the tiebreaker.
Ranked Robin passes vote-splitting clone independence: cloning a candidate can't make that candidate lose.
External links
- Ranked Robin thread on r/EndFPTP (starting November 1, 2021)
- Ranked Robin thread on Voting Theory Forum (starting October 25, 2021)
- Explanation of Ranked Robin from the Equal Vote Coalition
References
- ↑ G. Hägele & F. Pukelsheim (2001). "Llull's writings on electoral systems". Studia Lulliana. 41: 3–38.
- ↑ Maskin, Eric; Dasgupta, Partha (2004). "The Fairest Vote of All". Scientific American (290): 64–69 – via Harvard University.
- ↑ See Ballotpedia's "Intimidation of voters" article for more about voter coercion.