Condorcet Voting

Condorcet Voting:

Condorcet is a family of voting methods where the candidate who is preferred over all others always wins. If a single candidate who is preferred over all others doesn't exist, ballots are looked at more closely to find the most preferred candidate. 

Condorcet is almost universally regarded as the most fair, representative, and accurate way to tally ranked choice ballots for single winner elections, but it's worth noting that there are great Condorcet methods that work with 5 star ballots or rating ballots as well. 

On the advice of leading Condorcet experts and advocates, including Professor Nicolaus Tideman, we recommend the Minimax method, described below with the 'Smith Set' and 'Margins' variations for tallying ranked ballots.

 

Voting:

Voters rank candidates in order of preference. Equal rankings and skipped rankings are allowed. Candidates left blank are ranked last. 

 

Tabulation:

Step 1. For each pair of candidates, add up the number of voters who preferred each. 

The full preference data for the election can be recorded in a preference matrix:

2. Elect the Condorcet winner if possible. 

In the example above, Allison is the Condorcet winner because she is preferred over all others. If this doesn't narrow it down to only one winner you have a Condorcet cycle, or rock-paper-scissors type tie.

 

3. Determine the finalists.

The next step is to find the group of finalists (the Smith Set,) who are collectively preferred over all others, while eliminating the candidates who aren't preferred over any of the finalists. 

 

 

  • Find the candidate with the most wins and designate them as a finalist. If multiple candidates have the same number of match-ups won, the number of tied match-ups can be used to narrow it down further. (In this example Dre and Edith both won four of their match-ups, so at this stage both are considered finalists.)
  • For each remaining candidate, add them to the finalist roster if in their head-to-head match-up they tied with or beat at least one of the finalists. (Frank beat Dre so he is added to the finalist list.) 
  • For each new finalist added, add any candidates who beat or tied with the new finalist in their match-up. (Abby tied with Frank, so she can be added to the finalist list. Ben beat Abby, so he can also be added to the finalist list. Cici is the only candidate who didn't beat any of the finalists and can be eliminated.)
  • This can be performed on computers using Kosaraju-Sharir’s algorithm.



4. Calculate 
each finalist's margin of loss across all of their races. The finalist with the smallest margin wins:

The example above shows five finalists. In order to find a winner, we calculate the margin of each candidate's worst loss.

  • In each match-up a candidate lost, calculate the margin they lost by. To do this, subtract the losing votes from the winning votes for each match-up. (Dre lost to Frank by 47-20=27 votes. Edith lost to Dre by 39-23=16 votes. Frank lost to Edith by 37-24=13 votes, and Frank tied with Abby. Ben lost to Dre by 35-32=3 votes, Ben lost to Edith by 35-23=12 votes, and Ben lost to Frank by 35-32=3 votes. Abby lost to Dre by 40-28=12 votes, Abby lost to Edith by 43-19=24 votes, Abby tied with Frank, and Abby lost to Ben by 30-24=6 votes.)
  • Find each candidate's worst loss. (Dre, Edith, and Frank each only lost one race. Ben lost to Dre, Edith, and Frank, but his worst loss was to Edith with a 12 vote margin. Abby lost to Dre, Edith, and Ben, but her worst loss was to Edith with a 24 vote margin.)
  • The winner is the finalist whose worst loss had the smallest margin. (In this example, the smallest margin for any candidate’s worst loss was Ben by only 12 votes. Ben wins!)

 

 

 

Condorcet Advantages:

  • Highly accurate at electing representative winners. 
  • Highly resistant to strategic voting.
  • Allowing voters to give equal rankings helps voters express a more nuanced opinion and reduces cognitive load.
  • No wasted votes. All rankings are counted and skipped rankings are allowed. It would be very difficult to accidentally void or spoil your ballot. 
  • Fair and equal. The system doesn't play favorites and ensures an equally weighted vote. 
  • Condorcet voting was invented hundreds or even thousands of years ago and has been studied extensively. Peer review and other study consistently backs up claims made by Condorcet advocates. 
  • Condorcet is less complex to tabulate than other methods with long track records for real-world implementation.

 

Condorcet Disadvantages:

  • While the general concept of electing the candidate who was preferred over all others or who is most preferred is quite simple, ranked Condorcet methods are some of the more complex single-winner methods to explain in detail or tabulate. In the modern age, this is much less of a barrier than it used to be, as tabulation can be automated, but some expertise is needed to fully understand the details of the method and to verify that it's been coded correctly. 
  • A ranked ballot doesn't allow voters to express their level of support, and thus a 2nd choice ranking could be almost as good as a voter's favorite or almost as bad as their worst case scenario.

Note: Both of the issues above are eliminated if a five star ballot Condorcet method like Condorcet STAR (aka Smith//Score) is used.

 

Other Condorcet Methods:

There are a number of Condorcet voting methods with different variations on the tie-breaking protocol including high performing ranked options such as Minimax, Ranked Pairs, Schulze, and variations of each. Among experts and advocates in the field including Dr. Nicolaus Tideman, Department of Mathematics, Virginia Tech, and Professor Emeritus Richard Darlington, Department of Psychology, Cornell University, there seems to be a growing consensus around the Smith//Minimax option, which is the method described above. For rated or 5 star ballots, Smith//Score is another great option. Keep in mind that regardless of the Condorcet variation you prefer, the winners for most of the top-tier Condorcet methods will be the same the vast majority of the time.

A few Condorcet methods use flawed single-winner voting methods as the tie-breaking mechanism. We do not recommend Condoret paired with Instant Runoff Voting (VoteFair) because IRV ignores some voters’ rankings and thus can skew outcomes in close competitive races with multiple candidates. We also do not recommend the Condorcet Borda method because it can result in outcomes that are worse than random if voters are strategic, which is highly incentivized in Borda methods.

 

Condorcet as a metric for measuring voting method accuracy:

The Condorcet method is often used to measure the accuracy of other voting methods because it takes into account all the voter preferences from all ballots. For ranked methods, preference data is all you have to work with, so the Condorcet winner is always the correct winner. For methods that use score ballots, preference order and also level of support is shown, so Condorcet is still one good metric, but strength of support can also be taken into account. Regardless of the metrics used to measure accuracy, Condorcet methods tend to come out in the top tier.

 

Condorcet as a tiebreaker:

Condorcet, and in particular the Minimax method described above, was designed to be a highly effective and fair tiebreaker. Minimax is a great tie-breaking protocol to adopt for voting methods that allow voters to show their preference order.

 

The case for Condorcet Voting:

Condorcet Voting was first invented in the 13th century and is the original Ranked Choice Voting method, but in modern times the voting reform movement has been largely focused on the Instant Runoff Voting (IRV) version. In 1870 when IRV was invented, computers were not an option and tabulating Condorcet elections was time consuming. In contrast, IRV was easier to hand count and was logistically much more viable, despite the fact that Condorcet voting is significantly more accurate and more resistant to strategic voting. While both methods start with the same ranked ballot, Condorcet voting counts all ballot data and does not ignore or discard any rankings or any ballots. IRV on the other hand only counts some of the rankings given and up to a 1/3 of ballots may not be counted at all in the deciding round, which can skew election results. 

While this compromise may have been deemed necessary in 1870, we do not believe it's worth it, especially in the context of modern voting science. For voters who are sold on the idea of ranked ballots, we recommend the above method as a now viable option which offers much more fair and representative outcomes than the Instant Runoff method.

 

Try Condorcet:

To try Condorcet voting for yourself you can host elections online using the Condorcet Internet Voting Service.