Introduction

Condorcet voting methods take into account the entire ranking of the candidates by every voter, which means they have more information to use in picking the winner. But just as important as the use of rankings is how those rankings are used.

Condorcet methods were invented by Marie Jean Antoine Nicolas de Caritat, marquis de Condorcet, a French philosopher, mathematician, and political scientist, known for his contributions to the field of social choice theory and for his advocacy of the rights of women and minorities.

Condorcet elections are head-to-head elections because they use voter rankings to compare every candidate against every other in head-to-head contests. Intuitively, it would not make sense to choose a winner who would lose a head-to-head contest with some other candidate. Condorcet methods aim to prevent this outcome.

The voters are considered to collectively have a preference for candidate A over candidate B if A is ranked higher than B on more ballots than B is ranked higher than A on. If the voters prefer a candidate A to every other candidate, candidate A is the Condorcet winner (CW). An election method is a Condorcet method if it is guaranteed to elect the Condorcet winner, when one exists.

There is usually a Condorcet winner, especially when there are many voters. Statistics from 20 years of CIVS polls suggest that this happens around 99% of the time. However, it is possible that there isn't one. Preference cycles may exist. For example, it is possible that on most ballots, candidate A is preferred to candidate B, on most ballots B is preferred to C, and on most ballots C is preferred to A. Fortunately, cycles involving the top-ranked candidate don't seem to happen often. And there are good ways to resolving these cycles, called completion rules. If the goal of the election is simply to pick the top candidate, it rarely matters which completion rule is used.

In the CIVS election result report, the color coding of the final preference matrix tells you whether a completion rule was needed. If there are no red cells above the diagonal (or green cells below it), then there are no cycles for a completion rule to resolve, and it doesn't matter what completion rule is used.

Supported completion rules

CIVS currently supports six rules for Condorcet completion: Minimax-PM (the default rule), Schulze (also known as Beatpath Winner or Cloneproof Schwartz Sequential Dropping), Maximize Affirmed Majorities (MAM), a deterministic variant of MAM called CIVS Ranked Pairs, and two runoff-based Condorcet algorithms called Condorcet-IRV and Bottom-2 Runoff. The Schulze and MAM rules are described in the linked documents; Minimax-PM, CIVS Ranked Pairs and Condorcet-IRV are described below.

CIVS does not impose a completion rule; in fact, anyone viewing the results of an election can see what the results would have been with each of the rules. It is probably a good idea for the election supervisor to decide on an rule ahead of time, and include it in the election description. On the other hand, all five rules usually agree with each other, especially on the ranking of the first few candidates.

Which completion rule should I use?

Usually it doesn't matter which completion rule is used, because there is usually a Condorcet winner, in which case all the rules will agree. If all the rules agree, you can be confident that you're getting the right result. However, it's a good idea to commit to the rule you're going to use ahead of time, to avoid arguments later on.

The different rules have advantages and disadvantages. Minimax finds the candidate that in a reasonable sense is closest to being the Condorcet winner; it is also the cheapest to compute. Richard Darlington has written an in-depth comparison of Minimax against other preferential voting methods (both Condorcet and non-Condorcet), and concluded that Minimax is the best method. Schulze (Beatpath Winner) is based on finding the strongest chains of pairwise defeats that can be used to argue one candidate is preferred over another; it can also be computed fairly cheaply using the Floyd-Warshall all-pairs-shortest-paths algorithm. The two ranked-pairs rules (CIVS Ranked Pairs and MAM) are more expensive but are still usually fast enough to be practical. If there are n candidates, the ranked pairs algorithms are about n times slower than Schulze. This slowdown is only noticeable if there are many candidates (more than twenty). The rankings produced by minimax tend to be the most stable in the sense that a given voter's ballot does not affect the ordering much. The difference between the two ranked pairs methods, CIVS Ranked Pairs and MAM, is that MAM uses a random tie-breaking method, whereas CIVS Ranked Pairs is completely deterministic. Thus, the result of running MAM is not determined by just the ballots cast. Comparison of the results of MAM and CIVS RP will show if randomization was needed and used by MAM. If there is a lot of concern about strategic voting (particularly, burying attacks), Condorcet-IRV is a reasonable candidate.

Minimax-PM

Minimax (also known as Simpson–Kramer), orders candidates based on their weakest defeats. It is an attractive method because it finds the candidate who could become the Condorcet winner with the fewest number of additional ballots. Unlike some other methods (notably, Dodgson and Young) that also try to find candidates “close to” being Condorcet winners, Minimax is also inexpensive to compute.

There are several variants of Minimax, with subtly different properties. The version of Minimax implemented by CIVS, called Minimax-PM, was introduced by Richard Darlington. Minimax-PM orders candidates in a series of tie-breaking steps. Let W stand for the number of ballots ranking candidate 1 over candidate 2, and L the number of ballots with the opposite ranking. Note that since CIVS allows voters to leave some candidates tied, the sum W+L may be less than the total number of ballots. Minimax-PM orders defeats first by the margin W−L, and further by the ratio W/L when margins are tied. Further, two candidates are compared first by their weakest defeats; then, if tied on those, by their 2nd weakest defeats, and so on. Darlington's simulation-based study showed Minimax-PM to be the best Condorcet method.

CIVS Ranked Pairs

The CIVS Ranked Pairs completion rule is a deterministic variant of Eppley's MAM method; it is also related to other completion methods such as Tideman Ranked Pairs. In these algorithms, each of the pairwise preferences in the preference matrix is considered in in the order of the strength of the preference, and kept (affirmed) if it does not create cycles with previously kept preferences. Otherwise, the preference is ignored because it is in conflict with stronger preferences.

Affirming preferences

In the CIVS ranked pairs algorithm, as in MAM, one preference is stronger than another if it has more votes in favor, or if the number of votes in favor are equal, if the preference has fewer votes against. Of course, it is entirely possible that two preferences have exactly the same number of votes in favor and against. Like MAM and unlike Tideman, the ordering of preferences does not take margins into account.

The major difference between CIVS Ranked Pairs and MAM is the rule on when to keep a preference. In CIVS RP, a preference is kept exactly when it does not create any new cycles when considered in conjunction with strictly stronger, kept preferences. Thus, preferences of equal strength may be kept even though in conjunction they produce a new cycle, as long as individually they do not.

CIVS RP is a deterministic method that does not use randomness, unlike MAM (and some other voting methods). Voting methods that rely on randomness need to have a mechanism for generating randomness in a trustworthy way, because otherwise the voting system itself might cheat by generating randomness until the best possible outcome is achieved from the viewpoint of whoever controls the randomness.

Ranking the candidates

The algorithm for ranking the various candidates is to successively identify the Schwartz sets defined by the graph of kept preferences. The top-ranked candidates are the initial Schwartz set: the smallest set of candidates such that no candidates outside the set are preferred to any in the set. After these candidates are removed from the graph, the second tier of candidates are the Schwartz set in the new graph, and so on. Typically, the Schwartz set consists of a single candidate at every level; ties can only occur if there are preferences of equal strength. When a Schwartz set contains multiple candidates, there must be a cycle of kept preferences. In this case, the candidates within that Schwartz set are ranked based on the strength of the strongest preference against that candidate (note that preferences involving candidates from higher-ranked Schwartz sets are not germane for this comparison).

Condorcet-IRV

The Condorcet-IRV rule is a Condorcet completion rule that uses an IRV-like process to perform Condorcet completion. It is also known as Condorcet–Hare in the literature. Given a set of candidates, this algorithm finds the top-ranked candidate (or candidates) in the following way. If there is a Condorcet winner, that is the top-ranked candidate. Otherwise, for each candidate, the ballots are examined to see on how many of the ballots that candidate is the highest ranked among the candidates being considered. Call this number the top count for the candidate. The candidate with the smallest count is removed from consideration and the process repeats, looking for a CW among the remaining candidates. If multiple candidates tie for having the smallest top count, one is randomly picked for removal, the count of ballots where they have the second-rank is used to break the tie, and so on. Eventually, there will either be a Condorcet winner, or the remaining candidates all have the same count of ranks. The remaining candidate or candidates are then considered the top-ranked candidates among the set. CIVS repeats this algorithm to construct a ranking of all candidates.

Note that although this rule uses a runoff procedure to eliminate “weak” candidates who create a cycle in the preference graph, it is still a Condorcet election method, unlike IRV/STV. If there is a CW, it will always be the top-ranked candidate.

The advantage of Condorcet-IRV is that it is relatively resistant to certain kinds of strategic voting. In particular, it resists burying, an attack in which voters insincerely push strong competitors to their preferred candidates lower in the rankings. A weaker form of burying is truncation, in which voters do not express their full preference by giving some set of candidates the lowest possible rank. As long as burying does not create a preference cycle, it has no effect on any Condorcet method. However, it is possible for burying to create a preference cycle that many completion rules will then resolve in favor of the candidate of the voters who have voted insincerely.

Regardless of the completion rule, burying can easily backfire on voters who employ it, because it can result in weaker candidates appearing to be consensus candidates. A successful use of burying can be tricky to carry off; the best policy is to vote sincerely.

Bottom-Two Runoff

Bottom-two runoff operates similarly to IRV, and similarly aims to be resistant to strategic voting. The candidates are initially seeded according to their top counts (using lower ranks to break ties, as described for Condorcet-IRV). Then the two lowest-seeded candidates are compared head-to-head to see who is preferred. The loser of this head-to-head contest is removed from consideration and the process repeats with the remaining candidates until a winner is found. CIVS then repeats this entire runoff process repeatedly to obtain the overall ranking of all candidates. Unlike IRV, bottom-two runoff is a Condorcet-consistent method because the head-to-head runoffs guarantee that a CW can never be eliminated.