Condorcet methods take into account the complete ranking of choices from 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.
Condorcet election methods have been called Instant Round-Robin Voting because they use the voters' rankings to compare every choice against every other in head-to-head contests. Intuitively, it would not make sense if the winner lost in a head-to-head contest with some other choice. Condorcet methods aim to prevent such an outcome.
The voters are considered to collectively have a preference for choice A over choice B if A is ranked higher than B on more ballots than B is ranked higher than A on. If the voters prefer a choice A to every other choice, choice 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.
Although there is usually a Condorcet winner, especially when there are many voters, it is possible that there isn't one. In ranked-preference voting systems, preference cycles may exist. For example, it is possible that on most ballots, choice A is preferred to choice B, on most ballots B is preferred to C, and on most ballots C is preferred to A. Fortunately, cycles involving the top-ranked choice don't seem to happen often. And there are good methods for resolving these cycles, called completion rules. If the goal of the election is simply to pick the top choice, it usually doesn't matter 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.
CIVS currently supports five rules for Condorcet completion: Minimax (also known as Simpson–Kramer), Schulze (also known as Beatpath Winner and Cloneproof Schwartz Sequential Dropping), Maximize Affirmed Majorities (MAM), a deterministic variant of MAM called CIVS Ranked Pairs, and a runoff-based Condorcet algorithm called Condorcet-IRV. The first three rules are described elsewhere (follow the links); 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 choices.
Usually it doesn't matter which completion rule is used, because there is a Condorcet winner, in which case all the rules will agree. If all five 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 is the cheapest to compute and finds the choice that in one sense is closest to being the Condorcet winner. Beatpath Winner (Schulze) can also be computed fairly cheaply using the Floyd-Warshall all-pairs-shortest-paths algorithm. These two are probably the best known and most widely used methods. The two ranked-pairs rules (CIVS Ranked Pairs and MAM) are more expensive. If there are n choices, the ranked pairs algorithms are about n times slower than Beatpath Winner. This slowdown is only noticeable if there are many choices (more than twenty). Anecdotally, Beatpath Winner seems to be less stable than the other rules in the sense that adding one ballot to the set of ballots can have a large effect on the rankings, perhaps because one ballot can create a long new beatpath with global effects. Minimax tends to be the most stable in the sense that a given voter's ballot cannot affect the ordering as 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 choice.
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.
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.
The algorithm for ranking the various choices is to successively identify the Schwartz sets defined by the graph of kept preferences. The top-ranked choices are the initial Schwartz set: the smallest set of choices such that no choices outside the set are preferred to any in the set. After these choices are removed from the graph, the second tier of choices are the Schwartz set in the new graph, and so on. Typically, the Schwartz set consists of a single choice at every level; ties can only occur if there are preferences of equal strength. When a Schwartz set contains multiple choices, there must be a cycle of kept preferences. In this case, the choices within that Schwartz set are ranked based on the strength of the strongest preference against that choice (note that preferences involving choices from higher-ranked Schwartz sets are not germane for this comparison).
The Condorcet-IRV rule is a Condorcet completion rule that uses an IRV-like process to perform Condorcet completion. It was originally proposed by Thomas Hill of England's Electoral Reform Society. Given a set of choices, this algorithm finds the top-ranked choice (or choices) in the following way. If there is a Condorcet winner, that is the top-ranked choice. Otherwise, for each choice, the ballots are examined to see on how many of the ballots that choice is the highest ranked among the choices being considered. Call this number the top count for the choice. The choice with the smallest count is removed from consideration and the process repeats, looking for a CW among the remaining choices. (If multiple choices tie for having the smallest top count, one is randomly picked for removal.) Eventually, there will either be a Condorcet winner, or the remaining choices will all have the same top count. The remaining choice or choices are then considered the top-ranked choices among the set. CIVS repeats this algorithm to construct a ranking of all choices.
Note that although this rule uses a runoff procedure to eliminate “weak” choices 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 choice.
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 choices 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 choices 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 choice 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 choices appearing to be consensus choices. A successful use of burying can be tricky to carry off; the best policy is to vote sincerely.