The discussion in this article is technical, and nothing in this article is required reading for any future articles on this site. This section takes a deeper look into methods designed to resolve cycles that can occur in head-to-head methods.

In social choice theory, the concept of a Condorcet winner as a consensus or compromise candidate has existed for a long time. Even though cyclical results are uncommon, resolving them is still important. This has led to the development of systems such as Ranked Pairs and Schulze, which handle cyclical results differently.

Like Head-to-Head Record discussed in the previous article, Ranked Pairs begins with a list of head-to-head matchups between candidates. The matchups or "pairs" are then sorted by margin of victory, or "ranked" from strongest to weakest. Starting from the top of the list, victories are “locked” if they do not create cyclical loops, or skipped if they do.

We return to the Rail Corridor Commissioner example from the previous article:

The voters filled out their ballots as follows:

Those ballots produce the following head-to-head results:

This creates a cyclical result with no Condorcet winner. To apply Ranked Pairs, we first list the winning matchups and their margins of victory:

Next, the matchups are sorted from strongest victory to weakest victory:

Finally, the pairs are locked or skipped depending on whether they create a loop or not.

In this example, "Arnold over Bellingham" is skipped because "Bellingham over Cole" and "Cole over Arnold" have already been locked. Adding "Arnold over Bellingham" would complete a circular loop. The remaining locked pairs are:

This leaves "Bellingham above Cole", and "Cole above Arnold", making Bellingham the final winner under Ranked Pairs.

The Schulze method also begins with the same head-to-head matchups, but it approaches cycles differently. Instead of locking victories one at a time, Schulze compares candidates using "beat paths" and "strongest path strength". This works as follows:

• If a candidate defeats another candidate directly, this is considered a direct beat path. The path strength is the number of votes received by the winner in that head-to-head matchup.
• If a candidate loses to another candidate, but can still reach that candidate through a cycle, this is considered an indirect beat path. The path strength is determined by the weakest link in that path.
• If a candidate loses to another candidate and has no indirect beat path, then the path strength is 0.

A candidate may lose directly to another candidate, but still have a stronger path through an indirect route. A ranking based on strongest path strengths can then be used to determine the winner.

We will describe this using the same Rail Corridor Commissioner example. First, list the direct head-to-head victories and calculate strongest path strength:

Then, consider indirect paths between candidates, where the path strength is determined by the weakest link:

For each path, the strength of the path is determined by its weakest matchup. The strongest available path between two candidates is then used for comparison. Combining the direct and indirect paths produces the following strongest-path matrix:

The matrix can be used to decide winners using the higher strongest path strength:

Note that while "Arnold defeats Bellingham" in a direct head-to-head matchup, "Bellingham defeats Arnold" based on strongest path strength. Based on strongest path strength, Cole defeats Arnold, while Bellingham defeats both Cole and Arnold. Under the Schulze method, Bellingham is the winner.

While Condorcet-style methods produce the same winner when a Condorcet winner exists, they do not always produce the same winner when one does not. The next example is intentionally unusual and is designed to demonstrate how different methods can diverge. For this example, instead of candidate names we will use letters:

75 voters fill out the ballots as follows:

Using these ballots, we can calculate the head-to-head record for each candidate:

Candidate A finishes with the best overall head-to-head record at 3-1. However, there is still no Condorcet winner, because Candidate A does not defeat every other candidate head-to-head.

We now apply Ranked Pairs to the same ballots. First, extract the winning matchups and their victory margins:

Next, sort the victories from strongest to weakest:

The victories are then locked from the top down, skipping any matchup that would create a loop.

The final locked ordering becomes:

This produces the ordering B > C > D > E > A. Under Ranked Pairs, Candidate B becomes the winner.

We now apply the Schulze method to the same ballots. First, list the direct head-to-head victories between candidates and calculate strongest path strength:

Next, examine the indirect paths connecting candidates and calculate strongest path strength:

Finally, compare the strongest path strengths between all candidate pairs to determine which candidates defeat others based on path strength:

Using this matrix, the resulting ordering is C > B > E > D > A. Under the Schulze method, Candidate C becomes the winner.

One important takeaway is that while all three methods would produce the same Condorcet winner, they can produce very different winners when no Condorcet winner exists. Even though examples like this are uncommon, they demonstrate an important property of head-to-head systems. Different head-to-head methods can interpret the same set of ranked ballots differently when cyclical outcomes occur. There is much deeper mathematics surrounding cyclical results and Condorcet methods than what is covered here, and readers interested in these topics can find extensive academic discussion online.

Because of this, if a head-to-head method is going to be used in an election, the rules for resolving cycles and comparing candidates need to be defined before the election takes place. In the rare case that cyclical outcomes determine the election, the result should be determined by the system rather than by reactions to the outcome.