Head-to-head methods examine matchups between exactly two candidates from the full set on a ranked choice ballot. To determine a head-to-head winner between two candidates, all ballots are reviewed, and each ballot counts as one vote for whichever candidate is ranked higher. The candidate with more votes between the two wins that matchup. This process is repeated for every pair of candidates, producing a head-to-head record for each candidate.

First, consider the feline community using Head-to-Head Record:

The Feline Community

This community has 3 candidates, the Tiger, the Cheetah, and the Skunk:

There are 60 voters in the Jungle group who prefer the Tiger over the Cheetah, and 40 voters in the Plains group who prefer the Cheetah over the Tiger. In both groups, the Skunk is the least preferred candidate.

First, consider the case where everyone votes their preferences:

Now, looking at the three head-to-head matchups and calculating each candidate's record:

The Tiger finishes 2-0, and wins the race. Now consider the case where the Plains group attempts to improve their outcome by ranking the Skunk ahead of the Tiger:

The outcome remains the same. The Plains group can adjust their ballots all they want, but cannot overturn a candidate with a majority preference.

The Wilderness Example

We return to the wilderness from previous articles:

The 3 factions are the Foresters (who support the Bear), the Felines (who support the Lion), and the Avians (who support the Eagle). The factions fill out their ballots based on their true preferences:

The Eagle finishes 2-0 and is the winner. But there is a much more important aspect of this result. The Avians did not need to coordinate with the Felines to reach their best achievable outcome. The Foresters did not need to coordinate with the Avians to reach their best achievable outcome. Everyone was able to vote their true preferences, and everyone reached that outcome given how the other voters voted.

The Eagle won this election, despite having the fewest first-place votes. Now let's suppose a fourth candidate, the Dolphin, were to enter this election:

None of the three Wilderness factions favor the Dolphin as their top choice, but all three factions favor the Dolphin as their second choice:

45 Voters

35 Voters

20 Voters

With no first place votes, the Dolphin would have been the first eliminated in an IRV or runoff election. But when each head-to-head matchup is examined, and each candidate's record is calculated:

The Dolphin defeats every other candidate head-to-head, without receiving a single first-place vote. This can be very counterintuitive. But there is a deeper principle in play here: every voter was able to reach their best achievable outcome without having to vote strategically.

In social choice theory, a candidate who can beat every other candidate head-to-head is called the Condorcet winner. Many people associate the term with candidates who receive over 50% of first-place votes. It is correct that such a candidate will always be the Condorcet winner, but a candidate doesn't need any first-place votes to be a Condorcet winner. On this website, we will simply refer to such a candidate as the consensus candidate. In these examples, the Eagle was the consensus candidate in the 3-way race, while the Dolphin was the consensus candidate in the 4-way race.

A processing method that both allows voters to reach their best achievable outcome without voting strategically, and leaves no incentive to manipulate ballots to influence outcomes, is going to be a powerful way to process ranked ballots. However, there is a tradeoff, as the following example will show:

Rail Corridor Commissioner

This is a different type of election. These voters came for the mayor race at the top of the ballot. The ballot includes numerous down-ballot races that not all voters are familiar with.

One of those down-ballot races is the race for Rail Corridor Commissioner, which has three candidates.

There are three voting groups.

The Transit voters rank Bellingham first because of his experience. They rank Cole second because he has good intentions, but also see him as someone just looking for a job. They rank Arnold last because of his questionable intentions.

The Short-name voters came to this election because they had a strong opinion on the mayor race, but they know nothing about the candidates in this race. This group looks at the names on the ballot and ranks them by length, choosing the shortest names first. They rank Cole first, Arnold second, and Bellingham last.

The ABC-voters also came to this election because of the mayor race and have no opinion on this contest. They like completing the ballot, so they fill in ovals even in races where they have no opinion. In those cases, they simply vote in ballot order, ranking Arnold first, Bellingham second, and Cole last.

In an election with 40 Transit voters, 35 Short-name voters, and 25 ABC voters, the ballots look like:

The head-to-head results produce a three-way tie with no clear winner.

Each candidate has a reasonable claim to winning. Bellingham could argue he has the most first-place votes. Arnold could argue he beats Bellingham head-to-head. Cole could argue that with the fewest last-place votes, he is the "least hated," and therefore should be the consensus candidate. Depending on how the tie is broken, any of the three could win.

This result raises a natural question. Voters may not always produce perfectly consistent preferences, but an election system still needs a way to resolve the outcome.

In sports, teams frequently finish a season with the same record. Most leagues have built-in tiebreaker systems, which consider factors such as head-to-head results, point differentials, and strength of schedule. There have been many cases where tiebreaker methods across different leagues have been widely debated.

Head-to-head methods provide a powerful framework for processing ranked ballots. They preserve clear majority preferences in many situations and allow voters to express their true preferences without needing to vote strategically. In these cases, winners cannot be derailed by strategic voting, and voters do not need to vote strategically to reach their best achievable outcome. However, when preferences form a cycle, there may be no single candidate who clearly stands above the others. In those cases, the outcome depends on the rules used to resolve the tie, and different rules can produce different winners. The next section will look at some possible tiebreaker methods.