# Benham's method

**Definition of Benham's method:**

Do IRV, but before each elimination check if there is an un-eliminated candidate who pairwise beats all other un-eliminated candidates, and elect them if they exist. Between two candidates X and Y, X pairwise beats Y if more ballots rank X over Y than rank Y over X.

An alternative way of describing it is: "Elect the Condorcet winner (if there is one), otherwise eliminate the candidate ranked the highest by the fewest voters, and repeat". This can be further rephrased to also work as a variant of STV, in the following way: "Eliminate the candidate ranked the highest by the fewest voters unless that candidate is the Condorcet winner, in which case eliminate the candidate ranked highest by the second fewest voters".

When voters are allowed to equally rank candidates, Benham's method can either be implemented by equally splitting each voter's vote between each candidate they equally ranked highest, or giving each equally-highest-ranked candidate one vote. See the single transferable vote page for more information.

Benham's method is a Smith-efficient Condorcet method. This is because there will always be a point in the count where at least one Smith set member is uneliminated, and that candidate must beat all other candidates by virtue of being in the Smith Set. (It is also possible that once some members of the Smith set are eliminated, one of the multiple remaining members wins by virtue of beating every remaining candidate, including those in the Smith set. In fact, any voting method that operates by eliminating one at a time until there is a Condorcet winner is Smith-efficient.) Benham's method passes mono-add-plump, unlike several Condorcet-IRV hybrids. It fails ISDA, however.^{[1]}

Benham's method can be thought of as a Tideman alternative method that uses the Condorcet winner as its "set". Benham's method can also be thought of as an advanced version of IRV which interprets "majority" to mean "candidate who can win a majority pairwise against all other uneliminated candidates" rather than "majority's 1st choice among all uneliminated candidates". Because the latter is always equivalent to the former (a candidate who is a majority's 1st choice is always a Condorcet winner), Benham's method will never require more rounds of counting (eliminations, ignoring the discovery of the pairwise counting table) than plain IRV, and will often require none (when there is a Condorcet winner).

Example:

34: A>B>C

32: B>A

34: C>B>A

Regular IRV eliminates B and elects A here, whereas Benham elects B for being the Condorcet winner (pairwise beats A and C 66 to 34 each). This is an example of an averted center squeeze instance.

## References[edit | edit source]

- ↑ Green-Armytage, James. "Four Condorcet-Hare Hybrid Methods for Single-Winner Elections" (PDF).