# Benham's method

Benham's method is a variation of instant-runoff voting independently invented by Chris Benham, David Hill, and Robert Loring.[1][2][3] The method calls for tabulating the first-choice of all voters on all ballots (as done with instant-runoff), 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 Equal-ranking methods in IRV page for more information.

## Example

(This doesn't feature the actual ballots, but shows the IRV round-by-round breakdown and the pairwise table; for each IRV round, a column next to it is created to show how many votes all candidates from the bottom going up have combined; this allows you to see how candidates were batch eliminated. The winner is the candidate who pairwise beats all other candidates remaining in the round):

IRV round-by-round breakdown (candidates to be eliminated are italicized, while candidates who just barely avoided elimination are underlined)
Initial ordering of

candidates (1st)

1st (shows candidates' % of

# of other candidates

they pairwise beat)

Cumulative

Reordering of

candidates (2nd)

2nd
B 35% (2/4) A 40% (1/2)
A 25% (3/4) B 32% (0/2)
C 17% (3/4) 32%

(>= 25%)

C 28% (2/2)
D 8% (1/4) 15% (<17%)
E 7% (1/4) 7% (<8%)
Pairwise table (not reorganized) (wins are bolded, and eventually eliminated candidates' matchups are struck-out)
A B C D E
A --- 163 297 405 389
B 105 --- 156 378 401
C 328 157 --- 305 250
D 120 167 307 --- 250
E 57 61 127 278 ---

Note that in regular IRV, it's possible C could've lost because he had the fewest 1st choices, which would've resulted in A and B continuing to the next round and one of them winning.

## Notes

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.) Benham's method passes mono-add-plump, unlike several Condorcet-IRV hybrids. It fails ISDA, however.[4]

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). An alternative way of looking at it is that the regular IRV result can be found when a round is reached where a Condorcet winner with over 1/3rd of the 1st choice votes exists (because of the dominant mutual third property) and elects them, whereas Benham does the same but without the "over-1/3rd" requirement.

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. Note that had B had a few more 1st choices, they would've had over 1/3rd of all 1st choice votes, and thus been guaranteed to win in IRV as well.

Because Benham's method is just IRV with the possibility of ending in an earlier round, it is possible to figure out who the Benham winner would be if given the full results of an IRV election (the round-by-round breakdown) and the pairwise comparison table for that election.

Like other deterministic voting methods, Benham's method is vulnerable to tactical voting. The combination of Condorcet and IRV principles leads to what may be considered a conflicting mechanism: on the one hand, first place votes are ignored when there's a Condorcet winner; on the other, they're all that matters when there's no Condorcet winner. However, these principles may also cover each other's weak spots, thus in part explaining Benham's unusual strategy resistance.

Benham is cloneproof for the same reason that IRV is: suppose the winner is X and is cloned. Then these clones can't have higher first preference counts than X itself, and so all appear after X. When all of these clones but one has been eliminated, then X still wins. The introduction of clones may lead to Benham needing more rounds to determine the winner, however.

## References

1. Benham, Chris (2006-10-16). "Condorcet + IRV completion?". Election-methods mailing list. Retrieved 2022-03-28.
2. Ossipoff, Michael (2014-04-28). "Benham's method looks best, among the Smith + CD methods". Election-methods mailing list. Retrieved 2022-01-11.
3. Loring, Robert (2000-08-16). "Loring One Winner Rule". Accurate Democracy Voting Rules. Retrieved 2022-04-23.
4. Green-Armytage, James. "Four Condorcet-Hare Hybrid Methods for Single-Winner Elections" (PDF).