# Composite methods

Composite methods are voting methods constructed by combining multiple other methods. Usually sets like the Smith set or uncovered set are considered to be methods in this context.

Two common method compositions used on the election-methods list are the comma and double slash (or slash) compositions. These are as follows:

## Comma[edit | edit source]

The composition M1,M2 denotes taking the result of method M1, then breaking any ties in the ordering by using M2. For instance, for the following ballot set:

10: L>A>B>C>R 8: R>B>C>A>L 5: C>A>B>L>R

The first past the post ordering is L>R>C>A=B and the Smith set is {A, B, C}. Thus the method Smith,Plurality would return the ordering C>A=B>L>R.

This composition was first defined by Woodall in 2003.^{[1]}

## Slash[edit | edit source]

The composition M1//M2 denotes taking the result of method M1, eliminating everybody but the winners according to that method, and then giving the outcome of M2 on the reduced ballot set. For instance, the method that first eliminates every candidate not in the Smith set, then runs IRV on the remaining candidates, is Smith//IRV.

This composition was first defined on the EM list by Bruce Anderson in 1996.^{[2]}^{[fn 1]}

## Criterion compliances[edit | edit source]

If M1 and M2 are two methods, then M1,M2 is monotone if:

- M1 and M2 are both monotone
- Raising a candidate X can't add another candidate Y into the M1 set (or make Y tie for first if M1 is a method).

M1//M2 is monotone if:

- M1 and M2 are both monotone
- Raising a candidate X can neither add nor remove another candidate Y from the M1 set (or tied for first).

These are sufficient conditions: methods can be monotone even if they fail these conditions. For instance, Copeland//Borda (Ranked Robin) is monotone even though the conditions are not met.

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## Footnotes[edit | edit source]

- ↑ In Anderson's post, "Condorcet" refers not to the Condorcet winner but to Minimax Condorcet, which was called plain Condorcet in the early days of EM.

## References[edit | edit source]

- ↑ Woodall, D. R. (2003). "Properties of single-winner preferential election rules II: examples and problems (draft)". Cite journal requires
`|journal=`

(help) - ↑ Anderson, B. (1996-04-03). "Reply on EM to Mike's Reply on ER".
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