# Approval-seeded Maximal Pairings

**AMP** is a method designed to elect C in the following situation:

- 51 strategic voters with A(100) > C(52) > B(0)
- 49 strategic voters with B(100) > C(52) > A(0)

### Ballot

- Each voter marks one option as her "favourite" option and may name any number of "offers". An "offer" is an (ordered) pair of options (
*y,z*). by "offering" (*y,z*) the voter expresses that she is willing to transfer "her" share of the winning probability from her favourite*x*to the compromise*z*if a second voter transfers his share of the winning probability from his favourite*y*to this compromise*z*. (Usually, a voter would agree to this if she prefers*z*to tossing a coin between her favourite and*y*).

- Alternatively, a voter may specify cardinal ratings for all options. Then the highest-rated option
*x*is considered the voter's "favourite", and each option-pair (*y,z*) for which*z*is higher rated that the mean rating of*x*and*y*is considered an "offer" by this voter.

- As another, simpler alternative, a voter may name only a "favourite" option
*x*and any number of "also approved" options. Then each option-pair (*y,z*) for which*z*but not*y*is "also approved" is considered an "offer" by this voter.

### Tally

- For each option
*z*, the "approval score" of*z*is the number of voters who offered (*y,z*) with any*y*. - Start with an empty urn and by considering all voters "free for cooperation".
- For each option
*z*, in order of descending approval score, do the following:- Find the largest set of voters that can be divvied up into disjoint voter-pairs {
*v,w*} such that*v*and*w*are still free for cooperation,*v*offered (*y,z*), and*w*offered (*x,z*), where*x*is*v*'s favourite and*y*is*w*'s favourite. - For each voter
*v*in this largest set, put a ball labelled with the compromise option*z*in the urn and consider*v*no longer free for cooperation.

- Find the largest set of voters that can be divvied up into disjoint voter-pairs {
- For each voter who still remains free for cooperation after this was done for all options, put a ball labelled with the favourite option of that voter in the urn.
- Finally, the winning option is determined by drawing a ball from the urn.

(In rare cases, some tie-breaking mechanism may be needed in step 3 or 3.1.)