# Total approval chain climbing

Total approval chain climbing (TACC) is a Condorcet completion method using an approval component, invented by Forest Simmons. In the three-candidate case it is equivalent to electing the candidate in the cycle who defeats the approval loser pairwise. The motivation is to reduce burial incentive.

It works as follows:
1. Sort the candidates by increasing total approval. 2. Starting with an empty "chain of candidates", consider each candidate in the above order. When the candidate defeats all candidates already in the chain, add her at the top of the chain. The last added candidate wins.
Note that because of Condorcet cycles, it isn't always possible for one candidate to beat all others. Because of this, the procedure may stop before all candidates can be added to the chain.

Example:

34 A>B|>C 35 B|>C>A 31 C>A|>B

Approvals (using the approval threshold symbol "|") are A 65, B 69, C 31. C is added first into the chain because C has the lowest approval, and then B, not A, is added into the chain, because B is the only candidate who pairwise beats C. A can't then be added in, because A can only beat B and not also C. Therefore, B is the winner.

## Notes

TACC and its variants are notable for always electing from the Banks set, a subset of the uncovered set and thus the Smith set..

TACC is Smith-efficient because all members of the Smith set pairwise beat all candidates not in the Smith set, therefore once a Smith set member is added to the chain, a non-Smith set member can't be added to the chain, because they can't beat the Smith set member, and thus, can't beat everyone in the chain. Since this implies a Smith set member must be the last added candidate, only members of the Smith set can win.

Total Score chain climbing

Total Score chain climbing (TSCC) is a Condorcet completion method using an score-component. The motivation is to reduce burial incentive.

It works as follows:
1. Sort the candidates by increasing total score.

2. Starting with an empty "chain of candidates", consider each candidate in the above order and do a pairwise comparison: for each ballot:

- determine the maximum score in the "chain of candidates" for this individual ballot

- if the candidate has a higher score then this maximum score count this as a win,

- if the candidate has the same or a lower score then this maximum score count this as a loss

3. If there are more wins then losses add the candidate to the top of the chain.

4. The last added candidate wins.