# Condorcet//Approval

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The procedure for Smith//Approval.

Condorcet//Approval or C//A is an election method by which the Condorcet winner is elected if one exists, otherwise the approval winner is elected. Approval could be specified in various ways. The double-slash notation signifies that one eliminates all losers of the first step before performing the second step. It is also possible to limit contenders to members of the Smith or Schwartz set, resulting in Smith//Approval or Schwartz//Approval.

When approval is implemented such that it isn't possible to rank some candidate X over a candidate Y without approving candidate X, Condorcet//Approval and similar methods have good burial resistance. This mostly holds even in the case where it isn't possible to rank X above Y without approving X if you also approved Y.

## Example

#voters Their vote
6 A>C>B>D
2 B>C>A>D
3 B>A>C>D
2 C>D>A>B
2 C>B>A>D
5 D>C>A>B
1 D>A>C>B

The Condorcet winner is C.

#voters Their vote
6 A>B>D>C
2 B>C>A>D
3 B>A>C>D
2 C>D>A>B
2 C>B>A>D
5 D>C>A>B
1 D>A>C>B

The A-top voters' strategy would make A win in most Condorcet methods, but supposing every voter sets their approval threshold between C and A, C still wins in both Condorcet//Approval and Smith//Approval.

In this sense, Condorcet-cardinal hybrids can use cardinal information to allow voters to focus on the pairwise matchups most important to them, which are usually the ones between the frontrunners.

### Explicit/Implicit Approval

Approval can be designated by a cutoff/threshold ranked among alternatives (explicit approval). Alternatively, if truncation is allowed, all explicitly ranked candidates could be considered to be approved (implicit approval).

### Burial resistance

Condorcet methods are generally vulnerable to burying strategy. One faction buries a candidate by ranking him insincerely below other candidates. This is an attempt to give this candidate new or stronger pairwise defeats.

When all explicitly ranked candidates are considered approved, Condorcet//Approval makes burying strategy more risky than in other Condorcet methods. Burying is only effective when it prevents the targeted candidate from being the Condorcet winner. But a faction can't succeed in this task without then being counted as approving the candidate(s) beneath which the targeted candidate was insincerely ranked. This makes it quite likely that burying strategy will backfire, and cause a candidate to be elected who is actually liked less than the targeted candidate.

One drawback in proposing implicit approval of all explicitly ranked alternatives is that, in general elections, many "sincere" (naive) voters will feel entitled to rank all alternatives. Many may then bristle at the assertion that they approve of unsavory candidates by exercising their franchise to distinguish between lesser and greater "evils". This perception may be mitigated somewhat by a change in terminology. Even so, voters or legislators may balk at reform if they anticipate voter frustration.

### Favorite Betrayal criterion compliance

Approval voting's satisfaction of the Favorite Betrayal criterion can be preserved in Condorcet//Approval by using the tied at the top rule. This results in the Improved Condorcet Approval (ICA) method. However, this variant technically isn't Condorcet-compliant.

## Satisfied Criteria

Condorcet//Approval satisfies Condorcet and always elects a majority favorite, but doesn't satisfy Smith or the Majority criterion for solid coalitions. It satisfies monotonicity and, when truncation is permitted, Minimal Defense (and the Strong Defensive Strategy criterion) and the Plurality criterion.

It fails Clone-Winner, Participation, and Later-no-harm.

## Smith//Approval

The approval winner may be limited to the Smith set. This variation satisfies the Smith criterion and the Majority criterion for solid coalitions. It satisfies clone independence at least when clones are defined such that every voter approves either all or no members of a clone set.

Example:

25 A| >B>C

40 B>C| >A

35 C>A| >B

There is an A>B>C>A cycle, with the approvals being A 60, B 40, and C 75. C is elected for having the most approvals in the Smith set.

17 A>B>C|

17 B| >C>A

17 C>A| >B

5 A>B>C>D| >E

1 A|

25 D>E|

24 E>D|

Approvals are A 40, B 39, C 39, D 54, and E 49. The Smith set is A, B, and C, because there is an A>B>C>A Condorcet cycle and they each pairwise beat D and E by at least 56 votes to at most 49. Within the Smith set, A has the most approvals, so A wins. Note that the Approval winner and Condorcet//Approval winner is D, because there is no Condorcet winner and D has the most approvals overall.

The Smith//Approval ranking of the above candidates is (A>B=C)>(D)>(E) (parentheses used to indicate each Smith set in the Smith set ranking This is because A has the most approvals in the Smith set, with B and C tying for having the 2nd-most. The next two Smith sets have only one candidate in them each, so they are ranked consecutively.

## Explicit, Fully Ranked Smith//Approval

If this deserves a concise name, then use mine -- Jrfisher
Voters must rank all but one alternative, (disallow ties or truncation).
Therefore, use an approval cutoff (explicit approval).
Pick the Condorcet Winner (CW) if there is one. Otherwise...
Find the smallest set of alternatives having no defeats outside the set (Smith set).
Within that set, the alternative with the most winning votes against the approval bar is the winner.

## Notes

If a voter ranks A>B>C>D>E, and approves only A and B, but C, D, and E are the only candidates in the Smith Set, then this voter would have no influence over who wins in the Smith Set in Smith//Approval. Thus, a modification could be that every voter is assumed to approve their favorite candidate(s) in the Smith Set. Alternatively, if this voter had approved A, B, and C, and all 3 of them were the only candidates in the Smith Set, then again they'd have no influence over which Smith Set candidate wins. So it's also possible to assume every voter disapproves their least favorite(s) In the Smith Set.

Approval can be indicated on ranked or rated ballots with an approval threshold based on ranks or scores (i.e. a voter could approve everyone they scored a 5/10 and up or ranked 3rd and up). It could also be done by letting the voter mark approval for each individual candidate (in addition to being able to rank each one) or having the approval threshold itself be rankable i.e. a voter ranking A>B>threshold>C would on their ballots rank A 1st, B 2nd, C 3rd and the threshold would be another "candidate" that they rank as their 2nd choice, so that their 1st and 2nd choices A and B would be approved.

Condorcet-Approval hybrids are a specific case of Condorcet-Score hybrids, such as Smith//Score.

One thing that may work in favor of Condorcet-cardinal hybrids as opposed to other Condorcet methods is that the cycle resolution is simpler and put more into the voter's hands i.e. it is more intuitive to elect the candidate with the greatest overall support as explicitly indicated by the voters than to run more complex algorithms to determine the winner.

Here is a counting trick for Smith//Approval that should help most of the time, since the Approval voting ranking of the candidates is often very close to the Smith set ranking: if you order the candidates from most approvals to least, and are finding the Smith set, and the candidate with the most approvals is found to be one of the members of the Smith set, then you don't need to find the other members of the Smith set to find the winner, since it's guaranteed that if the candidate with the most approvals overall is in the Smith set, then they are the winner. The same applies for the candidate with the next-most approvals, etc.