Majority Choice Approval: Difference between revisions

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== How does it work? ==
== How does it work? ==


Voters rate candidates into a fixed number of rating classes. There are commonly 3, 4, 5, or even 100 possible rating levels. The following discussion assumes 3 ratings, called "preferred", "approved", and "unapproved".
Voters rate candidates into a fixed number of rating categories, e.g. "Good," "Neutral," and "Bad."


If one and only one candidate is preferred by an [[absolute majority]] of voters, that candidate wins. If not, approvals are added to preferences, and again if there is only one candidate with a majority they win.
If one and only one candidate is given the highest rating by an [[absolute majority]] of voters, that candidate wins. If not, the second-highest rating is added to each candidate's vote total; again, if there is only one candidate with a majority they win. This process continues until some candidate has a majority.


Unfortunately, unless there are many categories, and voters do not cluster at round numbers (e.g. multiples of 5 or 10), this procedure is likely to end up with multiple candidates reaching a majority at the same rating. Therefore, a tiebreaking procedure is needed. Some possible resolution methods include:
If the election is still unresolved, one of two things must be true. Either multiple candidates attain a majority at the same rating level, or there are no candidates with an absolute majority at any level. In either case, there are different ways to resolve between the possible winners - that is, in the former case, between those candidates with a majority, or in the latter case, between all candidates.

The possible resolution methods include:


* MCA-A: Most approved candidate (most votes above lowest possible rating). This is also called "Majority Top//Approval", or MTA.
* MCA-A: Most approved candidate (most votes above lowest possible rating). This is also called "Majority Top//Approval", or MTA.


* MCA-P: Most preferred candidate (most votes at highest possible rating)
* MCA-P: Most preferred candidate (most votes at highest possible rating).


* MCA-M: Candidate with the highest score at the rating level where an absolute majority first appears, or MCA-A if there are no majorities. This is the system closest to traditional [[Bucklin voting]].
* MCA-M: Candidate with the highest score at the rating level where an absolute majority first appears. This system is equivalent to traditional [[Bucklin voting]].


* MCA-S: [[Range voting|Range]] or Score winner. That is, the candidate with the highest score, counting (in the case of 3 ranking levels) 2 points for each preference and 1 point for each approval.
* MCA-S: [[Range voting|Range]] or Score winner. The candidate with the highest average (mean) score is declared winner, where candidates are given 0 points for the lowest rating, 1 point for the second-lowest, etc.


* MCA-R: Runoff - Two "finalists" are chosen by one or two methods, such as one of the methods above or a equality-allowed Condorcet method over the given ballots. The finalists are then measured against each other using one of the following methods:
* MCA-R: Runoff. Two finalists are chosen by one of the methods above or an equality-allowed Condorcet method over the given ballots. The finalists are then measured against each other using one of the following methods:
** MCA-IR: Instant runoff (Condorcet-style, using ballots): Ballots are recounted for whichever one of the finalists they rate higher. Ballots which rate both candidates at the same level are counted for neither.
** MCA-IR: Ballots are counted for whichever one of the finalists they rate higher.
** MCA-AR: Actual runoff: Voters return to the polls to choose one of the finalists. This has the advantage that one candidate is guaranteed to receive the absolute majority of the valid votes in the last round of voting of the system as a whole.
** MCA-AR: Actual runoff: Voters return to the polls to choose one of the finalists. This has the advantage that one candidate is guaranteed to receive the absolute majority of the valid votes in the last round of voting of the system as a whole.


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All MCA variants satisfy the [[Plurality criterion]], the [[Majority criterion for solid coalitions]], [[Monotonicity criterion|Monotonicity]] (for MCA-AR, assuming first- and second- round votes are consistent), and [[Minimal Defense criterion|Minimal Defense]] (which implies satisfaction of the [[Strong Defensive Strategy criterion]]).
All MCA variants satisfy the [[Plurality criterion]], the [[Majority criterion for solid coalitions]], [[Monotonicity criterion|Monotonicity]] (for MCA-AR, assuming first- and second- round votes are consistent), and [[Minimal Defense criterion|Minimal Defense]] (which implies satisfaction of the [[Strong Defensive Strategy criterion]]).


All of the methods are [[Summability criterion|summable]] for counting at the precinct level. Only MCA-IR actually requires a matrix (or, possibly two counting rounds), and is thus "[[Summability criterion|summable for k=2]]" ; the others require only O(N) tallies, and are thus "[[Summability criterion|summable for k=1]]".
All of the methods are [[Summability criterion|summable]] for counting at the precinct level. Only MCA-IR actually requires a matrix (or, possibly two counting rounds), and is thus "[[Summability criterion|summable for k=2]]"; the others require only O(N) tallies, and are thus "[[Summability criterion|summable for k=1]]".


The [[participation criterion]] and its stronger cousin the [[consistency criterion]], as well as the [[later-no-harm criterion]] are not satisfied by any MCA variant, although MCA-P only fails participation if the additional vote causes an approval majority.
The [[participation criterion]] and its stronger cousin the [[consistency criterion]], as well as the [[later-no-harm criterion]] are not satisfied by any MCA variant, although MCA-P only fails participation if the additional vote causes an approval majority.


Other criteria are satisfied by some, but not all, MCA variants. To wit:
Other criteria are satisfied by MCA variants with appropriate tiebreakers, including:


* [[Independence of irrelevant alternatives]]
* [[Strategic nomination|Clone Independence]] is satisfied by most MCA versions. In fact, even the stronger [[Independence of irrelevant alternatives]] is satisfied by MCA-A, MCA-P, MCA-M, and MCA-S. Clone independence is satisfied along with the weaker and related [[ISDA]] by MCA-IR and MCA-AR, if ISDA-compliant Condorcet methods (ie, [[Schulze]]) are used to choose the two "finalists". Using simpler methods (such as MCA itself) to decide the finalists, MCA-IR and MCA-AR are not strictly clone independent.


* The [[Condorcet criterion]] is satisfied by MCA-IR if the [[pairwise champion]] (aka CW) is visible on the ballots and would beat at least one other candidate by an absolute majority. It is satisfied by MCA-AR if at least half the voters at least approve the PC in the first round of voting. These methods also satisfy the [[Strategy-Free criterion]] if an SFC-compliant method such as [[Schulze]] is used to pick at least one of the finalists. All other MCA versions, however, fail the Condorcet and strategy-free criteria.
* The [[Condorcet criterion]] is satisfied by MCA-IR if the [[pairwise champion]] (aka CW) is visible on the ballots and would beat at least one other candidate by an absolute majority. It is satisfied by MCA-AR if at least half the voters at least approve the PC in the first round of voting. These methods also satisfy the [[Strategy-Free criterion]] if an SFC-compliant method such as [[Schulze]] is used to pick at least one of the finalists. All other MCA versions, however, fail the Condorcet and strategy-free criteria.
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