Majority Choice Approval
Majority Choice Approval (MCA) is a class of rated voting systems which attempt to find majority support for some candidate. It is closely related to Bucklin Voting, which refers to ranked systems using similar rules. In fact, some people consider MCA a subclass of Bucklin, calling it ERBucklin (for EqualRatings[allowed] Bucklin).
Contents
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".
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 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:
 MCAA: Most approved candidate (most votes above lowest possible rating). This is also called "Majority Top//Approval", or MTA.
 MCAP: Most preferred candidate (most votes at highest possible rating)
 MCAM: Candidate with the highest score at the rating level where an absolute majority first appears, or MCAA if there are no majorities. This is the system closest to traditional Bucklin voting.
 MCAS: 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.
 MCAR: Runoff  Two "finalists" are chosen by one or two methods, such as one of the methods above or a equalityallowed Condorcet method over the given ballots. The finalists are then measured against each other using one of the following methods:
 MCAIR: Instant runoff (Condorcetstyle, 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.
 MCAAR: 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.
A note on terminology
"Majority Choice Approval" was first used to refer to a specific form, which would be 3level MCAAR in the nomenclature above (specifically, 3MCAARM). Later, a voting system naming poll chose this term as a moreaccessible replacement for "ERBucklin" in general. This is also sometimes known as "median ratings", because it can be defined as "the highest median rating wins".
Criteria compliances
All MCA variants satisfy the Plurality criterion, the Majority criterion for solid coalitions, Monotonicity (for MCAAR, assuming first and second round votes are consistent), and Minimal Defense (which implies satisfaction of the Strong Defensive Strategy criterion).
All of the methods are summable for counting at the precinct level. Only MCAIR actually requires a matrix (or, possibly two counting rounds), and is thus "summable for k=2" ; the others require only O(N) tallies, and are thus "summable for k=1".
The Participation criterion and its stronger cousin the Consistency criterion, as well as the Laternoharm criterion are not satisfied by any MCA variant, although MCAP only fails Participation if the additional vote causes an approval majority.
Other criteria are satisfied by some, but not all, MCA variants. To wit:
 Clone Independence is satisfied by most MCA versions. In fact, even the stronger Independence of irrelevant alternatives is satisfied by MCAA, MCAP, MCAM, and MCAS. Clone independence is satisfied along with the weaker and related ISDA by MCAIR and MCAAR, if ISDAcompliant Condorcet methods (ie, Schulze) are used to choose the two "finalists". Using simpler methods (such as MCA itself) to decide the finalists, MCAIR and MCAAR are not strictly clone independent.
 The Condorcet criterion is satisfied by MCAIR 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 MCAAR if at least half the voters at least approve the PC in the first round of voting. These methods also satisfy the StrategyFree criterion if an SFCcompliant method such as Schulze is used to pick at least one of the finalists. All other MCA versions, however, fail the Condorcet and strategyfree criteria.
 The Laternohelp criterion and the Favorite Betrayal criterion are satisfied by MCAP. They're also satisfied by MCAAR if MCAP is used to pick the two finalists.
 MCAAR satisfies the Guaranteed majority criterion, a criterion which can only be satisfied by a multiround (runoffbased) method.
Thus, the MCA method which satisfies the most criteria is MCAAR, using Schulze over the ballots to select one finalist and MCAP to select the other. Also notable are MCAM and MCAP, which, as rated methods (and thus ones which fail Arrow's rankingbased Universality criterion), are able to seem to "violate Arrow's Theorem" by simultaneously satisfying monotonicity and independence of irrelevant alternatives (as well as of course sovereignty and nondictatorship).
An example
Imagine that Tennessee is having an election on the location of its capital. The population of Tennessee is concentrated around its four major cities, which are spread throughout the state. For this example, suppose that the entire electorate lives in these four cities, and that everyone wants to live as near the capital as possible.
The candidates for the capital are:
 Memphis on Wikipedia, the state's largest city, with 42% of the voters, but located far from the other cities
 Nashville on Wikipedia, with 26% of the voters, near the center of Tennessee
 Knoxville on Wikipedia, with 17% of the voters
 Chattanooga on Wikipedia, with 15% of the voters
The preferences of the voters would be divided like this:
42% of voters (close to Memphis) 
26% of voters (close to Nashville) 
15% of voters (close to Chattanooga) 
17% of voters (close to Knoxville) 





Assume half of voters in each city rate one city preferred, two cities approved, and one city unapproved; and half rate one preferred, one approved, and two unapproved.
City  Preferred  Approved 

Memphis  42  42 
Nashville  26  84 
Chatanooga  15  79 
Knoxville  17  45 
There is no preferred majority winner. Therefore approved votes are added. This moves Nashville and Chatanooga above 50%, so a winner can be determined. All the given resolution methods would pick Nashville, which is also the pairwise champion in this example.
Various strategy attempts are possible in this scenario, but all would likely fail. If the eastern and western halves of the state both strategically refused to approve each other, in an attempt by the eastern half to pick Chatanooga, Nashville would still win. If Memphis, Nashville, and Chatanooga all bulletvoted in the hopes of winning, most Knoxville voters would probably extend approval as far as Nashville to prevent a win by Memphis, and/or at least a few Memphis voters (>8% overall, out of 42%) would approve Nashville to stop Chatanooga from winning. Either one of these secondary groups would be enough to ensure a Nashville win in any of the MCA variants. It would take a conjunction of four separate conditions for Chatanooga to plausibly win: it could happen only if Knoxville voters also preferred Chatanooga, and Nashville voters (unstrategically) approved Chatanooga, and no Memphis voters preferred Nashville, and the MCAP variant were used.
General strategy and notes
If the electorate knows which two candidates are frontrunners, and the pairwise champion is indeed among those two, the stable strategy is for everyone to approve or prefer exactly one of those two, and fill out the rest of the ballot honestly and as expressively as possible. (In the example above, that would mean an east/west split, with Nashville winning 6832 approval over Chatanooga.) If everyone follows this strategy, the pairwise champion will win with the only absolute majority. And if even half of voters follow this strategy, multiple majorities are highly unlikely.
However, this twofrontrunner strategy does not mean that MCA is subject to Duverger's law. A pairwise champion who is not one of the perceived frontrunners still has a good chance of winning, especially if they have some strong supporters. This fact, in turn, will affect what "frontrunner" means; an extremist candidate with a strong but sharplylimited base of support  the kind of candidate who, using simple runoff voting, makes it into a runoff with a strong showing of 35% but then gets creamed with only 37% in the runoff  will never be perceived as a frontrunner in the first place.
Thus, overall, many elections should be resolved without need for a resolution method, and so all MCA methods should give broadly similar results. However, if resolution is needed, a lack of majorities is, overall, more likely than multiple majorities. Since the design intent is to minimize these situations, the resolution method chosen should be one which tends to encourage extending approvals; that is, one which comes "close" to fulfilling the Laternoharm criterion, so that extending approval is unlikely to harm one's favorite candidate. From simple to complex, such methods are: MCAS, MCAIR, and MCAAR.