By the way, my first message mistakenly said MCA fails the Summability criterion; I meant the Consistency criterion.
Here's the latest version of the criteria compliance, which is the same as before except for the above change and some editing and reformatting: [edit<http://wiki.electorama.com/wiki/index.php?title=Majority_Choice_Approval&action=edit§ion=3> ]Criteria compliance All MCA variants satisfy the Plurality criterion<http://wiki.electorama.com/wiki/Plurality_criterion>, the Majority criterion for solid coalitions<http://wiki.electorama.com/wiki/Majority_criterion_for_solid_coalitions> , Monotonicity <http://wiki.electorama.com/wiki/Monotonicity_criterion> (for MCA-AR, assuming first- and second- round votes are consistent), and Minimal Defense <http://wiki.electorama.com/wiki/Minimal_Defense_criterion> (which implies satisfaction of the Strong Defensive Strategy criterion<http://wiki.electorama.com/wiki/Strong_Defensive_Strategy_criterion> ). All of the methods are matrix-summable<http://wiki.electorama.com/wiki/Summability_criterion> for counting at the precinct level. Only MCA-IR actually requires a matrix (or, possibly two counting rounds), and is thus "summable for k=2<http://wiki.electorama.com/wiki/Summability_criterion>" ; the others require only O(N) tallies, and are thus "summable for k=1<http://wiki.electorama.com/wiki/Summability_criterion> ". The Participation criterion<http://wiki.electorama.com/wiki/Participation_criterion> and its stronger cousin the Consistency criterion<http://wiki.electorama.com/wiki/Consistency_criterion>, as well as the Later-no-harm criterion<http://wiki.electorama.com/wiki/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: - Clone Independence<http://wiki.electorama.com/wiki/Strategic_nomination> is satisfied by most MCA versions. In fact, even the stronger Independence of irrelevant alternatives<http://wiki.electorama.com/wiki/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<http://wiki.electorama.com/wiki/ISDA> by MCA-IR and MCA-AR, if ISDA-compliant Condorcet methods (ie, Schulze<http://wiki.electorama.com/wiki/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<http://wiki.electorama.com/wiki/Condorcet_criterion> is satisfied by MCA-IR if the pairwise champion<http://wiki.electorama.com/wiki/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<http://wiki.electorama.com/wiki/Strategy-Free_criterion> if an SFC-compliant method such as Schulze<http://wiki.electorama.com/wiki/Schulze> is used to pick at least one of the finalists. All other MCA versions, however, fail the Condorcet and strategy-free criteria. - The Later-no-help criterion<http://wiki.electorama.com/wiki/Later-no-help_criterion> and the Favorite Betrayal criterion<http://wiki.electorama.com/wiki/Favorite_Betrayal_criterion> are satisfied by MCA-P. They're also satisfied by MCA-AR if MCA-P is used to pick the two finalists. - MCA-AR satisfies the Guaranteed majority criterion<http://wiki.electorama.com/wiki/Guaranteed_majority_criterion>, a criterion which can only be satisfied by a multi-round (runoff-based) method. Thus, the MCA method which satisfies the most criteria is MCA-AR, using Schulze <http://wiki.electorama.com/wiki/Schulze> over the ballots to select one finalist and MCA-P to select the other. Also notable are MCA-M and MCA-P, which, as *rated* methods (and thus ones which fail Arrow's *ranking* -based Universality criterion<http://wiki.electorama.com/wiki/Universality_criterion>), are able to seem to "violate Arrow's Theorem<http://wiki.electorama.com/wiki/Arrow%27s_Theorem>" by simultaneously satisfying monotonicity and independence of irrelevant alternatives<http://wiki.electorama.com/wiki/Independence_of_irrelevant_alternatives> (as well as of course sovereignty and non-dictatorship).
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