[EM] MCA on electowiki

Jameson Quinn jameson.quinn at gmail.com
Mon Oct 18 16:56:57 PDT 2010


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&section=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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