[EM] MCA on electowiki

[Jameson Quinn]

I edited Electowiki to essentially replace the Bucklin-ER article with a
new, expanded MCA article. In this article, I define 6 MCA variants. I find
that as a class, they do surprisingly well on criteria compliance. Please
check my work:

http://wiki.electorama.com/wiki/Majority_Choice_Approval#Criteria_compliance

<http://wiki.electorama.com/wiki/Majority_Choice_Approval#Criteria_compliance>I
also put a mention of the pre-Napoleonic use of Bucklin in Geneva on the
Bucklin page.

Here's a copy of the definitions and compliances for MCA:

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<http://wiki.electorama.com/wiki/index.php?title=Absolute_majority&action=edit&redlink=1>
of
voters, that candidate wins. If not, the same happens if there is only one
candidate approved by a majority.

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


   - MCA-P: Most preferred candidate


   - 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.


   - MCA-S: Range or Score winner, using (in the case of 3 ranking levels) 2
   points for preference and 1 point for approval.


   - MCA-R: Runoff - One or two of the methods above is used to pick two
   "finalists", who 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-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.

[edit<http://wiki.electorama.com/wiki/index.php?title=Majority_Choice_Approval&action=edit&section=2>
]A note on the term MCA

"Majority Choice Approval" was at first used to refer to a specific form of
MCA, which would be 3-level MCA-AR in the nomenclature above. Later, a
voting system naming poll <http://betterpolls.com/v/1189> chose it as a
more-accessible replacement term for ER-Bucklin in general.
 [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>
).

The Condorcet criterion<http://wiki.electorama.com/wiki/Condorcet_criterion> is
satisfied by MCA-VR if the pairwise champion (PC, aka CW) is visible on the
ballots. It is satisfied by MCA-AR if at least half the voters at least
approve the PC in the first round. Other MCA versions fail this criterion.

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 to decide the
finalists, MCA-IR and MCA-AR are not clone independent.

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.

The Participation <http://wiki.electorama.com/wiki/Participation_criterion>
 and Summability
criterion<http://wiki.electorama.com/wiki/Summability_criterion> are
not satisfied by any MCA variant, although MCA-P only fails Participation if
the additional vote causes an approval majority.

None of the methods satisfy
Later-no-harm<http://wiki.electorama.com/wiki/Later-no-harm_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>
".

Thus, the 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), 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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