[EM] MCA on electowiki

[Kathy Dopp]

James,

Why is failure of the "participation criteria" not equivalent to
failure of monotonicity?

Thanks.
Kathy

> Date: Mon, 18 Oct 2010 14:26:06 -0500
> From: Jameson Quinn <jameson.quinn at gmail.com>
> To: election-methods <election-methods at electorama.com>,
>        electionsciencefoundation <electionscience at googlegroups.com>
> Subject: [EM] MCA on electowiki
> Message-ID:
>        <AANLkTimGdVNrtAZ9VHn2jqJbAd2wXO7vYHz_NhxUSTR8 at mail.gmail.com>
> Content-Type: text/plain; charset="iso-8859-1"
>
> 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).



-- 

Kathy Dopp
http://electionmathematics.org
Town of Colonie, NY 12304
"One of the best ways to keep any conversation civil is to support the
discussion with true facts."

Fundamentals of Verifiable Elections
http://kathydopp.com/wordpress/?p=174

Realities Mar Instant Runoff Voting
http://electionmathematics.org/ucvAnalysis/US/RCV-IRV/InstantRunoffVotingFlaws.pdf

View some of my research on my SSRN Author page:
http://ssrn.com/author=1451051