Maximize Affirmed Majorities: Difference between revisions
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'''Maximize Affirmed Majorities''' (MAM) is a [[
A simple summary of it is that it looks at all [[Pairwise matchup|pairwise matchups]], orders them from strongest (largest) to weakest (smallest), and starting from the first defeat, fixes the winner of the pairwise matchup higher in the [[Order of finish|order of finish]] than the loser, and repeats this for all successive defeats (if doing this for a particular defeat would contradict a previous fixed ranking, then that defeat is ignored). The winner is the candidate or candidate(s) who are in 1st place in the constructed order of finish.
MAM is a [[Condorcet method]]. If there is a candidate X who, for every other candidate Y, is ranked by a majority over Y, then X will be elected. In other words, MAM will place X atop the order of finish. Such a candidate is a Condorcet winner, and requiring it be elected (when it exists) is the Condorcet criterion.
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Like many Condorcet methods, MAM always elects a candidate in the Smith set (a.k.a. top cycle), which is the smallest non-empty subset of the candidates such that, for each candidate x in the subset and each candidate y outside the subset, a majority rank x over y. (When there is a Condorcet winner, it alone is in the Smith set.) Requiring that the winner be in the Smith set generalizes the Condorcet criterion to the case where there is not necessarily a Condorcet winner and has a similar justification. With MAM, all candidates in the Smith set finish over all candidates outside the Smith set.
MAM is very similar to the "[[Ranked Pairs]]" method invented by [[Nicolaus Tideman]]. However, the there are key differences between the methods regard whether ties are allowed (since MAM allows voters to express ties at any tier of preference) and how defeats are calculated.
MAM satisfies numerous desirable criteria, including Nicolaus Tideman's independence from clone alternatives, Peyton Young's local independence from irrelevant alternatives, Mike Ossipoff's strong defensive strategy criterion, weak defensive strategy criterion and truncation resistance, and Eppley's immunity from majority complaints. (Only MAM and extremely similar variations satisfy immunity from majority complaints.) ▼
== Criteria ==
▲MAM satisfies numerous desirable criteria, including [[Nicolaus Tideman]]'s independence from clone alternatives, Peyton Young's local independence from irrelevant alternatives, [[Mike Ossipoff]]'s strong defensive strategy criterion, weak defensive strategy criterion and truncation resistance, and Eppley's immunity from majority complaints. (Only MAM and extremely similar variations satisfy immunity from majority complaints.)
MAM is highly deterministic, satisfying the resolvability and reasonable determinism criteria. To a casual observer, MAM may appear less deterministic than it is. When sorting the majorities into the largest-to-smallest order of precedence, the order in which same-size majorities are sorted would depend on chance. That often won't affect the order of finish. Even when it does affect the order of finish, it often doesn't affect which alternative finishes on top, which is the common standard for evaluating determinism.
=== Thwarted majorities ===
Originally MAM was defined as the voting method that finds the order of finish that minimizes the largest "thwarted" majority. (A majority is thwarted if the order of finish does not place their more preferred candidate over their less preferred candidate.) It was originally named "Minimize Thwarted Majorities." The original definition is equivalent (same order of finish) to the quicker procedure with which MAM is more commonly described. (The equivalence is helpful when proving some properties of MAM.) The name was soon changed to MAM to emphasize the positive rather than the negative.
MAM has some similarity to the Kemeny-Young voting method (KY), which is also defined in terms of finding the "best" order of finish but defines "best" differently. KY finds the order of finish that maximizes the sum over all pairings of the voters who ranked the pair the same as in the order of finish. Unlike MAM and RP, KY satisfies Young's reinforcement criterion, which requires that if two collections of votes produce the same order of finish, then the order of finish given all the votes must be the same. (This is weaker than the same-named criterion satisfied by the Borda count, which requires that if two collections of votes produce the same winner, then the winner given all the votes must be the same.) KY fails independence of clones in a way that could induce nominations of large numbers of clones, which arguably is much more important than MAM's failure to satisfy reinforcement. A small minority can easily manipulate failures of clone independence since it doesn't take many people to nominate alternatives, whereas election rules can easily prevent a minority from manipulating failures of reinforcement.▼
Some people treat MAM as a variation of the [[Ranked Pairs]] (RP) voting method invented by Tideman in 1987 and refined by Tideman & Zavist in 1989. However, some differences cause MAM and RP to behave significantly differently. The two most important differences are: (1) RP expects votes to be linear orderings whereas MAM allows weak orderings. (That is, a vote can rank two or more candidates as equals to express indifference between them.) (2) RP subtracts the size of the opposing coalition when measuring the size of each pair, whereas MAM does not subtract the size of the opposing minority when measuring the size of each majority. These differences allow MAM to satisfy desirable criteria that RP fails, making it risky and misleading to treat MAM as just a variant of RP.▼
== Comparisons ==
As a Condorcet method, Maximize Affirmed Majorities is often compared to other election methods
=== Ranked Pairs ===
▲MAM has some similarity to the Kemeny-Young voting method (KY), which is also defined in terms of finding the "best" order of finish but defines "best" differently. KY finds the order of finish that maximizes the sum over all pairings of the voters who ranked the pair the same as in the order of finish. Unlike MAM and RP, KY satisfies Young's reinforcement criterion, which requires that if two collections of votes produce the same order of finish, then the order of finish given all the votes must be the same. (This is weaker than the same-named criterion satisfied by the Borda count, which requires that if two collections of votes produce the same winner, then the winner given all the votes must be the same.) KY fails independence of clones in a way that could induce nominations of large numbers of clones, which arguably is much more important than MAM's failure to satisfy reinforcement. A small minority can easily manipulate failures of clone independence since it doesn't take many people to nominate alternatives, whereas election rules can easily prevent a minority from manipulating failures of reinforcement.
{{seealso|Ranked Pairs}}
:''See also: [[Ranked Pairs]]''
Some people treat MAM as a variation of the [[Ranked Pairs]] (RP) voting method invented by Tideman in 1987 and refined by Tideman & Zavist in 1989. However, some differences cause MAM and RP to behave significantly differently. The two most important differences are:
# The "Ranked Pairs" method relies on votes to be linear orderings whereas MAM allows weak orderings. (That is, a vote can rank two or more candidates as equals to express indifference between them.)
▲
There are also some minor differences between MAM and RP:
# When RP sorts the pairs into its order of precedence, it includes all N<sup>2</sup>-N ordered pairs, including tied pairs and minorities, whereas MAM sorts only the majorities and ignores minorities & ties completely (except in rare cases, when sorting majorities that are the same size). When votes can be weak orderings, ties may be larger than majorities, which means tied pairs could affect the outcome if included in the order of precedence.
# Since RP includes ties and minorities in its order of precedence, RP does more tiebreaking during sorting and none later, whereas with MAM there may be a tiebreaking step after the sorted majorities are considered (if more than one alternative is atop the order of finish).
# Since RP requires votes to be linear orderings, a linear tiebreak order can be had by randomly picking a single vote, whereas MAM, which allows votes to be weak orderings, may need to pick multiple votes to construct the linear tiebreak order.
# The details of how MAM uses the tiebreak order differ subtly from how Zavist & Tideman's Ranked Pairs uses their tiebreak order, so that MAM completely satisfies the strong Pareto criterion.
=== Other methods ===
Two other voting methods that can be considered close variations of MAM are [[Maximum Majority Voting]] (MMV) and River.
== Immunity from second place complaints ==
One of the main "competing" Condorcet methods is the [[Schulze method
== Procedure ==
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In some situations, a very large majority may be required to enact certain changes. For instance, a 3/4 supermajority might be needed to amend a group's charter. Continuing the status quo is a highly privileged alternative. What voting method should be used if more than one change is proposed, and the changes conflict so they can't all pass? They can be voted on using MAM. (Continuing the status quo would be an alternative too.) If the status quo tops the order of finish, or if the alternative that tops the order of finish isn't ranked over the status quo by a sufficiently large supermajority, then the status quo wins. Otherwise the alternative atop the order of finish wins (and becomes the new status quo).
== Mike Ossipoff's restatement of MAM ==
[[User:MichaelOssipoff]] proposed the following definition of MAM, which is copied here from the for [[MAM]] page
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1. X beats Y iff the number of ballots ranking X over Y is greater than the number of ballots ranking Y over X.
2. If X beats Y, then there is said to be a "defeat" of Y, by X. "XY" refers to that defeat.
3. XY is "stronger" than AB iff the number of ballots ranking X over Y is greater than the number of ballots ranking Y over X.
...but if those numbers are equal, then XY is stronger than AB iff the number of ballots ranking B over A is greater than the number of ballots ranking Y over X.
If neither XY nor AB is stronger than the other, then they're "equal".
4. A "cycle" is a sequence of defeats in which each defeat's defeated candidate is the defeating candidate in the next defeat,
and in which the sequence's initial defeating candidate is the defeated candidate in the sequence's last defeat.
5. A defeat "contradicts" a set of other defeats iff it is in a cycle consisting only of it and them.
6. A defeat is a "discarded defeat" iff it contradicts a set of not-discarded defeats each of which is either stronger than it, or equal to it and above it in the "dominance-order".
7. How to determine two defeats' dominance-order:
...a) Randomly choose one of the election's ballots. It will be referred to as the "comparison-ballot", and used as such for entire count, for all defeat-dominance-order comparisons.
...b) If XY and AB are equal (neither stronger than the other, as defined above), then XY is above AB in the dominance order iff B is ranked higher than Y on the comparison ballot.
...c) ...but if the comparison-ballot ranks Y and B equal (at the same rank-position), then XY is above AB in the dominance-order iff X is ranked higher than A on the comparison-ballot.
...d) If XY and AB are equal, and if, by b) and c), neither XY nor AB is above the other in the dominance-order, then randomly choose another ballot as the comparison-ballot, for the purpose of comparing XY and AB with regard to dominance-order.
8. A candidate wins if s/he has no not-discarded defeats.
== Notes ==
The MAM ranking is a [[Smith set ranking]]. This is because all candidates in the n-th Smith set and higher Smith sets will have pairwise victories over all candidates in lower Smith sets, therefore, none of the candidates in lower Smith sets have majorities or ties that would fix them above candidates in higher Smith sets in the order of finish, and all candidates in the n-th Smith set must thus finish higher than all candidates in lower Smith sets.
MAM passes [[ISDA]]. This is because candidates not in the Smith set can't finish above any candidate in the Smith set, therefore they can't impact how candidates in the Smith set are ranked in relation to each other.
== External Resources ==
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* [http://condorcet.org/rp Ranked Pairs]
* [http://accurate-democracy.com/ Accurate Democracy]
* Maximum Majority Voting page on https://RadicalCentrism.org (archived copy: [https://web.archive.org/web/20070928062036/http://radicalcentrism.org/majority_voting.html
** "Maximum Majority Voting" seems to be a slightly different system.
* [
* [http://fc.antioch.edu/~jarmyta@antioch-college.edu/voting.htm Voting methods resource page]
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{{fromwikipedia}}
[[Category:Defeat-dropping Condorcet
[[Category:Smith-efficient Condorcet methods]]
[[Category:Voting system criteria]]
[[Category:Majority–minority relations]]
[[Category:Monotonic electoral systems]]
[[Category:Clone-independent electoral systems]]
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