[EM] Someone thinks that Approval should meet the Mutual Majority Criterion

Abd ul-Rahman Lomax abd at lomaxdesign.com
Thu Jun 6 11:41:10 PDT 2013

At 11:18 AM 6/6/2013, Michael Ossipoff wrote:
>Someone recently accused me of contriving the definition of sincere
>voting so that Approval would fail MMC (which specifies sincere voting
>in its premise).

Michael has a bit of misunderstanding floating in his head about 
this. I pointed out that definitions have been manipulated with 
certain voting systems criterion, in general, in order to generate 
"failures." And, of course, they can be manipulated in the other 
direction. This was on the Center for Election Science mailing list, 
which Michael just unsubscribed from, saying there were no open issues.

So he brings one here? Ah, well, to each his own.

He did clearly misunderstand the issue. First of all, the definition 
of "sincere vote" was not the issue. In Approval Voting, there is a 
large set of possible sincere votes. Rather, the issue is about the 
question of whether or not a vote that does not express a preference 
that is at the foundation of a voting system criterion satisfies the 
precondition for applying the test of the criterion.

Michael, here, did not give the relevant definitions. That is 
irritatingly typical. There were two definitions examined:

 From Wikipedia:
>The mutual majority criterion is a criterion used to compare voting 
>systems. It is also known as the majority criterion for solid 
>coalitions and the generalized majority criterion. The criterion 
>states that if there is a subset S of the candidates, such that more 
>than half of the voters strictly prefer every member of S to every 
>candidate outside of S, this majority voting sincerely, the winner 
>must come from S. This is similar to but stricter than the majority 
>criterion, where the requirement applies only to the case that S 
>contains a single candidate.
>The Schulze method, ranked pairs, instant-runoff voting, Nanson's 
>method, and Bucklin voting pass this criterion.
>The plurality vote, approval voting, range voting, the Borda count, 
>and minimax fail this criterion.

The point I have made is that "strictly prefer" must require that the 
voter actually vote the preference. Otherwise we have what I call 
Space Alien Failure. Under Space Alien Failure, Plurality fails the 
Majority criterion, because the voter, being informed by Space Aliens 
that their favorite cannot win the election, vote for someone else. 
No, the voter must *actually vote the preference* to set up the 
condition for the test.

Otherwise we are led to a series of preposterous conclusions.

The term "stricly prefer" is ambiguous, unless we interpret "prefer" 
as "act to prefer." With Approval voting, a voter acts to prefer A to 
B by voting for A and not for B. The voter acts to prefer a set by 
voting for every member of the set and not for every non-member.

If a voter votes for a nonmember, in addition to voting for members 
of the set, the voter is no longer "strictly preferring" the set, so 
the precondition fails to apply.

Notice that this does not, at all, reduce the criterion to mincemeat. 
Approval voting passes, but range voting does not. That is, by the 
way, not necessarily the fault of range voting, because Range voting 
fails in a situation where the failure may improve the outcome, in 
ways where *every voter might agree.*

>One way to answer his objection is to ask him to compare Approval with
>methods that meet MMC, and ask himself if he notices a difference.

That's not an answer, it's a socratic question, and it requires me to 
study matters where I have little interest.

I see no sign that Michael has understood the objection.

>The methods that I recommend for the Green scenario, IRV, but
>especiallly Woodall or maybe Benham, also are free of chicken dilemma,
>and so maybe comparing Approval with them would be unfair--because
>we're only trying to show the benefit of MMC.
>So then, let's compare Approval with a method that has chicken
>dilemma, but passes MMC. Let's compare Approval with Beatpath:
>The A voters and the B voters all prefer both A and B to C. The A
>voters and the B voters are, together, a majority of the voters.
>They are a mutual majority, and {A,B} is their MM-preferred set.
>Let's assume that there is no chicken dilemma. The A voters and the B
>votes are co-operative and amicable. None of them are inclinded to
>defect against eachother. The A voters and the B voters have no
>chicken dilemm need to not rank eachother's candidate.
>So, voting sincerely, the voting looks like this:
>Because the A voters and the B voters add up to a majority, C is defeated.
>The A voters and the B voters succeeded in getting a winner from their
>majority-preferred set by merely ranking sincerely.
>If the A voters are more numerous than the B voters, then A will win
>instead of B. The A voters can gain that, while still fully supporting
>B againist C.
>Can they do that in Approval?

It's entirely a different question. Approval does not allow ranking 
beyond two ranks. The method that is analogous could be Bucklin, 
which is "instant runoff approval." That allows ranking. The voters 
vote the same way as described above, and the winner must come, under 
the stated conditions, from (A,B). Voters may choose to 
second-preference their less-favorite, or may drop this candidate to 
third rank (original Bucklin).

What about raw Approval? What can the voters do?

Well, they could simply vote for the members of the set (A,B). If 
their preference for A over B or the reverse is weak, this is a 
rational strategy.

However, we know that this is unlikely. 2-round top-two approval will 
handle this one. In the primary, they may vote for their favorite, 
simple, like that. By the conditions (which imply that C has stronger 
factional support), A or B will make it into the runoff with C, and, 
at that point, the majority may pick their winner. It will be the one 
of A and B that has stronger support, which is proper.

Bucklin does the job more reliably if used in two rounds, and 
probably accomplishes the same with a single round.

>MMC measures for something of practical importance that Beatpath, IRV,
>Woodall, Benham, and Schwartz Woodall have, but  which Approval
>doesn't have.

Michael has a point, but it should not be based on mind-reading. It's 
obvious that Approval cannot fully handle three-candidate elections 
in a single round, where what he calls the "chicken dilemma" applies. 
The chicken dilemma is simply a strategic choice:

1. Vote only for the favorite and risk a loss to C, the least-liked.
2. Vote for favorite and second best and risk loss of favorite to second best.

Hence I don't suggest Approval voting as a full solution to voting 
system problems. Approval merely make an improvement. I'd rather have 
that strategic choice than less choice, the condition under 
Plurality, where I may need to vote for second best as if he or she 
were my favorite.

It's crazy to compare Approval, a very simple voting system, with far 
more complex systems that allow full or enhanced ranking. 

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