More on MAS (version 3.0)

[Jameson Quinn]

I've been refining a 3-slot system for several weeks now. Let me be clear that I'm only working on one system, even though I've gone through various names as I refine it. The current name is MAS, Majority Acceptable Score. Here's my latest definition. Note that I've tweaked the default rule so that it can be said in one sentence. Mathematically it's trickier, but I think it makes some intuitive sense, as explained in the last sentence.

Here’s how MAS works: you can give each candidate 0, 1, or 2. Any candidate that gets a majority of 0’s is eliminated, unless that would eliminate everyone. Of the remaining candidates, highest score wins. 

Blank votes for a candidate are read as 0’s or 1’s; the proportion that count as 0’s is equal to the proportion between the voters that didn't give the candidate in question a 2, and those that gave a 2 to a candidate with a higher explicit score. Basically, that rule assumes that a voter would want to give 0s to they left blank if those candidates were weaker than their favorite, but 1s if those candidates were stronger.

Here's a scenario to illustrate:

Candidate	2 votes	1 votes	0 votes	Blank votes	Explicit score
A	30	0	0	70	60
B	25	25	0	50	75
C	42	0	55	3	84
D	8	42	0	50	58

(Note: I think that a scenario like the above, where one candidate got many more explicit 1-votes, would only happen in cases of center squeeze; that is, B's 1-votes probably come primarily from C voters. Thus, B is almost-certainly, but not quite provably, the CW here.)

Candidate A has 70 blank votes, and 70 voters who didn't give them a 2. 67 voters gave 2 to a candidate with a higher explicit score (C or B). 67 of A's blank votes count as 0s, leaving 3 1's. A gets a total score of 63, and is eliminated for a majority of 0's.

B has 50 blank votes, and 75 voters who didn't give them a 2. 42 voters gave 2 to a candidate with a higher explicit score (C). So 28 of the blank votes count as 0, 22 count as 1; B gets a score of 97. 

C is eliminated by explicit 0s. D has all their blank votes count as 0 since the number of 2-votes for explicitly stronger candidates is greater than the number who didn't vote for them. They are not quite eliminated. 

So B wins this scenario. If B had gotten 9 or fewer explicit 1-votes, A would have had a higher explicit score, and after assigning blank votes, A would have won.

This default rule does cause the system to technically fail FBC, because giving extra 2-votes to eliminated candidates can change how blank votes are assigned for uneliminated candidates. However, constructing an FBC-violating scenario would be nontrivial; I don't think it would ever happen in practice.

[Jameson Quinn]

Still working on refining this. Here's version 3.1. I expect the final version to be version 4.0, at which point the earlier versions and numberings will be only a historical curiosity.

Here’s how MAS works: you can give each candidate 0, 1, or 2. Any candidate that gets a majority of 0’s is eliminated, unless that would eliminate everyone. Of the remaining candidates, highest score wins. 

Blank votes for a candidate are read as 0’s or 1’s. The proportion that count as 0’s is equal to the proportion between the voters that didn't give the candidate in question a 2, and those that gave a 2 to a candidate with more 2's. 

This default rule gives exactly the result you'd get if blank votes were counted as 0 only for voters who preferred a stronger candidate, under some simple assumptions about which votes come from where: explicit votes of 1 for a given candidate are spread evenly among all voters who didn't give them a 2; explicit votes of 0 come only from voters who preferred a stronger candidate; every voter gives a 2 to exactly one "serious" candidate; and all "nonserious" candidates get fewer 2's than "serious" ones. You need simplifying assumptions like that so that counting can work by simply tallying the votes of each type, without recording how they are combined on each ballot.

[Jameson Quinn]

Still working on refining this. Here's version 3.2. I expect the final version to be version 4.0, at which point the earlier versions and numberings will be only a historical curiosity.

Here’s how MAS works: you can give each candidate 0, 1, or 2 points. Then, any candidate that gets a majority of 0’s is eliminated, unless that would eliminate everyone. Of the remaining candidates, highest score wins. 

Blank votes for a candidate count as 1 point in the same percentage as that candidate gets of 2-point votes. Otherwise, they count as 0.

Here's some examples of how that default rule works out:

	2-votes	1-votes	0-votes	Blanks	Total 0-votes	Score

A

30

0

0

70

0+70*.7=49	81
B	25	19	0	52	0+52*.75=39	83
C	40	0	40	20	40+20*.6=52	(88)
D	25	7	0	68	0+68*.75=51	(74)

As you can see, it takes just under 30% support to save a candidate from elimination if nobody explicitly downvotes them; and at around that level, it takes just under 4 explicit 1-votes to make up for a deficit in explicit 2-votes.

Since this rule looks at only one candidate at a time, it's easy to implement, and it easily passes FBC and participation. It fails consistency, but only in Simpson's-paradox-like situations, in which arguably consistency is actually a bad idea. It passes a weakened form of Frohnemayer balance:

[Jameson Quinn]

The table I sent had mistakes in the B row. Here it is, fixed:

	2-votes	1-votes	0-votes	Blanks	Total 0-votes	Score
A	30	0	0	70	0+70*.7=49	81

B

25

19

0

56

0+56*.75=42

83

[Ted Stern]

It would help to have a column for "Total 1-votes" also.

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[Jameson Quinn]

2016-10-14 17:40 GMT-04:00 Ted Stern <dodec...@gmail.com>:

It would help to have a column for "Total 1-votes" also.

Yes, but I couldn't fit more columns in my email (obviously, it depends what reader you use). 

I didn't say what the weakened Frohnmayer balance that it passes is. Here goes: for any given election outcome, and for any given ballot X, there is a ballot anti-X such that if you add both X and anti-X, the outcome order is preserved. (This differs from standard Frohnmayer balance in the underlined part. But basically the only part of the election outcome that matters is the list of which candidates are eliminated.)

I am pretty satisfied with this rule. It's summable; it works for reasonable chicken-dilemma and center-squeeze scenarios; it can be stated in one sentence, and has some intuitive appeal. So I think that MAS 3.2 will probably become MAS 4.0, that is, canonical MAS. Going once.... (that is, once I say this two more times, it will be definitive.)

[Andy Jennings]

On Fri, Oct 14, 2016 at 1:34 PM, Jameson Quinn <jameso...@gmail.com> wrote:

Here’s how MAS works: you can give each candidate 0, 1, or 2 points. Then, any candidate that gets a majority of 0’s is eliminated, unless that would eliminate everyone. Of the remaining candidates, highest score wins. 

Blank votes for a candidate count as 1 point in the same percentage as that candidate gets of 2-point votes. Otherwise, they count as 0.

Equivalently, for each candidate, calculate the fraction of voters who gave that candidate a 2.  It will be some number between 0 and 1.  Blank votes for that candidate count as this real number.

Eric Sanders would be proud.  (https://groups.google.com/d/msg/electionscience/SpLDc1Z0hzE/3NNFOT0oMvwJ)

[Jameson Quinn]

No, that's not equivalent, because a majority of 0s includes implicit 0s. (this would be easy to say if we used +1, 0, -1; but I don't want to use negative numbers.)

[Andy Jennings]

You're right.  It doesn't do the majority-zero elimination step properly, though for candidates that survive, it does the point calculation correctly.

[Andy Jennings]

On Fri, Oct 14, 2016 at 3:15 PM, Andy Jennings <abjen...@gmail.com> wrote:

You're right.  It doesn't do the majority-zero elimination step properly, though for candidates that survive, it does the point calculation correctly.

You could re-phrase the majority-zero elimination step to make it fit within this framework.  Here is how you could describe the whole system:

If there are any blanks for any candidates, they are filled in with the fraction of voters who gave that candidate a 2 (a number between 0 and 1).

Then, pretend all the 2-votes are 1-votes and calculate each candidate's average.  If anyone gets an average over 0.5, then eliminate everyone who got an average below 0.5.

If more than one candidate remains, count the 2-votes as 2-votes and highest average wins.

I think this is worse than your original verbiage.  But it could be useful as an "alternative formulation" for people trying to understand the system.


 

[Jameson Quinn]

How about this:

Blank votes are counted as 1 or 0 using the percentages of voters who gave or did not give the candidate in question a 2. So if 30% of voters gave a certain candidate a 2, and they got 1000 blank votes, those would count as 30 points and 70 votes of 0.

[William Waugh]

Have you reduced the tallying algorithm to practice as a Google spreadsheet?

On Thursday, October 13, 2016 at 6:48:07 AM UTC-4, Jameson Quinn started the discussion at https://groups.google.com/forum/?fromgroups#!topic/electionscience/OGDZPQO13DQ

[Jameson Quinn]

Here's a MAS spreadsheet: https://docs.google.com/spreadsheets/d/1siFG6XmOZokygY-86EhAKgv8YwzKtTET6AJopyXRqu0/edit?usp=sharing

2016-10-15 10:13 GMT-04:00 William Waugh <2knuw...@snkmail.com>:

Have you reduced the tallying algorithm to practice as a Google spreadsheet?

On Thursday, October 13, 2016 at 6:48:07 AM UTC-4, Jameson Quinn started the discussion at https://groups.google.com/forum/?fromgroups#!topic/electionscience/OGDZPQO13DQ

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[Warren D Smith]

One possibly-good feature of MAS or systems like it is, it is asymmetric --
treats the score "0" differently. Not like the max score for example.

Why is this good? Well, it probably is not good in some
mathematical idealization, but
in reality, it is a known psychological fact/flaw in humans
that they heavily overuse the score "0".

So if we had some model of human voters which knew about that, and put
that model in a Bayesian Regret simulation, then
asked which voting systems perform well in terms of BR...
then the results (1) might be interesting, (2)
might favor some MAS-like system more than you might
have at first thought.

Or not. For example, MAS might get into major trouble
precisely because of strategic voter overuse of 0 and might
indeed worsen that problem. But even if that does happen,
that does not mean the overarching idea of trying
to design a voting system that tries to make lemonade out
of the asymmetric "0" lemons,
was bad -- it just would mean MAS was bad.

Another interesting thing about MAS is it sort of
is a hybrid of approval & score voting.


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Warren D. Smith
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