[EM] XA

[Jameson Quinn]

>
>
> Jameson--
>
> The Name "Majority-Score" isn't nearly descriptive enough.
>

The highest average score among those given a score by a majority? Would
you prefer "minority-rejected score"? (I'm not saying I'd rename it to that
solely on your say-so, but if various people prefer that name I'd probably
take up the suggestion.)


>
> I like "SARA" much better. Yes, the letters are out of order, but that's
> ok, because "Abstain" is signifiantly different from the other ratings.
> "Support", "Accept", and "Reject" all cast a vote for or against the
> candidate. Support & Reject also give points.  Abstain does nothing, and
> that distinguishes it from the others, justifying having it last in the
> acronym.
>

Actually, the rating that's qualitatively different is "reject". The other
three are effectively just numbers. In Majority Score, I've changed the
labels to be:
support, assist, accept, reject
rather than:
support, accept, abstain, reject.

I think that "abstain" was a misnomer, given that an abstention counted
against rejection and reduced average score. Doing two "opposite" things
isn't the same as doing nothing; it's still strictly nicer than "reject"
and strictly meaner than the other middle rating (which was "accept" and is
now "assist").

>
> Forest--
>
> I ask you what I asked Andy Jennings:  What's the motivation that led to
> XA? That would clarify what's going on with the method, and why the
> percentages are used to refer both to percentage of voters and percentage
> of max rating. And it would surely lead to the best way of introducing &
> explaining the method to the public.
>
> Michael Ossipoff
>
> On Thu, Oct 27, 2016 at 5:56 PM, Forest Simmons <fsimmons at pcc.edu> wrote:
>
>> It turns out that Chiastic Approval is a good method in the context of
>> the Chicken Dilemma, much better than ordinary Approval, Majority Judgment,
>> or plain Range.
>>
>>
>> Ballots are score/range style ratings.  Let x be the greatest number for
>> which there is some candidate that is given a rating of at least x percent
>> on at least x percent of the ballots.  Elect the candidate X that is
>> given a rating of at least x percent on the greatest number of ballots.
>>
>>
>> The Greek letter Chi corresponds to the Roman letter X,, hence the name
>> Chiastic Approval or XA for short.  Furthermore, when the method is
>> described graphically, the value of x is found by intersecting two graphs
>> whose union looks like the letter Chi.
>>
>>
>> Andy Jennings came up with XA while thinking about how to improve
>> Majority Judgement.  Since we were both familiar with ancient literary
>> structures called Chiasms (identified in the Book of Mormon about 15
>> decades after its first publication) the name came naturally.
>>
>>
>> Skip the following technical paragraph unless you are very curious about
>> the graphical description.
>>
>>
>> [Let f be the function given by f(x) = the percentage of ballots on which
>> X is given a rating of at least x percent.  Then f is a decreasing
>> function whose graph looks like the downward stroke of the letter Chi.  The
>> graph of y = x looks like the stroke with positive slope.  These two
>> graphs cross at the point (x, x) which yields the Chiastic Approval cutoff
>> x.]
>>
>>
>> Now consider the following ballot profile …
>>
>> 41 C
>>
>> 31 A>B(33%)
>>
>> 28 B>A(50%)
>>
>>
>> Note that A is the only candidate with a rating of at least 50% on at
>> least 50% of the ballots, so A is the XA winner.
>>
>>
>> We could lower the 50% to 42%, and raise the 33% to 40%, and A would
>> still be the XA winner, as the only candidate with a rating of at least 42%
>> on at least 42% of the ballots.
>>
>>
>> In fact we could go further than that by splitting up the the 28 B>A
>> faction with some die hard defectors:
>>
>> 41 C
>>
>> 31 A>B(40%)
>>
>> 11 B>A(42%)
>>
>> 17 B
>>
>>
>> Candidate A is still the only candidate given a rating of at least 42% on
>> at least 42 percent of the ballots.
>>
>>
>> But if two more B faction voters defect, then C is elected as the only
>> candidate given a rating of at least 41 percent on at least 41 percent of
>> the ballots:
>>
>> 41 C
>>
>> 31 A>B(40%)
>>
>> 8 B>A(42%)
>>
>> 20 B
>>
>>
>> In the general CD set up we have three factions with sincere preference
>> profiles
>>
>> P: C
>>
>> Q: A>B
>>
>> R: B>A
>>
>>
>> Where P > Q > R>0, and P+Q+R=100
>>
>>
>> Under Chiastic Approval there is a Nash equilibrium that protects the
>> sincere CW candidate A :
>>
>> P: C
>>
>> Q: A>B(33%)
>>
>> R: B>A(50%)
>>
>>
>> Candidate A is the only candidate rated at a level of at least 50% on at
>> least 50% of the ballots.
>>
>>
>> As in the first example, the equilibrium is preserved if the 33% is
>> raised to any value less than P%, and/or the 50% is lowered to any value
>> greater than P percent.
>>
>> P: C
>>
>> Q: A>B(P%-epsilon)
>>
>> R: B>A(P%+epsilon)
>>
>>
>> Furthermore part of the B>A faction can defect without destroying this
>> equilibrium:
>>
>>
>>
>> P: C
>>
>> Q: A>B(P%-epsilon)
>>
>> R1: B>A(P%+epsilon)
>>
>> R2: B
>>
>> For R=R1+R2 as long as R1 > P – Q .
>>
>>
>> So we see that XA has a rather robust Nash equilibrium that protects the
>> CWs in the context of a Chicken Dilemma threat.  The threatened faction
>> down-rates the candidate of the potential defectors to any value less than
>> P%.  Since (in this context) P is always greater than 33 (otherwise it
>> could not be the largest of the three factions), the 33 percent rating can
>> always be safely used to deter the defection.  Mainly psychological
>> reasons would make it more satisfactory to raise that 33% closer to P%.
>>
>>
>> So we see that high resolution ratings are not needed. Four levels will
>> suffice nicely if they are 0, 33%, 50%, and 100%.  Grade ballots like those
>> used for Majority Judgement could be adapted to XA.
>>
>>
>> As an approval variant like Bucklin, XA has no vulnerability to burial
>> tactics.
>>
>>
>> Unlike MMPO it also satisfies Plurality.
>>
>>
>> It is monotone and clone independent (in the sense that Approval and
>> Range are clone independent).
>>
>>
>> It is efficiently summable, but is it precinct consistent? i.e. does a
>> candidate that wins in every precinct win over-all?
>>
>>
>> Does it satisfy Participation?
>>
>>
>> We need to explore it, and learn how to explain it as simply as possible,
>> so we can persuade people to use it.
>>
>>
>> Forest
>>
>>
>>
>>
>>
>>
>>
>>
>
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://lists.electorama.com/pipermail/election-methods-electorama.com/attachments/20161028/0d745bc0/attachment.htm>