[EM] XA

Forest Simmons fsimmons at pcc.edu
Mon Oct 31 16:12:25 PDT 2016


Good suggestion.  Here's my first crack at it:

If you write 100 next to a candidate's name, your ballot will contribute
max support (one point) to that candidate's approval, unconditionally.

If you write zero next to the name of candidate X, your ballot will not
contribute anything to X's approval.

If you write 70 next to a candidate, your ballot will contribute one point
to that candidate's approval unless more than 70 percent of the voters
write a larger number next to her name.

[Your ballot does not contribute to over zealous support.]

If you write 50 next to the name of candidate X, your ballot will
contribute one point to X's approval, unless moe than 50% of the ballots
write a number larger than 50 next to her name.

If you write 20 next to the name of candidate X, your ballot will
contribute one point to X's approval, unless moe than 20% of the ballots
write a number larger than 50 next to her name.

Etc.

The candidate with the greatest total approval is elected.




On Sat, Oct 29, 2016 at 5:09 PM, Michael Ossipoff <email9648742 at gmail.com>
wrote:

> It seems to me that if there's a way to introduce & explain XA to people,
> it starts out like this:
>
> "With XA, when  you assign a number, it isn't just a merit-rating.
>
> "If you write ".9" next to a candidate's name, you're saying that you want
> for .......to........"
>
> .That's as far as I got.
>
> The left-out parts should refer to something directly affecting the matter
> of who wins. It should be brief & simple, for a clear, easy, natural &
> intuitive introduction & explanation.
>
> 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
>>
>>
>>
>>
>>
>>
>>
>>
>
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