[EM] XA

[Michael Ossipoff]

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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