[EM] XA

[Michael Ossipoff]

It occurred to me that, to go with 0, 1/3, 1/5 & 1, there should be a 2/3
rating for symmetry.

Also, then you could use the letter-grades, A, B, C, D, F.

But the 1/3 would be used in chicken-dilemma situations, a time when
diplomacy is a good idea. A "D" rating doesn't sound very diplomatic.

So, maybe just call the ratings 0, 1/3, 1/2, 2/3, & 1.

Or else: Top, Very Good, Middle, Ok, & Bottom.

An "Ok" rating is more diplomatic than a "D".

Jameson--

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

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.

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
>
>
>
>
>
>
>
>
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