[EM] Fwd: XA

Forest Simmons fsimmons at pcc.edu
Thu Oct 27 14:56:45 PDT 2016

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

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

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

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.

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