FS
Forest Simmons
Thu, Oct 27, 2016 9:56 PM
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
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
JQ
Jameson Quinn
Thu, Oct 27, 2016 11:01 PM
I do see that this is better than Bucklin, as it is more robust to burial.
However, all your scenario-building still relies on relatively
high-information voters. Specifically, voters have to know which CD faction
is the honest winner. Without that knowledge, the larger faction will (for
safety) have to give enough cooperation to the larger to enable a betrayal.
Is there any way to fix this? There might be... I've had some thoughts, and
none of them has worked yet, but I'm not convinced that none will.
Still, I think that the majority score (formerly known as SARA) solution to
the CD is better. This is to eliminate candidate C (the minority threat)
because it is rejected by a majority, and then compare candidates A and B
(the subfactions) using some measure in which minimal cooperation cannot be
distinguished from rejection.
(One could make majority score voting more robust to rejection by one CD
subfaction, by doing eliminations in order from most- to least-rejected,
and stopping before the final elimination. But I think this "robustness" is
merely an illusion, because it would by making betrayal strategy safer, it
could make it more common.)
Note that majority score has an additional safeguard against CD betrayal: a
faction smaller than 25% cannot possibly win through such a betrayal. This
means that some (exploitable) cooperation may needed in case of CD scenario
where the majority is split more than two ways; but I think that's
tolerable.
How often will there be a CD faction that is considering offensive strategy
(that is, feels it might be smaller) yet is confident that it is more than
25%? I think that if the margin of error in polling is in the neighborhood
of 4%, then such a faction would need to be at least 29%, and the opposing
faction would need to be around 33%, leaving just 38% for the minority
threat; and from 38% down to a non-threatening 33% is not that big a gap.
So I think such situations would be rare and unstable; not worth the
trouble of organizing strategy around.
I do see the allure of XA; it has the "no zero-information exaggeration
incentive" property that Bucklin has, with this additional CD resistance
that Forest points out. But I still think that majority score is better.
(Of course, I also still think that SODA has the best CD resistance of any
well-defined single-winner method I know. But SODA requires participation
by the candidates, which is not in all cases possible; and I think its
strangeness makes it overall tougher to sell than something like majority
score.)
2016-10-27 17:56 GMT-04:00 Forest Simmons fsimmons@pcc.edu:
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
Election-Methods mailing list - see http://electorama.com/em for list info
I do see that this is better than Bucklin, as it is more robust to burial.
However, all your scenario-building still relies on relatively
high-information voters. Specifically, voters have to know which CD faction
is the honest winner. Without that knowledge, the larger faction will (for
safety) have to give enough cooperation to the larger to enable a betrayal.
Is there any way to fix this? There might be... I've had some thoughts, and
none of them has worked yet, but I'm not convinced that none will.
Still, I think that the majority score (formerly known as SARA) solution to
the CD is better. This is to eliminate candidate C (the minority threat)
because it is rejected by a majority, and then compare candidates A and B
(the subfactions) using some measure in which minimal cooperation cannot be
distinguished from rejection.
(One could make majority score voting more robust to rejection by one CD
subfaction, by doing eliminations in order from most- to least-rejected,
and stopping before the final elimination. But I think this "robustness" is
merely an illusion, because it would by making betrayal strategy safer, it
could make it more common.)
Note that majority score has an additional safeguard against CD betrayal: a
faction smaller than 25% cannot possibly win through such a betrayal. This
means that some (exploitable) cooperation may needed in case of CD scenario
where the majority is split more than two ways; but I think that's
tolerable.
How often will there be a CD faction that is considering offensive strategy
(that is, feels it might be smaller) yet is confident that it is more than
25%? I think that if the margin of error in polling is in the neighborhood
of 4%, then such a faction would need to be at least 29%, and the opposing
faction would need to be around 33%, leaving just 38% for the minority
threat; and from 38% down to a non-threatening 33% is not that big a gap.
So I think such situations would be rare and unstable; not worth the
trouble of organizing strategy around.
I do see the allure of XA; it has the "no zero-information exaggeration
incentive" property that Bucklin has, with this additional CD resistance
that Forest points out. But I still think that majority score is better.
(Of course, I also still think that SODA has the best CD resistance of any
well-defined single-winner method I know. But SODA requires participation
by the candidates, which is not in all cases possible; and I think its
strangeness makes it overall tougher to sell than something like majority
score.)
2016-10-27 17:56 GMT-04:00 Forest Simmons <fsimmons@pcc.edu>:
> 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
>
>
>
>
>
>
>
>
> ----
> Election-Methods mailing list - see http://electorama.com/em for list info
>
>
MO
Michael Ossipoff
Fri, Oct 28, 2016 12:44 AM
Interesting.
Being rating-methods, both of these 2 methods meet FBC, don't they?
That's neat, four-slot XA, with ratings of 0, 1/3, 1/2, & 1.
That has a lot of appeal, and the ratings give it a kind of simplicity.
I've just now found the postings. I've only just read them, preliminarily,
so this is just a quick preliminary reply.
Both methods, of course, would be a bit difficult to get most people to
accept the definition of, to get someone to sit still for the entire
definition.
XA is remarkable, in how it uses the rating information in a completely
different way that, I guess, had never been considered until Jennings
suggested it. But that great unfamiliarity won't help its acceptance.
That doesn't mean that offering it isn't possible, but, as you said, there
will be the task of finding the easiest explanation and way of wording the
definition.
I was wondering before, if XA's unique & unusual way of using the
information could result in its having especially desirable properties that
are otherwise difficult to attain, and evidently it does.
SARA, like 4-slot XA, has 4 ratings, but they're qualitatively different
ratings, each with a different kind of meaning. That's a complication for
the public, knowing which of the four completely separate things to do.
Without chicken dilemma, someone could just use the 4 XA ratings as sincere
grades if they wanted to. I'd guess that, with any rating method, including
those, the best strategy is to top-rate one's top-set.
The difference and the problem is what to do in a chicken dilemma
situation. No doubt they both make that sitluation easier than Approval
would.
Yes, continuing examination is needed, but it's promising. Maybe MMPO has a
rival or two.
Michael Ossipoff
On Thu, Oct 27, 2016 at 7:01 PM, Jameson Quinn jameson.quinn@gmail.com
wrote:
I do see that this is better than Bucklin, as it is more robust to burial.
However, all your scenario-building still relies on relatively
high-information voters. Specifically, voters have to know which CD faction
is the honest winner. Without that knowledge, the larger faction will (for
safety) have to give enough cooperation to the larger to enable a betrayal.
Is there any way to fix this? There might be... I've had some thoughts,
and none of them has worked yet, but I'm not convinced that none will.
Still, I think that the majority score (formerly known as SARA) solution
to the CD is better. This is to eliminate candidate C (the minority threat)
because it is rejected by a majority, and then compare candidates A and B
(the subfactions) using some measure in which minimal cooperation cannot be
distinguished from rejection.
(One could make majority score voting more robust to rejection by one CD
subfaction, by doing eliminations in order from most- to least-rejected,
and stopping before the final elimination. But I think this "robustness" is
merely an illusion, because it would by making betrayal strategy safer, it
could make it more common.)
Note that majority score has an additional safeguard against CD betrayal:
a faction smaller than 25% cannot possibly win through such a betrayal.
This means that some (exploitable) cooperation may needed in case of CD
scenario where the majority is split more than two ways; but I think that's
tolerable.
How often will there be a CD faction that is considering offensive
strategy (that is, feels it might be smaller) yet is confident that it is
more than 25%? I think that if the margin of error in polling is in the
neighborhood of 4%, then such a faction would need to be at least 29%, and
the opposing faction would need to be around 33%, leaving just 38% for the
minority threat; and from 38% down to a non-threatening 33% is not that big
a gap. So I think such situations would be rare and unstable; not worth the
trouble of organizing strategy around.
I do see the allure of XA; it has the "no zero-information exaggeration
incentive" property that Bucklin has, with this additional CD resistance
that Forest points out. But I still think that majority score is better.
(Of course, I also still think that SODA has the best CD resistance of any
well-defined single-winner method I know. But SODA requires participation
by the candidates, which is not in all cases possible; and I think its
strangeness makes it overall tougher to sell than something like majority
score.)
2016-10-27 17:56 GMT-04:00 Forest Simmons fsimmons@pcc.edu:
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
Election-Methods mailing list - see http://electorama.com/em for list
info
Interesting.
Being rating-methods, both of these 2 methods meet FBC, don't they?
That's neat, four-slot XA, with ratings of 0, 1/3, 1/2, & 1.
That has a lot of appeal, and the ratings give it a kind of simplicity.
I've just now found the postings. I've only just read them, preliminarily,
so this is just a quick preliminary reply.
Both methods, of course, would be a bit difficult to get most people to
accept the definition of, to get someone to sit still for the entire
definition.
XA is remarkable, in how it uses the rating information in a completely
different way that, I guess, had never been considered until Jennings
suggested it. But that great unfamiliarity won't help its acceptance.
That doesn't mean that offering it isn't possible, but, as you said, there
will be the task of finding the easiest explanation and way of wording the
definition.
I was wondering before, if XA's unique & unusual way of using the
information could result in its having especially desirable properties that
are otherwise difficult to attain, and evidently it does.
SARA, like 4-slot XA, has 4 ratings, but they're _qualitatively_ different
ratings, each with a different kind of meaning. That's a complication for
the public, knowing which of the four completely separate things to do.
Without chicken dilemma, someone could just use the 4 XA ratings as sincere
grades if they wanted to. I'd guess that, with any rating method, including
those, the best strategy is to top-rate one's top-set.
The difference and the problem is what to do in a chicken dilemma
situation. No doubt they both make that sitluation easier than Approval
would.
Yes, continuing examination is needed, but it's promising. Maybe MMPO has a
rival or two.
Michael Ossipoff
On Thu, Oct 27, 2016 at 7:01 PM, Jameson Quinn <jameson.quinn@gmail.com>
wrote:
> I do see that this is better than Bucklin, as it is more robust to burial.
> However, all your scenario-building still relies on relatively
> high-information voters. Specifically, voters have to know which CD faction
> is the honest winner. Without that knowledge, the larger faction will (for
> safety) have to give enough cooperation to the larger to enable a betrayal.
>
> Is there any way to fix this? There might be... I've had some thoughts,
> and none of them has worked yet, but I'm not convinced that none will.
>
> Still, I think that the majority score (formerly known as SARA) solution
> to the CD is better. This is to eliminate candidate C (the minority threat)
> because it is rejected by a majority, and then compare candidates A and B
> (the subfactions) using some measure in which minimal cooperation cannot be
> distinguished from rejection.
>
> (One could make majority score voting more robust to rejection by one CD
> subfaction, by doing eliminations in order from most- to least-rejected,
> and stopping before the final elimination. But I think this "robustness" is
> merely an illusion, because it would by making betrayal strategy safer, it
> could make it more common.)
>
> Note that majority score has an additional safeguard against CD betrayal:
> a faction smaller than 25% cannot possibly win through such a betrayal.
> This means that some (exploitable) cooperation may needed in case of CD
> scenario where the majority is split more than two ways; but I think that's
> tolerable.
>
> How often will there be a CD faction that is considering offensive
> strategy (that is, feels it might be smaller) yet is confident that it is
> more than 25%? I think that if the margin of error in polling is in the
> neighborhood of 4%, then such a faction would need to be at least 29%, and
> the opposing faction would need to be around 33%, leaving just 38% for the
> minority threat; and from 38% down to a non-threatening 33% is not that big
> a gap. So I think such situations would be rare and unstable; not worth the
> trouble of organizing strategy around.
>
> I do see the allure of XA; it has the "no zero-information exaggeration
> incentive" property that Bucklin has, with this additional CD resistance
> that Forest points out. But I still think that majority score is better.
>
> (Of course, I also still think that SODA has the best CD resistance of any
> well-defined single-winner method I know. But SODA requires participation
> by the candidates, which is not in all cases possible; and I think its
> strangeness makes it overall tougher to sell than something like majority
> score.)
>
> 2016-10-27 17:56 GMT-04:00 Forest Simmons <fsimmons@pcc.edu>:
>
>> 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
>>
>>
>>
>>
>>
>>
>>
>>
>> ----
>> Election-Methods mailing list - see http://electorama.com/em for list
>> info
>>
>>
>
MO
Michael Ossipoff
Fri, Oct 28, 2016 2:47 AM
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@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
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@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
>
>
>
>
>
>
>
>
JQ
Jameson Quinn
Fri, Oct 28, 2016 4:01 AM
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@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
>
>
> 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@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
>>
>>
>>
>>
>>
>>
>>
>>
>
TP
Toby Pereira
Fri, Oct 28, 2016 3:56 PM
I think Chiastic Approval would fail participation, assuming I've done this right. Take the following ballots with scores out of 100:
2 voters: A=50, B=401 voter: A=50, B=60
A would have a score of 50. B would have a score of 40. Everyone gives B a score of at least 40, and only a third give B a score higher. Now imagine there are two extra voters and we have these ballots:
2 voters: A=50, B=401 voter: A=50, B=602 voters: A=100, B=60
A still has a score of 50, but B now has a score of 60. So these two ballots cause B to overtake A despite them both preferring A to B.
From: Forest Simmons <fsimmons@pcc.edu>
Does it satisfy Participation?
I think Chiastic Approval would fail participation, assuming I've done this right. Take the following ballots with scores out of 100:
2 voters: A=50, B=401 voter: A=50, B=60
A would have a score of 50. B would have a score of 40. Everyone gives B a score of at least 40, and only a third give B a score higher. Now imagine there are two extra voters and we have these ballots:
2 voters: A=50, B=401 voter: A=50, B=602 voters: A=100, B=60
A still has a score of 50, but B now has a score of 60. So these two ballots cause B to overtake A despite them both preferring A to B.
From: Forest Simmons <fsimmons@pcc.edu>
Does it satisfy Participation?
FS
Forest Simmons
Fri, Oct 28, 2016 10:32 PM
It looks like we'll have to settle for mono-add-plump.
On Fri, Oct 28, 2016 at 8:56 AM, Toby Pereira tdp201b@yahoo.co.uk wrote:
I think Chiastic Approval would fail participation, assuming I've done
this right. Take the following ballots with scores out of 100:
2 voters: A=50, B=40
1 voter: A=50, B=60
A would have a score of 50. B would have a score of 40.
Everyone gives B a score of at least 40, and only a third give B a score
higher. Now imagine there are two extra voters and we have these ballots:
2 voters: A=50, B=40
1 voter: A=50, B=60
2 voters: A=100, B=60
A still has a score of 50, but B now has a score of 60. So these two
ballots cause B to overtake A despite them both preferring A to B.
From: Forest Simmons fsimmons@pcc.edu
Does it satisfy Participation?
It looks like we'll have to settle for mono-add-plump.
On Fri, Oct 28, 2016 at 8:56 AM, Toby Pereira <tdp201b@yahoo.co.uk> wrote:
> I think Chiastic Approval would fail participation, assuming I've done
> this right. Take the following ballots with scores out of 100:
>
> 2 voters: A=50, B=40
> 1 voter: A=50, B=60
>
> A would have a score of 50. B would have a score of 40.
>
Everyone gives B a score of at least 40, and only a third give B a score
> higher. Now imagine there are two extra voters and we have these ballots:
>
> 2 voters: A=50, B=40
> 1 voter: A=50, B=60
> 2 voters: A=100, B=60
>
> A still has a score of 50, but B now has a score of 60. So these two
> ballots cause B to overtake A despite them both preferring A to B.
>
> ------------------------------
> *From:* Forest Simmons <fsimmons@pcc.edu>
>
>
> Does it satisfy Participation?
>
>
>
FS
Forest Simmons
Fri, Oct 28, 2016 10:36 PM
It occurred to me that, to go with 0, 1/3, 1/2, & 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".
Yes, I was worried about the psychological effect of a "D" grade option. I
like your suggestion much better.
On Thu, Oct 27, 2016 at 7:47 PM, Michael Ossipoff <email9648742@gmail.com>
wrote:
> It occurred to me that, to go with 0, 1/3, 1/2, & 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".
>
Yes, I was worried about the psychological effect of a "D" grade option. I
like your suggestion much better.
MO
Michael Ossipoff
Sat, Oct 29, 2016 9:19 PM
So XA fails Participation. So what? Participation is just one of those
"could-look-bad" criteria, like the Plurality Criterion.
We all know that if you want anything that Approval & Score don't offer,
then you have to pay for it, with "could-look-bad" criterion-failures.
And the more you want, the more "could-look-bad" you have to accept.
Plain MMPO is tops in important strategy criteria and properties, and so,
it has the most could-look-bad.
...which has been shown to not significantly wrong anyone or be a genuine
problem.
As I've been saying, improvements on Approval are mostly or entirely
illusory.
wv strategy could be some strategy improvement, but it's a bit
questionable. But, in addition to FBC & wv strategy, Plain MMPO offers full
reliable Weak CD compliance. (Its failure of strong CD is mitigated by its
wv strategy's burial-deterrence.).
It's difficult to improve on Approval, but, if any method can, in some
meaningful way, Plain MMPO is the method that best qualifies for that claim.
You get what you pay for.
Michael Ossipoff
On Fri, Oct 28, 2016 at 6:32 PM, Forest Simmons fsimmons@pcc.edu wrote:
It looks like we'll have to settle for mono-add-plump.
On Fri, Oct 28, 2016 at 8:56 AM, Toby Pereira tdp201b@yahoo.co.uk wrote:
I think Chiastic Approval would fail participation, assuming I've done
this right. Take the following ballots with scores out of 100:
2 voters: A=50, B=40
1 voter: A=50, B=60
A would have a score of 50. B would have a score of 40.
Everyone gives B a score of at least 40, and only a third give B a score
higher. Now imagine there are two extra voters and we have these ballots:
2 voters: A=50, B=40
1 voter: A=50, B=60
2 voters: A=100, B=60
A still has a score of 50, but B now has a score of 60. So these two
ballots cause B to overtake A despite them both preferring A to B.
From: Forest Simmons fsimmons@pcc.edu
Does it satisfy Participation?
So XA fails Participation. So what? Participation is just one of those
"could-look-bad" criteria, like the Plurality Criterion.
We all know that if you want anything that Approval & Score don't offer,
then you have to pay for it, with "could-look-bad" criterion-failures.
And the more you want, the more "could-look-bad" you have to accept.
Plain MMPO is tops in important strategy criteria and properties, and so,
it has the most could-look-bad.
...which has been shown to not significantly wrong anyone or be a genuine
problem.
As I've been saying, improvements on Approval are mostly or entirely
illusory.
wv strategy could be some strategy improvement, but it's a bit
questionable. But, in addition to FBC & wv strategy, Plain MMPO offers full
reliable Weak CD compliance. (Its failure of strong CD is mitigated by its
wv strategy's burial-deterrence.).
It's difficult to improve on Approval, but, if any method can, in some
meaningful way, Plain MMPO is the method that best qualifies for that claim.
You get what you pay for.
Michael Ossipoff
On Fri, Oct 28, 2016 at 6:32 PM, Forest Simmons <fsimmons@pcc.edu> wrote:
> It looks like we'll have to settle for mono-add-plump.
>
>
> On Fri, Oct 28, 2016 at 8:56 AM, Toby Pereira <tdp201b@yahoo.co.uk> wrote:
>
>> I think Chiastic Approval would fail participation, assuming I've done
>> this right. Take the following ballots with scores out of 100:
>>
>> 2 voters: A=50, B=40
>> 1 voter: A=50, B=60
>>
>> A would have a score of 50. B would have a score of 40.
>>
> Everyone gives B a score of at least 40, and only a third give B a score
>> higher. Now imagine there are two extra voters and we have these ballots:
>>
>> 2 voters: A=50, B=40
>> 1 voter: A=50, B=60
>> 2 voters: A=100, B=60
>>
>> A still has a score of 50, but B now has a score of 60. So these two
>> ballots cause B to overtake A despite them both preferring A to B.
>>
>> ------------------------------
>> *From:* Forest Simmons <fsimmons@pcc.edu>
>>
>>
>> Does it satisfy Participation?
>>
>>
>>
>
MO
Michael Ossipoff
Sun, Oct 30, 2016 12:09 AM
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@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
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@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
>
>
>
>
>
>
>
>
