Steepest Ascent Tweak
A few months ago I suggested an "afterburner" finisher that could enhance
any election method (when retrofitted with the afterburner) by making the
tweaked method Landau efficient.
A few days ago I repeated the suggestion in response to a pitiful tweak
(warmed over Baldwin) publicly proposed by Foley and Maskin in an op ed
response to the Alaska RCV/IRV debacle, a repeat of the Burlington Vermont
IRV fiasco that will become an endemic plague if the tide of RCV/IRV
adoptions is not stemmed.
In explaining the amazing universal applicability and robust effectiveness
of the simple afterburner tweak, I compared it to the method of steepest
ascent ... a well known general purpose method for optimization of
multivariate functions.
For that reason I would like to change the name of the procedure from
"afterburnner" to "steepest ascent tweak".
I want to elaborate my brief geometric explanation of the steepest ascent
technique in the voting context. The best visualization is a "Yee/Bolson"
[Bolson for Brian Olson who first reported his and Yee's discovery to the
EM listserv] diagram that shows candidate positions relative to a smooth
multivariate Gaussian distribution of the voter positions. The Cartesian
graph of a Gaussian distribution density function in three dimensions
(x,y,z) is a smooth hill with the top of the hill hovering above the mean,
median, and mode point MMM of the distribution in the x/y plane.
If the third dimension z is suppressed, the circles centered on the
pointMMM in the (x, y, 0) plane, will form a contour map of the
distribution.
At every point p=(x,y,0) in the x/y plane, the gradient grad(x y) is an
arrow perpendicular to the contour curve passing through p, pointed towards
the center MMM point. That is the direction of max increase of the density
function (the Gaussian in our example). If that arrow is lifted and rotated
vertically until it is tangent to the surface of the 3D graph, then it will
point in the (local) direction of steepest ascent up the hill (surface).
In the Yee/Bolson diagram, a candidate at point C will cover a candidate at
point W iff C is closer to the MMM point than W is.
And of all the possible candidate points on a disc D centered on MMM while
excluding W, the candidate point C closest to W is the one to which W has
the least pairwise opposition.
To see this, let PBWC be the perpendicular bisecteor of the segment WC.
The total density of the points on the W side of the bisector is the
pairwise opposition preferring W to C.
Holding W fixed while varying C inside the disc D, we see that when C is
closest to W, directly between W and MMM, the total density of points
(votes) supporting W over C will be minimized.
The directed segment from W to C is parallel to the gradient at p, which is
the direction pointing to MMM.
Of course, the distribution of voters will not, in general be a nice
symmetrical Gaussian, and there may not be any candidate C directly between
W and MMM. In fact, Mean. Median, and Mode may be three different points.
But the geometric understandings of covering, and of minimum pairwise
opposition are still valid, and help explain why the afterburner tweak
works so effectively, and helps us appreciate the new moniker "steepest
ascent tweak."
The step from W to C is one step of the tweak. If the endpoint C of the
step is covered, the step is repeated ... starting over with the variable W
updated as the output of the previous step.
This stepwise process is continued until an uncovered (i.e. Landau)
candidate is reached.
How do we know this will happen?
Because this process cannot cycle. Why? Because the covering relation is
transitive, unlike the (mere) defeat relation. If C2 covers C1=W2, and C1
covers W , then C2 also covers W. Each successive C covers all of the
previous ones, so the process cannot proceed indefinitely unless there are
infinitely many candidates.
That's the foundation for the application of this steepest ascent tweak ...
one of which (the afterburner enhancement) we have already advertised.
We will explore other equally interesting applications in subsequent
messages of this thread.
To be continued ....
-Forest
In this installment I want to give an instructive example of the corrective
power of the steepest ascent tweak before going on to some of its uses in
election method design.
Consider this ballot profile in conjunction with the simple MinMax method:
30 C1>C2>C3
30 C2>C3>C1
30 A>C3>C1
6 A>C1>C2
4 A>C2>C3
Minimax elects A, the Condorcet Loser, because its max pairwise defeat is
only 60 to 40, whereas each of the C's is defeated by a greater majority in
favor of some other C ... a case of Clone Winner failure.
Does this bad result daunt the steepest ascent tweak?
Let's see. ... the tweak starts by setting the variable W to the value A,
the output of the flawed method.
Let's check to see if W is covered: yes, in fact each of the three C clones
(cyclones?) covers A, because no beatpath goes from A to any other
candidate, let alone a short beatpath to C1, C2, or C3.
Of these, which is the C to which W=A offers the least opposition? Well,
they are tied in this regard, because the common opposition is the forty
percent of voters that prefer W=A over any of the clones.
How do we break the tie?
We go with the tied candidate that has the most core support ... the most
losing votes in its worst pairwise failure.
C1 defeats C2 66 to 34
C2 defeats C3 66 to 34
C3 defeats C1 64 to 36
C1's core support is 36 ... which is greater than 34, the vore support of
each of the other tied candidates.
So C1 is the winner as corrected by the steepest ascent tweak.
Note that if the non clone A were out of the picture C1 would have won, so
at least in this example, this tweak has done a good job of restoring the
rightful clone to the throne, so to speak.
-Forest
Next time ... steepest ascent as an organic part of decloned MinMax, not
just an afterthought tweak to cleanup the mess made by regular clone
infested MinMax ...
On Fri, Nov 4, 2022, 10:05 PM Forest Simmons forest.simmons21@gmail.com
wrote:
A few months ago I suggested an "afterburner" finisher that could enhance
any election method (when retrofitted with the afterburner) by making the
tweaked method Landau efficient.
A few days ago I repeated the suggestion in response to a pitiful tweak
(warmed over Baldwin) publicly proposed by Foley and Maskin in an op ed
response to the Alaska RCV/IRV debacle, a repeat of the Burlington Vermont
IRV fiasco that will become an endemic plague if the tide of RCV/IRV
adoptions is not stemmed.
In explaining the amazing universal applicability and robust effectiveness
of the simple afterburner tweak, I compared it to the method of steepest
ascent ... a well known general purpose method for optimization of
multivariate functions.
For that reason I would like to change the name of the procedure from
"afterburnner" to "steepest ascent tweak".
I want to elaborate my brief geometric explanation of the steepest ascent
technique in the voting context. The best visualization is a "Yee/Bolson"
[Bolson for Brian Olson who first reported his and Yee's discovery to the
EM listserv] diagram that shows candidate positions relative to a smooth
multivariate Gaussian distribution of the voter positions. The Cartesian
graph of a Gaussian distribution density function in three dimensions
(x,y,z) is a smooth hill with the top of the hill hovering above the mean,
median, and mode point MMM of the distribution in the x/y plane.
If the third dimension z is suppressed, the circles centered on the
pointMMM in the (x, y, 0) plane, will form a contour map of the
distribution.
At every point p=(x,y,0) in the x/y plane, the gradient grad(x y) is an
arrow perpendicular to the contour curve passing through p, pointed towards
the center MMM point. That is the direction of max increase of the density
function (the Gaussian in our example). If that arrow is lifted and rotated
vertically until it is tangent to the surface of the 3D graph, then it will
point in the (local) direction of steepest ascent up the hill (surface).
In the Yee/Bolson diagram, a candidate at point C will cover a candidate
at point W iff C is closer to the MMM point than W is.
And of all the possible candidate points on a disc D centered on MMM while
excluding W, the candidate point C closest to W is the one to which W has
the least pairwise opposition.
To see this, let PBWC be the perpendicular bisecteor of the segment WC.
The total density of the points on the W side of the bisector is the
pairwise opposition preferring W to C.
Holding W fixed while varying C inside the disc D, we see that when C is
closest to W, directly between W and MMM, the total density of points
(votes) supporting W over C will be minimized.
The directed segment from W to C is parallel to the gradient at p, which
is the direction pointing to MMM.
Of course, the distribution of voters will not, in general be a nice
symmetrical Gaussian, and there may not be any candidate C directly between
W and MMM. In fact, Mean. Median, and Mode may be three different points.
But the geometric understandings of covering, and of minimum pairwise
opposition are still valid, and help explain why the afterburner tweak
works so effectively, and helps us appreciate the new moniker "steepest
ascent tweak."
The step from W to C is one step of the tweak. If the endpoint C of the
step is covered, the step is repeated ... starting over with the variable W
updated as the output of the previous step.
This stepwise process is continued until an uncovered (i.e. Landau)
candidate is reached.
How do we know this will happen?
Because this process cannot cycle. Why? Because the covering relation is
transitive, unlike the (mere) defeat relation. If C2 covers C1=W2, and C1
covers W , then C2 also covers W. Each successive C covers all of the
previous ones, so the process cannot proceed indefinitely unless there are
infinitely many candidates.
That's the foundation for the application of this steepest ascent tweak
... one of which (the afterburner enhancement) we have already advertised.
We will explore other equally interesting applications in subsequent
messages of this thread.
To be continued ....
-Forest
Let satlp(X) be the probability that the top ranked candidate on a randomly
drawn ballot gets sent to X by the steepest ascent tweak.
In other words satlp(X) is the steepest ascent tweak lottery probability of
choosing candidate X.
This lottery probability function on the set of candidates is the basis of
the following steepest ascent election method (SAEM):
Elect argmin(SA(Y)), where SA(Y) is ...
Sum{satlp(X)|X defeats Y pairwise}
This steepest ascent based method is an improvement on the friendly voting
method that uses the random ballot favorite lottery probabilities instead
of the steepest ascent tweak lottery probabilities ... an improvement
because it avoids the problem that arises when most of the first place
candidates are weak fantasy candidates that defeat few or none of the Smith
candidates.
The first place lottery is an example of a proportional lottery, which has
it support scattered all over the candidate space including on the outer
fringes. But the viable candidates are the ones nearer the center, so the
proportional lotteries encourage compromising strategy ... voters cannot
safely vote their favorites ahead of their viable compromises.
That's where the steepest ascent tweak comes in. It moves the support of
proportional lotteries inward... giving this method a built in DSV
(Designated Strategy Voting) feature that takes the burden of compromise
strategy off of the voters.
The method is also resistant to burial strategy, because of its close
relationship with Friendly Voting ... specifically designed to be burial
resistant.
Note also that SAEM is a one pass, precinct summable method.
It can be thought of as the natural result of decloning Minimax, by
replacing Max with Sum, and then replacing the defeat margins with
corresponding lottery probabilities.
Continuing our example from last time will clear up these cryptic comments
...
To Be Continued ...
On Fri, Nov 4, 2022, 10:05 PM Forest Simmons forest.simmons21@gmail.com
wrote:
A few months ago I suggested an "afterburner" finisher that could enhance
any election method (when retrofitted with the afterburner) by making the
tweaked method Landau efficient.
A few days ago I repeated the suggestion in response to a pitiful tweak
(warmed over Baldwin) publicly proposed by Foley and Maskin in an op ed
response to the Alaska RCV/IRV debacle, a repeat of the Burlington Vermont
IRV fiasco that will become an endemic plague if the tide of RCV/IRV
adoptions is not stemmed.
In explaining the amazing universal applicability and robust effectiveness
of the simple afterburner tweak, I compared it to the method of steepest
ascent ... a well known general purpose method for optimization of
multivariate functions.
For that reason I would like to change the name of the procedure from
"afterburnner" to "steepest ascent tweak".
I want to elaborate my brief geometric explanation of the steepest ascent
technique in the voting context. The best visualization is a "Yee/Bolson"
[Bolson for Brian Olson who first reported his and Yee's discovery to the
EM listserv] diagram that shows candidate positions relative to a smooth
multivariate Gaussian distribution of the voter positions. The Cartesian
graph of a Gaussian distribution density function in three dimensions
(x,y,z) is a smooth hill with the top of the hill hovering above the mean,
median, and mode point MMM of the distribution in the x/y plane.
If the third dimension z is suppressed, the circles centered on the
pointMMM in the (x, y, 0) plane, will form a contour map of the
distribution.
At every point p=(x,y,0) in the x/y plane, the gradient grad(x y) is an
arrow perpendicular to the contour curve passing through p, pointed towards
the center MMM point. That is the direction of max increase of the density
function (the Gaussian in our example). If that arrow is lifted and rotated
vertically until it is tangent to the surface of the 3D graph, then it will
point in the (local) direction of steepest ascent up the hill (surface).
In the Yee/Bolson diagram, a candidate at point C will cover a candidate
at point W iff C is closer to the MMM point than W is.
And of all the possible candidate points on a disc D centered on MMM while
excluding W, the candidate point C closest to W is the one to which W has
the least pairwise opposition.
To see this, let PBWC be the perpendicular bisecteor of the segment WC.
The total density of the points on the W side of the bisector is the
pairwise opposition preferring W to C.
Holding W fixed while varying C inside the disc D, we see that when C is
closest to W, directly between W and MMM, the total density of points
(votes) supporting W over C will be minimized.
The directed segment from W to C is parallel to the gradient at p, which
is the direction pointing to MMM.
Of course, the distribution of voters will not, in general be a nice
symmetrical Gaussian, and there may not be any candidate C directly between
W and MMM. In fact, Mean. Median, and Mode may be three different points.
But the geometric understandings of covering, and of minimum pairwise
opposition are still valid, and help explain why the afterburner tweak
works so effectively, and helps us appreciate the new moniker "steepest
ascent tweak."
The step from W to C is one step of the tweak. If the endpoint C of the
step is covered, the step is repeated ... starting over with the variable W
updated as the output of the previous step.
This stepwise process is continued until an uncovered (i.e. Landau)
candidate is reached.
How do we know this will happen?
Because this process cannot cycle. Why? Because the covering relation is
transitive, unlike the (mere) defeat relation. If C2 covers C1=W2, and C1
covers W , then C2 also covers W. Each successive C covers all of the
previous ones, so the process cannot proceed indefinitely unless there are
infinitely many candidates.
That's the foundation for the application of this steepest ascent tweak
... one of which (the afterburner enhancement) we have already advertised.
We will explore other equally interesting applications in subsequent
messages of this thread.
To be continued ....
-Forest