[EM] Steepest Ascent Tweak

[Forest Simmons]

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
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