[EM] Steepest Ascent Tweak

[Forest Simmons]

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 at 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
>
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