[EM] Steepest Ascent Tweak

[Forest Simmons]

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 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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