[EM] A Decisive, Landau Efficient, Generalized Median, Single Winner Method

[Forest Simmons]

It is now obvious to me that the simplest, adequate measure of beatpath
cost, and the one that completes the "friendliness" measure in our
longstanding foA-fpC thread, is simply the number of steps in the
beatpath.  The fewer the number of steps in the beatpath from X to Y, the
friendlier Y is to X.

Time to summarize the method that has culminated in this thread. I'm
tempted to call it the Munsterhjelm Venzke Heitzig or MVH method after the
principal contributors of the key ideas. Many other EM List members from
(Demorep to Lung) have contributed in ways too numerous to enumerate.  I
can truly say that anybody who made regular contributions to the EM List in
the form of questions, criticisms, or clarifications over any period of a
year or more has contributed to my understanding of voting methods ... for
which I am very grateful. Special mention to Benham, Ossipoff, Rob LeGrand,
Andrew Jennings, Joe Weinstein, Bart Ingles, etc. for tolerating my newbie
efforts.

HVM: Elect the candidate X with the smallest total of beatpath costs from X
to the ballot favorites.

In other words, for each ballot B, and each candidate X, let Cost(X,fpB) be
the minimum number of steps in a beatpath from X to fpB, the first place
favorite of ballot B. Note that if X=fpB, then the beatpath from X to F(B
has zero steps, so the cost is zero.
Let SumfpB(X) be the sum over all ballots B of Cost(X,fpB).
Elect argmin SumfpB(X)

It is not hard to show that this simple, decisive, Universal Domain
election method is clone independent, Landau efficient, and monotone.

If these claims hold up to close scrutiny, and MVH truly does generalize
fpA-fpC, then why not propose it for public elections?

-Forest

On Fri, Oct 7, 2022, 6:54 PM Forest Simmons <forest.simmons21 at gmail.com>
wrote:

> The median X of a finite set of distinct points arranged along a straight
> line segment will always minimize the sum of distances from it to the other
> points. [If the points are not distinct, a weighted sum does the job.]
>
> Consequently one way to generalize the concept of "median" in a general
> metric space is by minimization of (weighted) sums of distances.
>
> Thus, the Kemeny-Young method chooses the "finish order" that minimizes
> its sum of distances to the ballots, i.e to their respective rank orders.
>
> In this context, the distance from a ballot order to a potential finish
> order is their Kendall-tau distance, the total number of basic order
> reversals necessary to convert one order into the other.
>
> There are two unnecessary difficulties associated with Kemeny-Young:
>  (1) The number of finish orders that need to be checked grows
> exponentially with the number of candidates,  even when all we need is the
> winner of a single winner election.
> (2) The method is clone dependent ... a fatal flaw in the context of
> electoral politics. The basic spoiler problem that sparked election method
> reform in the first place was a failure of clone independence. Even IRV
> with all of its other problems, is clone independent.
>
> The method we propose is both clone independent and computationally
> efficient.
>
> The key innovation is that we gauge the distance from ballot B to a
> potential winner X by the cost of the least expensive beatpath from X to
> the candidate f(B) that is favored above all others on ballot B.
>
> I'm going to break here to let this idea sink in a little before filling
> in the few remaining details.
>
> To be continued...
>
>
>
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