[EM] The Quintessential UD Generalized Median Method?

[Forest Simmons]

We should call this method MEPO for .Min Expected Pairwise Opposition in
comparison with Min Max Pairwise Opposition MMPO.

The main defect of MMPO is Plurality failure, which cannot afflict MEPO as
long as E is the Random Favorite Lottery Expectation, i.e. the Benchmark
Lottery Expectation.

By the way, when Kevin first posted about MMPO, he based it on the same
geometry that I used to describe MEPO:
As X moves further from Y, the number of ballots that prefer Y over X
increases.

-Forest

On Wed, Oct 12, 2022, 11:29 PM Forest Simmons <forest.simmons21 at gmail.com>
wrote:

> A generalized median voting method  elects the alternative that minimizes
> the total distance to the ballots. But how do we gauge the distance from an
> alternative to a ballot in the Universal Domain context?
>
> One piece of the puzzle is that the position of a ballot B in issue space
> is most simply represented by the position of the most favored alternative
> on that ballot Y=f(B).
>
> So we need a metric d(X,Y), for the distances between the various possible
> positions of the respective alternatives (X), and the fixed positions (Y)
> of the ballot favorites.
>
> The simplest metric I can think of in the Universal Domain context for the
> distance from a moving (i.e. adjusting towards minimality) alternative X to
> a fixed alternative Y, is the number of ballots on which Y is preferred
> over X.
>
> This makes sense, because as X moves directly away from stationary Y, the
> number of ballots on which Y is preferred over X can only increase.
>
> Put these pieces of the puzzle together and we can model the total
> distance from X to the ballots as the sum ..
>
> S(X)=Sum(over Y) of d(X, Y)*f(Y),
>
> where d(X,Y) is the number of ballots on which Y outranks X, and f(Y) is
> the percentage of ballots on which Y is the favorite alternative.
>
> So the purest median method I can come up with in the UD context is to
> elect argmin S(X).
>
> If I am not mistaken, this method satisfies the FBC.
>
> If you prefer that the winner be uncovered, you can trade in the FBC for
> Landau efficiency by attaching a Landau afterburner:
>
> While argmin S(X) is covered, eliminate X, and replace it with the
> remaining alternative closest to X in its value of S among the alternatives
> that cover X.
>
> -Forest
>
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