[EM] The Quintessential UD Generalized Median Method?

[Forest Simmons]

Let's look at a Chicken example:

49 C
26 A>B
25 B (sincere B>C)

The pairwise matrix M is

[[0, 26, 26]
[25,0,51
][49,49,0]]

The random ballot favorite row vector V  is
[[26, 25, 49]]

The matrix product VM is

[[25^2+49^2,26^2+49^2,26^2+25×51]]

Evidently the third component of this vector is the smallest so
argmin S(X) is candidate C.... the B faction Chicken ploy failed.

Now, let's look at ...

40 A
10 A=C
10 B=C
40 B

We have M equal to
[[0,50,40]
[50,0,40]
[10,10,0]]

The random ballot favorite row vector is
[[45,45,10]]

The matrix product VM is the row vector
[[45*50+10^2,45*50+10^2,45^2+45^2]]

Evidently, argmin S(X) is {A,B}, so A and B are tied for the win.

Our method MEPO seems to respect Plurality, unlike MMPO.

More and more, I like the idea of MEPO with a Landau afterburner. Stitching
on the afterburner leaves a slight seam in the rare cases it needs to be
applied to achieve Landau efficiency. [None of our examples have needed it,
so far.]

-Forest

On Thu, Oct 13, 2022, 9:14 AM Forest Simmons <forest.simmons21 at gmail.com>
wrote:

> 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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