[EM] The Quintessential UD Generalized Median Method?

[Forest Simmons]

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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