[EM] Fwd: New Method?
Forest Simmons
fsimmons at pcc.edu
Sun Dec 27 22:06:57 PST 2020
Excuse the lack of examples; soon I will have an actual computer keyboard
for typing messages (and more) thanks to a generous gift from a big hearted
Santa Claus helper🧙♂️🎅🤶
... so soon you will see illustrative examples galore!
-------- Mensaje original --------
De: Forest Simmons <fsimmons at pcc.edu>
Fecha: 27/12/20 9:29 p. m. (GMT-08:00)
A:
CC: Kevin Venzke <stepjak at yahoo.fr>
Asunto: Fwd: New Method?
Below I have defined the directed distance d(x, y) from x to y as the
length of the shortest directed path from x to y. (where length of a
directed path is the sum of the weights of each step, and in this context
the weight of a directed edge is the number of losing votes, i.e.
resistance against the defeat represented by the edge).
A basic (directed distance) closed neighborhood centered at x, of radius r,
denoted N(x, r), is ...
the set of alternatives y such that d(x, y) is no greater than r.
New Method:
Elect the alternative z for which the set undefeatedby(z) is contained in
the smallest radius neighborhood possible.
The set undefeatedby(z) is the set of all alternatives that are not
defeated by z, i.e. the alternatives that pairwise beat or tie z.
For each set H, let rad(H) = min{r | for some alternative t, the basic
neighborhood N(t, r) contains H}.
So we can restate this method thusly:
Elect from the set W(x) = argmin(over x)
of rad( undefeatedby(x)).
Note that if z1 covers z2,
then rad(undefeated(z1)) is no greater than
rad(undefeated(z2)),
So there is always an uncovered alternative in W(x).
In particular, if W(x) is a singleton (the generic case), then the unique,
un-tied winner is uncovered, i.e. a Landau alternative.
It will be interesting to compare this method with the better known
Beatpath description of CSSD.
---------- Forwarded message -------
Anybody Seen This Before?
On Sunday, December 27, 2020, Forest Simmons <fsimmons at pcc.edu> wrote:
> Start by building a directed graph whose vertices are the alternatives
> under consideration, and edges are directed from winners to losers.
> Pairwise ties are represented by double arrows. The edges are weighted with
> the losing or tied votes.
>
> The length of a directed path is the sum of the weights of the traversed
> edges.
>
> Define the directed distance d(x,y) from alternative x to alternative y as
> the length of the shortest directed path from x to y, if there is one,
> else infinity.
>
> The radius R(x, S) from x of a subset S of alternatives is the max (over y
> in S) of d(x, y). When S is the entire set of alternatives we abbreviate
> its radius from x as R(x).
>
> Let T(x) be the number of ballots that do not rank x.
>
> The Least Resistance Winner is the candidate that minimizes the sum T(x) +
> R(x).
>
> Remark: for any uncovered candidate this sum is less than twice the number
> of ballots.
>
> Comment: the truncation term T(x) in the sum is there to ensure that the
> method satisfies the Plurality Criterion.
>
>
>
>
>
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