[EM] Fwd: Paths of Least Resistance

[Forest Simmons]

Here's a method similar to but simpler than Beatpath ... it defines the
length of a beatpath as the sum of the losing votes along the path ... the
total resistance put up by the losing teams along the path.

Enjoy!

---------- Forwarded message ----------
From: *Forest Simmons* <fsimmons at pcc.edu>
Date: Sunday, December 27, 2020
Subject: Paths of Least Resistance
To: Forest Simmons <fsimmons at pcc.edu>


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 maximizes 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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