# [EM] Paths of Least Resistance

Forest Simmons fsimmons at pcc.edu
Sun Dec 27 17:40:27 PST 2020

```Should be "minimizes"

On Sunday, December 27, 2020, Forest Simmons <fsimmons at pcc.edu> wrote:

> 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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