# [EM] Beatpath(Unc, wv), a version of Beatpath that always picks from the Uncovered set

Forest W Simmons fsimmons at pcc.edu
Tue Mar 13 18:39:57 PDT 2007

```Here's a version of Beatpath that always picks from the uncovered set:

As in Beatpath(margins) or Beatpath(wv) we define a relation R
on the candidates as follows:

X R Y  iff and only if there is a stronger beatpath from X to Y than
from Y to X.

This relation is transitive, and when there are no tied strengths, as
in large scale public elections, it totally orders the candidates.

The strength of a beatpath is the strength of the weakest link in the
beatpath.

Different versions of Beatpath differ in how they measure the strength
of a pairwise defeat.  Beatpath(wv), Beatpath(margins) and use winning
votes and margins, respectively.  DMC is a version of Beatpath that
uses the approval of the pairwise victor to measure the strength of
defeat.

Like the former two versions, Beatpath(Unc,wv) does not require
approval information, though it could be easily modified to take that
information into account when it is available.

Let N be the number of ballots. In Beatpath(Unc, wv) the strength of a
pairwise victory of X over Y is N+1 if X covers Y, else it is the
number of ballots on which X is ranked above Y.

That's it, but here are some (intended to be helpful) comments for the
benefit of readers not used to thinking about coverings in relation to
beatpaths:

No beatpath can have a strength greater than N+1.

Candidate X is covered if and only if there is a beatpath to X with
strength N+1.

A beatpath of strength N+1 can always be shortened to a beatpath of one

It is impossible to have beatpaths of strength N+1 both from A to B and
from B to A, since covering is a non-reflexive, transitive relation.

This last property ensures that the existence of many paths of strength
N+1 is no impediment to the total ordering of the candidates by the R
relation.

Here's an example where Beatpath(wv) and Beatpath(Unc,wv) differ:

Suppose that the wv defeat strengths are

A 55 B 55 C 54 D 51 B,  C 54 A,  and  D 51 A.

(where the number between the two letters gives the wv for the left
letter over the right letter).

Then the beatpath induced total order is (with path strengths)

A 55 B 55 C 54 D .

On the other hand, the (Unc,wv) strengths are

A 55 B 55 C 54 D 51 B, C 54 A, and D (N+1) A.

Through beatpaths this induces the total order (with path strengths)

D (N+1) A 55 B 55 C.

The Beatpath(wv) winner A is covered by the Beatpath(Unc,wv) winner D.

The strongest beatpath from A to D is A 55 B 55 C 54 D, and that is
also the shortest path from A to D.

So this version of Beatpath puts high value on short paths via the
covering relation.

I hope some of you find this to be an interesting point of departure
for further innovations.

Forest

```