[EM] Pairwise Shored-up Coalitions

Forest Simmons fsimmons at pcc.edu
Tue Feb 23 23:33:46 PST 2021


A solid coalition on a ballot must include all of the alternatives ranked
equal to or above any other member of the coalition, at least for
traditional solid ballot coalitions as defined by Woodall. But it seems a
shame when one alternative disrupts an otherwise perfectly good ballot
coalition ... is it possible to fortify the concept to make it more robust?
In particular, what if the interloper is beaten pairwise by every member of
the coalition that is ranked behind it? Could we mentally sort it downward
to get it out of the way for the purpose of identifying a likely de-facto
coalition?

That's our motivation for the following definition of a "pairwise supported
ballot coalition" (a PSBC):

A subset of the ranked alternatives is a PSBC if every alternative outside
the coalition (not a member of the PSBC subset) is pairwise beaten by any
coalition member not ranked ahead of it.

This allows the coalition to "close ranks" by over-riding the ballot order
with pairwise order, similar to the way pairwise order can trump approval
order when sorting the approval list pairwise, whether by sink sort, bubble
sort, or approval margins priority***

This concept of coalition can be used for the random ballot coalition
lottery as described in my recent message ... using subsequent randomly
drawn ballots to narrow the coalition to a singlton the same way we do in
random ballot approval.

***An aside on ASM approval Sorted Margins ... the margin in the name
refers to the (absolute) difference in approval, but there might be an
advantage to using pairwise margins ... for keeping clones solid in the
sorted order ... or one for nominal use and the other for tie breaking.
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