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.