[EM] coherent, house monotonic, droop proportional

[Gustav Thorzen]

On Fri, 24 Apr 2026 16:18:35 -0500
Ross Hyman <rossahyman at gmail.com> wrote:

> Thanks Gustav,
> 
> In the paragraph before Claim 4 I wrote:
> 
> "For the proofs of claims 4-9, which constitute a proof of Droop
> proportionality, consideration is restricted to fully ranked ballots,
> and for simplicity, to ballot sets for which there are no ties for
> elections or exclusions."
> 
> I should have moved the part about "no ties' to apply to all the
> claims.Thanks for pointing that out.

I think the formulation would have been fine if the ballot format
had been clarified explicitly earlier, rather then infered,
because then the 'no ties' would clearly not be about the ballot format.
Also would have been nice if the paragraph was placed before the claims.

Actually simply having a bullet list like overview for definitions and
clarification with background and motivations separatly would have
made things much easier to go through, 
rather then the current state where I felt I had to infere the procedure
and every relevant term by repeatedly rereading everything over and over,
since a lot of details were brought up only in passing,
worried I would get something wrong by missing something importaint.
Phragmen priorites especially at first,
since you explicitly provided the definition for approval ballots,
but left out the rankorder case, the case your method use.

> I am asking if my proof of Droop proportionality is correct for ballot
> sets for which there are no ties.

I can't really say the proofs are correct without clarification about what
'exceeds' is meant to represent, since it explicitly means strictly,
but all counterexamples to your claims I could come up with were the kind
that resulted in random tiebreaks.
So for what it is worth, I can't manage to disprove the claims as now clear intended,
but (3) about Later-No-Help+Harm needs clarifications to the claim and/or method,
to avoid autofailure (I think providing explicit definitions is simplest).

Personally I really don't see how a scenario where one candidate
have first place preferences strictly outnumbering two or more quotas
only getting 1 seat/unit of voting power can be considered proportional,
but that is not the definition used here,
so your claims on Droop proportionality appears to holds as far as I can tell.

For what it's worth
Gustav