[EM] Request for method with certain properties

[Kristofer Munsterhjelm]

I'm fiddling around with an old idea of mine: that multiwinner methods
that are Droop proportional can't be summable in data whose bit count is
a polynomial in n and log(V) - the number of candidates and the
logarithm of the number of voters, respectively - and be able to return
a Droop-proportional outcome for any number of winners/seats with the
same data.

I have found out that one can make a Droop proportional method that's
summable with a bit count proportional to n and V (not log V) in this
manner; maybe I'll write about that later. But I haven't found a general
impossibility result yet for log(V), as it seems to be very hard.

I've reduced one strategy down to that if there exists a method that
fulfils the following properties, then that strategy can't work:

- It can produce an one-seat or two-seat outcome using the same
data/summary chunk derived from the actual ballots.
- Two elections' data chunks can be combined to produce a data chunk for
the combined election set.
- The data may not necessarily be polynomial in anything.
- It is impossible to use the data chunk to determine the number of
first preference votes for the candidates, if the number of candidates
is three.
- The method is Droop-proportional.

Can anyone think of a method that has these properties?

For single-winner methods, I already have:

- The single-winner (one seat) analog of Droop proportionality is the
mutual majority criterion.
- There do exist methods that pass mutual majority, but yet it's
impossible to infer the candidates' first preference counts in the
general case. For instance, Schulze.
- But if there are only two candidates (i.e. the number of winners plus
one), then any Condorcet method (including those passing mutual
majority) will give up the first preference counts. A's count is just
A>B and B's count is just B>A.

So I'm wondering if this holds for three candidates: that if a method
passes the DPC, returns results for both one and two seats using the
same data, and the number of candidates is the max number of seats plus
one (here 2+1=3), then it's always possible to determine the number of
first preferences for all the candidates, no matter what the data is set
up to be.

If it does, then I'm onto something. If not, then I'd have to find
another strategy.