[EM] Further investigations into Set Webster

[Kristofer Munsterhjelm]

So I finally made my source code interface with my multiwinner 
evaluation program. Doing so, I could have the method construct solid 
coalitions directly, instead of having to do it manually. After 
experimenting, a few things become apparent:

The method is better than STV on really large assemblies/councils. It's 
significantly worse than STV on small councils, and atrocious at 
single-winner scenarios.

The method does not reduce to DAC/DSC. A simple example is this:
	100: A = B = C = D
	1:   A
A should win, but there's no reasonable divisor that doesn't set "A" to 
0, thus making the outcome at best a tie between A, B, C, and D, and at 
worst actually harming A. If I want Set Webster to reduce to DAC or DSC, 
I have to find some way of fixing that; otherwise, I'll just have to 
accept that it doesn't.

The previous example also shows why it's so bad when making small 
councils. I call it the "phantom penalty" effect. Say that nobody votes 
for A; then a council that contains A doesn't face any direct 
contradictions (only through denying the council of a seat that could be 
used by another who would lower the penalty). However, now say that a 
single person bullet-votes for A. Then we have:
	1: A
and this will get rounded down, by any reasonable divisor, into

	{A} pick exactly zero.

which means that the A-voter makes councils with A attain a higher 
penalty than before. Nonmonotonicity!

How can that be solved?

The simplest way is to interpret the constraint not as "the outcome 
should have exactly this many", but as "the outcome should have this 
many, or more". However, why use divisors at all? The easiest way would 
be to have a very large divisor, so that each coalition says "0 or 
more", meaning that the tiebreaker must favor outcomes with fewer ties 
in that case. Still, ties may persist, and I'm not sure this solution is 
a good one.
If we have to break ties, a reasonable way of doing so might be: "pick 
the candidate that will produce least contradiction penalty for a 
council of this size plus one", keeping house monotonicity in mind. How 
to actually implement that is another question entirely.

If we're not to relax the constraint, what options do we have? Any other 
option, I think, must be based on the observation that any constraint on 
"the outcome must have a maximum of this many from the solid coalition" 
must be arranged so that a greater support for that coalition weakens 
the penalty from breaking the constraint, whereas "the outcome must have 
a minimum of this many from the set" must be arranged so that a greater 
support for the coalition increases the penalty from breaking the 
constraint.

More investigation is required. If any of you have any ideas on either 
how to make it reduce to DAC/DSC while retaining the divisor nature, or 
on fixing the phantom penalty problem, go ahead :-)

------------------------------------------------------------------------

Results given by my program for Set Webster as it is:

(The first number is the mean, the second is the median. Lower is better)

Single winner:

0.09825 0        STV
0.08546 0        STV-ME (Plurality)
0.07478 0        STV-ME (Schulze)
0.10895 0        Quota Bucklin(restart, Droop)
0.15286 0        QPQ(div Sainte-L, sequential)
0.1496  0        QPQ(div Sainte-L, multiround)
0.71029 0.78258  Set Webster

Reductions: "STV" is IRV, "STV-ME" reduces to Condorcet compliant 
methods (BTR-IRV), "Quota Bucklin" is plain old Bucklin, and I'm not 
sure what QPQ reduces to. PSC-CLE variant also reduces to a Condorcet 
compliant method.

Not very good, huh?

Small councils (mean council size: 27.8):
0.25401 0.25408  STV
0.27747 0.26511  STV-ME (Plurality)
0.28471 0.25898  STV-ME (Schulze)
0.25985 0.23442  Quota Bucklin(restart, Droop)
0.34727 0.27134  Set Webster

The discrepancy is less, but STV is still considerably better.

Large councils (mean council size: 45.2)

0.31809 0.26744  STV
0.35163 0.30711  STV-ME (Plurality)
0.38558 0.32804  STV-ME (Schulze)
0.34317 0.30969  Quota Bucklin(restart, Droop)
0.3333  0.2375   Set Webster

Now Set Webster is in second place as far as regards the mean, and first 
as regards median.

(Due to an unresolved bug, I haven't been able to get QPQ results for 
these runs.)