[EM] Range PAV results

[Kristofer Munsterhjelm]

fsimmons at pcc.edu wrote:
>> Also of note is that even though STV only has access to ordinal
>> information, none of the cardinal methods manage to dominate it.
> 
> In STV the candidates represent particular voters, and get no additional credit
> for pleasing some of the other voters.  In PAV the voters that get satisfaction
> from other voters' representatives are considered already partially covered.
> 
> Consider, for example, the two winner election profile
> 
> 10 A
> 40 A>B(99%)
> 45 C>A(99%)=B(99%)
>  5  C
> 
> If I am not mistaken, the set {A,C} would be the STV winner, but {A,B} would be
> the PAV winner, whether greedy or exhaustive.
> 
> The STV set has perfect proportionality, but the PAV set gives double coverage
> to 85 percent of the voters at the expense of 5 percent getting nothing.
>  
> Consider that (1) in a two winner election a faction of 5% has no guarantee of a
> representative, (2) even so in this election if all of the C supporters got
> their act together they could have gotten C into the junta, but
> (3) they are better off as a whole taking the double coverage.
> 
> It seems to me that on the whole, {A,B} is a more representative set than {A,C}.
> 
> How does the automatic rating system of the simulation compare them?

The automatic rating system can't determine the proportionality of an 
arbitrary ballot set. The way the system works, it first generates a 
model (in this case, every candidate and voter having n binary issues 
which are randomly yes or no according to a predetermined fraction of 
ayes for each), and then it derives the ballots based on that (voters 
preferring candidates closer to them, rating them proportional to the 
number of bits by which their opinions agree).

That said, I think my system would give a scenario like the above a 
better regret score at the cost of (exact) proportionality, because the 
majority (the 85%) get a better outcome with the 5% getting nothing.

If the scores are:

10: A(100%) > B = C (0%)
40: A(100%) > B(99%) > C (0%)
45: C(100%) > A = B (99%)
  5: C(100%) > A = B (0%)

then A+B has a utility sum of 100*10 + 100*40 + 99*40 + 99*45 + 99*45 = 
17870 whereas A+C's utility sum is 100*10 + 100*40 + 45*100 + 5*100 = 10000.

Thus, {A,B} would probably get a better regret score than {A,C}, but a 
somewhat worse proportionality score.

-

Your example makes use of preferences that are very close to 100. Would 
an ordinary (rank-ballot) multiwinner method (satisfying DPC etc) elect 
{A,B} if the preferences that are very close to 100 were to be brought 
to 100? I.e. if

10: A > B = C
40: A = B > C
45: (null)
  5: C > A = B

would an ordinary method elect {A,B}? I guess what I'm asking is whether 
Range PAV electing {A,B} is because of a property of Range PAV as 
compared to other (reasonable) cardinal multiwinner methods, or of 
cardinal multiwinner methods in comparison to ordinal ones.