[EM] Preliminary Range PAV results

[Kristofer Munsterhjelm]

Here are the results for Range PAV from my simulator so far. The first 
number is proportionality (normalized SLI), and the second number is 
normalized Bayesian regret. Except for the Cardinal-* methods, the 
scores being used are raw, i.e. not quantized in any way, but since the 
number of opinions are less than 20, it might not make much of a difference.

For electing 2 out of 4, 18000 rounds, all other parameters as on my page:

AS   0.20769   0.05808   Range-Auction(Cumul)
AS   0.30988   1e-05     Range-Auction(Ordinary)
AS   0.31306   0.00022   Range PAV(D'Hondt)
AS   0.31201   0.00088   Range PAV(Sainte-Lague)
AS   0.31289   0         Maj[Cardinal-20]
AS   0.30829   0.00591   Maj[Cardinal-20(norm)]
AS   0.1144    0.09555   Schulze STV

For electing 3 out of 5, 18000 rounds, all other parameters as on my page:

AS   0.19659   0.07225   Range-Auction(Cumul)
AS   0.39676   7e-05     Range-Auction(Ordinary)
AS   0.40555   0.0002    Range PAV(D'Hondt)
AS   0.4045    0.00076   Range PAV(Sainte-Lague)
AS   0.40612   0         Maj[Cardinal-20]
AS   0.39792   0.00602   Maj[Cardinal-20(norm)]
AS   0.08153   0.15948   Schulze STV

For electing 4 out of 10, 1448 rounds, all other parameters as on my page:

AS   0.25681   0.08738   Range-Auction(Cumul)
AS   0.34258   7e-05     Range-Auction(Ordinary)
AS   0.34919   0.00029   Range PAV(D'Hondt)
AS   0.34953   0.00096   Range PAV(Sainte-Lague)
AS   0.35293   0         Maj[Cardinal-20]
AS   0.34812   0.01027   Maj[Cardinal-20(norm)]
AS   0.03903   0.20502   Schulze STV

Unless there is a bug or the number of rounds shown aren't enough, these 
methods seem very majoritarian and are consistently beaten by Warren's 
Range Auction method (even though that, too, is majoritarian by the SLI 
measure).

My experience with trying to achieve PR with rated ballots is that it's 
very hard. Both birational voting and LPV0+ also show as majoritarian 
(although it's been some time since I've last checked them).

However, given Jameson's claim that his method satisfies a Droop 
criterion for approval ballots and gives results similar to greedy PAV 
(not tested here), I'm not entirely sure that what I'm seeing is not a 
bug. If someone on the list could give an example where, say, 2-seat PAV 
(not greedy) gives a significantly different result than just electing 
the k Approval winners (particularly a case where the PAV winner set and 
the k Approval winner set is disjoint), that would be useful for testing 
purposes.

-

Also, for those who would want to implement the Range generalization of 
PAV with "continuous divisors" (the integrals referred to in earlier 
posts), note that the simple approximation to H_n fails when n < 1 
(since it uses a logarithmic term). It's better to relate H_n to digamma 
and use the recurrence of (digamma(x) = digamma(x+1) - 1/x) to get into 
an area where the approximation error is not too great.