[EM] Fwd: Preliminary Range PAV results

[Kristofer Munsterhjelm]

Jameson Quinn wrote:
> I'm resending the message I sent to Kristofer because I think it's 
> generally interesting. I redid the formula for an STV-like Range-based 
> proportional system, and it's actually simpler than my previous (totally 
> broken) formula. When electing candidate A, just multiply all the 
> ballots by 1-r(A)D/S(A) (unless it's negative), where
> r(A) = ballot score for candidate A
> D = Droop quota
> S(A) = Sum of squares of scores for candidate A
> 
> (detail - you need to either normalize the scores to [0,1], or multiply 
> the droop quota by the top allowed score)
> 
> Note that my (still unpublished) summability trick can summably give 
> results which are the same as this method with a high probability. (I 
> developed the summability trick for approval ballots, but to use it for 
> Range, you just divvy up the ballot into approval-style "slices" which 
> approve all candidates which score higher than a given total. For 
> instance, in range(3), A3 B1 C0 would be equivalent to 2 (A) ballots and 
> one (AB) ballot.)
> 
> Kristofer - have you been able to get results for this formula? If you 
> send me the source code, I can try myself.

As you said you thought my "Quadratic Range STV" method was the same as 
yours, I just ran the simulator on that. I also added a few other 
methods: CFC-Range to show that CFC-Kemeny dominates it, hence the 
latter must derive some of its performance not just from CFC-* but also 
from -Kemeny; SAV (Maj[Cumul. ratings]) by request, as well as 
elimination methods based on them; and some others for comparison.

The results are:

    NAME                                           DISPROP REGRET
    ====================================           ======= =======
PA_CFC-Range(0,_exhaustive)                       0.07213 0.22535
PA_CFC-Range(0,_greedy)                           0.07565 0.22082
PA_Birational_(exhaustive)                        0.09482 0.15814
PA_Birational_(greedy)                            0.10152 0.15148
PA_Range_PAV(Sainte-Lague,_exhaustive)            0.10933 0.1305
PA_Range_PAV(Sainte-Lague,_greedy)                0.11316 0.12829

PA_STV                                            0.11902 0.1004

PA_Maj[Eliminate-Cumul._ratings]                  0.15335 0.06465
PA_Range_PAV(D'Hondt,_exhaustive)                 0.15439 0.05968
PA_Range_PAV(D'Hondt,_greedy)                     0.15639 0.059
PA_Q-Range_STV                                    0.19099 0.03127
PA_Maj[AVGEliminate-Cumul._ratings]               0.20509 0.07762
PA_Linear_Range_STV                               0.20928 0.01991
PA_Maj[Cumul._ratings]                            0.2104  0.07667
PA_Range-Auction(Ordinary)                        0.30966 1e-05
PA_Maj[Cardinal-20(norm)]                         0.31014 0.00776
PA_Maj[Cardinal-20]                               0.31348 0

where DISPROP is the disproportionality measure (1 = worst possible, 0 = 
best possible) and REGRET is Bayesian regret (again, 1 = worst possible, 
0 = best possible).

"Eliminate-Cumul. ratings" is the continuous cumulative vote version of 
IRV: eliminate the loser, then normalize only among winners, then 
eliminate the loser, etc.. Maj[x] just picks the two who rank first in 
the social ordering and constructs a council from that - in this case, 
the two candidates that are eliminated last.

If the above is IRV, then "AVGEliminate-Cumul. ratings" is Carey.

Q-Range STV is Quadratic Range STV. See my past post on this thread for 
an explanation.

Linear Range STV is based on Jameson's first suggestion, as quoted in 
http://lists.electorama.com/htdig.cgi/election-methods-electorama.com/2010-June/026505.html 
. This is the variant where N = number of voters voting this way, not 
size of entire electorate.

Range-Auction is Warren's Range Auction method, and Birational is also a 
method of his.

CFC-Range is the Range version of CFC-Kemeny. For every council, one 
runs through a linear program that assigns each voter a weight for each 
"camp" so that the sum of weights over camps for each voter is 1, and 
the sum of weights for each camp is equal. The "A"-camp gets as many 
points from voter x as x's rating of A times the A-camp weight on x. The 
linear program optimized weights so that the sum of the A camp's score 
of A plus B camp's score of B plus ... is maximized. The council 
combination that maximizes this measure is elected.

But for some reason, CFC-Kemeny beats CFC-Range (latter scores from my 
page):

    NAME                                           DISPROP REGRET
    ====================================           ======= =======
PA_CFC-Range(0,_exhaustive)                       0.07213 0.22535
PA_CFC-Kemeny_(EXP)                               0.07001 0.18467