Hello all,
I've rewritten my program that tests the proportionality of PR methods
by assigning binary issue profiles to voters and candidates and
comparing the council's proportion of candidates in favor of each issue
with the proportions of the people.
There were some bugs in my previous version. For one, I incorrectly
implemented IRV so that it got a higher score than should be the case.
The new version now puts that which is to IRV as SNTV is to Plurality as
one of the best methods.
The full results (of those better than the average randomly chosen
assembly) are:
0.176552 QPQ(div 0.1, multiround)
0.176552 QPQ(div 0.1, sequential)
0.191093 QPQ(div Sainte-L, sequential)
0.209409 QPQ(div Sainte-L, multiround)
0.230898 STV
0.248373 Maj[Eliminate-Plurality]
0.259064 QPQ(div D'Hondt, sequential)
0.26736 Meek STV
0.280724 QPQ(div D'Hondt, multiround)
0.314229 Maj[Plurality]
0.318016 Maj[AVGEliminate-Plurality]
0.358127 Maj[Eliminate-Heisman Trophy]
0.362992 ReweightA[Heisman Trophy]
0.391753 Maj[AVGEliminate-Heisman Trophy]
0.391753 Maj[Heisman Trophy]
0.393261 -- Random candidates --
That's 23 rounds, RMSE, normalized for each round so that 0 is best of
ten thousand and 1 worst of ten thousand random assemblies. The other
end (of those most majoritarian) has:
0.70004 Maj[Borda]
0.705089 Maj[AVGEliminate-Borda]
0.709886 Maj[Cardinal-20(norm)]
0.718258 Maj[ER-QLTD]
0.722618 Maj[Cardinal-20]
0.731794 Maj[ER-Bucklin]
0.750181 Maj[Eliminate-VoteForAgainst]
0.758258 Maj[Schulze(wv)]
0.761436 Maj[AVGEliminate-Antiplurality]
0.8394 Maj[Eliminate-Antiplurality]
Some notes on the terminology: Eliminate-X is loser elimination.
AVGEliminate-X is like Carey's "Q method", only generalized: it
eliminates all of those with worse than average scores. Maj[X] is the
simple porting of single-winner X to a multiwinner system, where one
just picks the n (for a council of n) highest ranked in the social
ordering. "Heisman Trophy" is the positional system 2, 1, 0, 0, .... 0.
VoteForAgainst is 1, 0, ..., -1. Antiplurality is 1, 1, 1, ..., 0.
ReweightA[X] is like RRV, only with positional scores (of positional
method X) instead of range scores.
The really strange thing here is that my method seems to have a
substantial small-state (small-party) bias. For instance, QPQ with a
divisor of 0.1 is scored much better than QPQ with a divisor of 0.5
(Webster/Sainte-Lague) or with 1 (D'Hondt).
I don't know why that happens, as it's not obvious from the idea
(generate hidden binary issue profiles, generate ranks of all candidates
based on Hamming distance to each candidate, run through election
method, compare proportions of TRUEs in the issue profiles of the
assembly to the proportions among the people). Could it be something
related to the assumption that people vote sincerely? Or is the variety
of positions so large that in order to get a lower score, it's better to
elect the one that supports your opinion than one who would deprive you
of the opinion while smoothing out all other opinions a little bit? But
if so, then using RMSE to measure party proportionality in multiparty
states would be flawed, and someone would have written about it.
Other odd results: Ordinary STV scores better than Meek STV (Meek is
usually considered better) and single-round QPQ scores better than
multi-round QPQ (where the latter is usually considered better). Some
methods that fail Droop proportionality score better than ones that pass
it: namely, IRV (Elimination-Plurality) scores better than Meek STV.
Perhaps the better PR rules do, well, better most of the time, but there
are some instances where they do much worse. To detect that reliably,
I'll have to add Pareto-domination tests or median (instead of/in
addition to average) scores.
It may also be tha 23 rounds is far from enough, but I've run some of
these longer (up to 500 rounds) and the general position isn't that far
off. Some times, IRV even gets ahead of STV.
(Here's an example with only QPQ being tested, with 76 rounds:
0.223125 QPQ(div D'Hondt, sequential)
0.231456 QPQ(div D'Hondt, multiround)
0.153442 QPQ(div Sainte-L, sequential)
0.16047 QPQ(div Sainte-L, multiround)
0.146262 QPQ(div 0.1, sequential)
0.146262 QPQ(div 0.1, multiround)
The small-party bias still seems to hold. And here's a 423-round test
with Meek and ordinary STV, and IRV:
0.203673 Maj[Eliminate-Plurality]
0.208361 STV
0.220629 Meek STV.)
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