[EM] Proportionality vs utility: redoing 2008 with better units

[Kristofer Munsterhjelm]

I've been incorporating some very old code for multiwinner method 
testing (see my very first post) into Quadelect, my general purpose 
voting program. And I've run some simulations and got an indication of 
what different multiwinner methods prioritize between proportionality 
and broad support. So I thought I'd write a post about the results.

The code tests the proportionality and utilitarian quality of a winner 
set in a similar way to the VSE-like proportionality measure I posted 
about on July 31. It just uses binary issues instead of a normal spatial 
distribution.

That is, each voter has a binary position on one of a set number of 
issues. Each voter's utility for a given candidate being elected is the 
number of issues where they agree.

The program determines proportionality by checking the similarity of the 
proportion of elected candidates who hold a yes position on each issue, 
with the proportion of the voters who do so. It quantifies the 
difference by using a proportionality measure like the Sainte-Laguë 
index. So far, just a copy of my approach in 2008 (it's the same code, 
after all).

But instead of the somewhat on-the-spot approach to combining the 
utility and disproportionality results, it uses VSE. This gives what 
seems to me to be more sensible results.

I've attached the results as pictures. The distribution parameters are:

Number of voters per round: 5 ... 260
Number of candidates: 5 ... 11
Number of issue dimensions: 1 ... 11
Number of winners: 3 ... 10,

and I'm using the Sainte-Laguë index as error measure.

The VSE zero points are anchored to the expected value 
(disproportionality and utility) of just picking candidates at random 
until the council is full. The vertical black dotted line on the detail 
plot thus separates methods that are more proportional than random 
candidate from those that are less.

Each purple dot (plus sign) corresponds to one method. Some methods take 
parameters that adjust their proportionality, e.g. most of Warren's 
cardinal methods take a D'Hondt-Sainte-Laguë adjustment parameter. Those 
are represented by curves which sweep either over the full range or a 
large area of it. (Since Harmonic voting is relatively good; I've run it 
outside its usual range to see what happens. The results for the 
out-of-range values are marked with a dashed line.)

Clearly there's a trade-off between proportionality and utilitarian 
efficiency. That's not very surprising, because if you have a large 
number of winners, proportionality tells you that you should elect one 
representative for each major issue position, but VSE tells you that you 
should fill the council with candidates close to the median voter.

But other things are more surprising. One is just how weak the Droop 
proportionality criterion constraint is. Both PSC-CLE and PSC-(Plurality 
loser) pass it, but they're quite far from each other on the plot.

Second, there doesn't seem to be any "cardinal superiority" here. 
Harmonic is the best cardinal method of those tested, and it intersects 
STV pretty cleanly. The concentrated nature (and comparatively lower 
performance) of Psi voting compared to Harmonic is also pretty surprising.

At first I didn't expect Schulze STV to be more proportional than 
ordinary STV; for Schulze vs IRV it's the other way around. But my 
Kemeny clustering methods (not shown here as they're far too slow) also 
fall much more on the proportional than utilitarian side of things. Pure 
Condorcet-based multiwinner methods might just be particularly good at 
identifying the nth quantile candidates, the way Condorcet does the 
median candidate; if so, such nth quantile candidates would be 
proportionality-focused rather than utility-focused.

But on the left side of the detail plot, why is minmax so far off 
Schulze? They're based on the same logic, after all.

Some of the names may be unfamiliar, so here are some references:

- Random dictator: strategyproof, elect the first k candidates listed on 
a random voter's ballot.
- D'Hondt without Lists: a method by Juho Laatu. 
http://lists.electorama.com/pipermail/election-methods-electorama.com/2008-October/121082.html 
My variant multiplies the quotients (1/2, 1/3 etc), as mentioned in that 
post, instead of adding them.
- Coombs, Borda, etc: these are all bloc voting: elect the k highest 
scoring candidates.
- Random ballots: this is also strategyproof (I think?). Pick a voter at 
random, elect the voter's first preference, eliminate that candidate 
from every ballot, remove the lucky voter's ballot, and repeat.
- Harmonic voting, Psi voting: see 
https://rangevoting.org/QualityMulti.html.
- QPQ: Quota-preferential by Quotient, a method by Woodall. 
https://www.votingmatters.org.uk/ISSUE17/INDEX.HTM
- Birational, LPV0+: Cardinal methods proposed by Forest and Warren. See 
https://rangevoting.org/WarrenSmithPages/homepage/multisurv.pdf. Ratings 
are counted as fractional approval votes.
- Isoelastic: Cardinal method based on a function proposed by Peter 
Zbornik. 
http://lists.electorama.com/pipermail/election-methods-electorama.com/2010-May/124448.html

-km
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