[Election-Methods] Determining representativeness of multiwinner methods

[Steve Eppley]

Hi,

I prefer a definition of "representativeness" that differs from 
Kristofer's.  To me, the more similar the *decisions* of a legislature 
are to the decisions the people themselves would make collectively in a 
well-functioning direct democracy, the more representative is the 
legislature. 

Given my definition, a non-proportional legislature comprised solely of 
centrist majoritarian compromise candidates may be very representative, 
since the people themselves would reach centrist compromises on the 
issues in a well-functioning direct democracy.  It might be more 
representative than a proportional legislature, since the proportional 
legislator could match her constituents' favorite position on every 
issue yet fail to match the way they would compromise.

Why should anyone care more about the legislature's proportionality than 
about their decisions?

Regards,
Steve
----------------------------------------
Kristofer Munsterhjelm wrote:
> Hello all,
>
> (says the newcomer.)
>
> I think I have found a metric for comparing multiwinner systems, at 
> least as these pertain to proportional representation, when all votes 
> are honest.
>
> The advantage of the metric is that, if what it measures is desirable, 
> it gives an idea of how good the system performs - how representative 
> it is - and thus its best case performance. In contrast, criterion 
> failure shows how bad a system can get in the worst case.
>
> The broad idea is this: The most proportional assembly is the one 
> which reflects the population on all issues. In other words, if a 
> fraction p of the population is of a certain position on a binary 
> opinion, it is better (ceteris paribus) for a council to have, of that 
> opinion, a fraction close to p than one far away from it.
>
> Thus we could make a simulation. First, set that there are n binary 
> issues. Each of the voters then have an issue profile which consists 
> of n booleans. Set these randomly with different biases for each issue 
> (so that, for instance, on the first issue, 70% may hold the "true" 
> position, while on another, only 23% do).
>
> Counting the proportion that hold the true-position for each issue 
> gives the popular issue profile. In general, the issue profile of a 
> certain subset takes the form of n numbers (for n issues), where each 
> number is equal to the proportion that holds the true-position for the 
> issue in question.
>
> Then a perfectly representative assembly has an issue profile that is 
> equal to the issue profile of the people. So now we have a measure of 
> how well the assembly or council represents the people: the more its 
> issue profile differs from that of the people, the less representative 
> it is.
>
> However, this presents a problem. How does one aggregate the 
> difference on each issue into a single score? Is a one-percent 
> difference on a single issue better than 1/n percent difference on all 
> issues? One way to solve this is to just settle on an aggregation 
> measure (like root-mean-square) and hope the results can be 
> generalized across; another is to use Pareto-domination as a measure 
> instead, in saying that councils produced by a method A is better than 
> councils produced by a method B to the extent that A-councils lie 
> strictly closer to the population profile than does B. That approach 
> can give no information on the cases where some issues are closer by 
> method A and some are closer by B (mutual nondomination).
>
> Putting all of the pieces together, to figure out the scores, a 
> simulation would do something like this:
>     - Generate issue vectors for all of the people, and get the
>         popular issue profile.
>     - Choose a subset of the people as candidates.
>     - Generate ballots for each voter of all the candidates.
>     - For a great number of random assemblies:
>         - Get the issue profile of this assembly, and calculate
>             the similarity measure for that with regards to
>             the popular issue profile.
>         - If the similarity measure is more similar or less
>             similar than any random assembly we've seen so
>             far, update the worst (respectively best)
>             record.
>
>     - For each multiwinner election system:
>         - Feed the ballots into the system.
>         - Get the issue profile of the elected assembly, and
>             calculate the similarity measure for that with
>             regards to the popular issue profile.
>         - Normalize the similarity measure with regards to the
>             worst and best random councils.
>         - Add the normalized similarity measure to that system's
>             running    total.
>
> To be robust, it would do this a lot of times with various population 
> sizes, council sizes, and issue numbers (n). With a similarity 
> measure, 0 would be perfect (impossible most of the time), and 1 (or 
> infinity, depending on the measure) be the worst possible.
>
> The only thing remaining is to find out how to generate ballots for 
> each voter. A reasonable assumption is that voters are going to prefer 
> the candidates who agree with them on many issues to those that agree 
> with them on a few. For binary issues, Hamming distance works: in the 
> simple model, voters rank (or rate) the candidates inversely of 
> Hamming distance.
>
> -- 
>
> I have made a program that does this. It is simple, does not use equal 
> ranks (randomizing preferences instead), but the results are interesting.
>
> Worst of the lot are the majoritarian systems ported to multiwinner 
> systems. Those would, for a council of size k, just pick the k first 
> in the social order of the single-winner method. This result shouldn't 
> be surprising, because the straight port excludes minority opinion. Of 
> some curiosity, however, is that IRV does the best among those; maybe 
> it reflects IRV's origins as the multiwinner method STV? Or maybe 
> noise (as resulting from nonmonotonicity and the likes) bring it 
> closer to the results gained by just picking a random assembly.
>
> Then come the vote-reweighted methods, like RRV. Vote-reweighted 
> methods can be generalized as: run a single-winner method, then 
> reweight those who voted for the winner, according to some function 
> that does not take the number of seats into account. Then run again, 
> and disregarding those that have already been elected, pick the next 
> member as the one who is closest to the top in the social ordering 
> output.
>
> Best of all were the "proper" methods implemented: STV (with 
> Senatorial rules) and QLTD-PR, which uses Woodall's QLTD instead of 
> IRV as its basis: it adds fractional votes until someone gets above 
> the quota, then reweights the voters who contributed to that one, 
> basing the weighting on the candidate's surplus.
>
> According to the RMSE scores:
>     Majoritarian assemblies:
>         Borda:         0.871528    *Plurality:       0.256192
>         Antiplurality: 0.73616    Nauru-Borda:      0.599807
>         IRV:           0.362097 Cardinal ratings: 0.894351
>
>     Vote-reweighted assemblies:
>         Borda:         0.376745    *Plurality:       0.260454
>         Antiplurality: 0.401539    Nauru-Borda:      0.406815
>         RRV (k = 1.0): 0.682116    RRV (k = 0.5):    0.644339
>     
>     Quota:
>         *STV:          0.193959    *QLTD-PR:         0.121693
>                     QLTD-PR (rated):  0.417813
>
>     Other:
>         Random Cands:  0.364437
>
>     STV-QLTD Pareto dominance: QLTD: 236, STV: 237, nondomin: 674
>
>     "Plurality" is the weighted positional system of {1, 0, 0....}
>     applied to ranked ballots.
>
>     (* marks those that are better than a random assembly, on
>      average)
>
> Some of the results may be due to artifacts in the voting pattern - 
> the simulator was a proof of concept, after all. I think that 
> Plurality benefits by that everyone votes sincerely, and that the 
> ballots are complete, for instance. Yet patterns emerge.
>
> If anyone wants to experiment with the simulation program, it is here: 
> http://munsterhjelm.no/km/raw/pr_elect.zip . QLTD is called "Quota 
> Bucklin" there, as I sort of independently discovered it while trying 
> to make a quota-proportional form of Bucklin.
>
> -- 
>
> On a second thought, it shouldn't be so surprising that 
> vote-reweighted methods, in general, do worse than quota-based ones. 
> Consider the following situation:
>
>     20: Left > Center > Right
>     20: Right > Center > Left
>      1: Center > Left = Right
>
> Condorcet would pick Center in the single-winner case. In the 
> situation of an assembly of two, the reasonable choice (which CPO-STV 
> picks) would be Left and Right.
>
> However, vote-reweighted methods based on Condorcet would have to 
> start off by picking Center, since all voters start off with equal 
> weights. After it has done so, there is not enough room on the 
> assembly to permit an even division of Left and Right, and thus either 
> Left or Right will be favored, assuming Center supports both sides 
> equally.
>
> Vote-reweighted methods that aren't based on Condorcet may pick Left 
> and Right, but they can only do so if they would pick either Left or 
> Right in the single-winner case.
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