Topics created by Toby Pereira

@toby-pereira said in Optimal cardinal proportional representation:

By the way, since PAV with infinite clones passes core (which it doesn't with a limited number of candidates), I presume the optimal version probably is properly proportional (passes perfect representation). I might update my paper with this in at some point.

I have updated the paper to mention the proportionality of Optimal PAV (with variable candidate weight allowed), which allows for a proper comparison with COWPEA - these two methods being the main candidates for a truly optimal cardinal PR method (practicalities aside).

@toby-pereira

Yes, but the big winner, so far, in the burial-deterrence category, is RP(wv), which isn't a new-invention method, but, rather is a traditional, classical, immensely popular & high-prestige method.

There's also a discussion of different PR philosophies on the wiki. It doesn't really cover all the ground though, as COWPEA or COWPEA Lottery wouldn't be included, for example. But perhaps non-deterministic methods deserve their own entry on there.

Optimised PAV Lottery is another non-deterministic method. In this you work out the optimum amount of weight each candidate should have (by e.g. infinitely cloning all candidates and running an election with a very large number of seats), and then elect candidates probabilistically according to these weights. (Though you would have to work out the distribution again every time a candidate is elected). Unlike deterministic PAV, it is thought (though not known) to be proportional by passing the Perfect Representation In the Limit criterion.

I also think that sequential methods generally fail participation (for a suitable multi-winner definition), whereas optimal elect-all-at-once methods are computationally infeasible. However, I think non-deterministic sequential methods can get around this failure. Optimised PAV Lottery is computationally infeasible anyway, but COWPEA Lottery is easily runnable.

Just to clarify on the Phragmén thing:

If you just have a fixed quota then the voters that get their candidates early can get a bad deal because they pay a whole quota, whereas later on, the might not be a candidate with a whole quota of votes and yet you have to elect one anyway, so the voters of this candidate get their candidate more "cheaply".

So you might then look for a quota that distributes the cost more evenly, and that's all Phragmén really does. It distributes the load or cost across the voters as evenly as it can.

I see subtractive fixed-quota methods as a cheap hack really.

Some things that came out of the paper:

If A, B, C and D are parties fielding multiple candidates, then with the following ballots, what proportion of the seats should each party win?

250: AC
250: AD
250: BC
250: BD
2: C
2: D

COWPEA would say that A and B would each win just under 1/4, and C and D would each win just over 1/4.

However, Optimised PAV would elect C and D with half the weight each, and A and B would not get any weight. The multi-winner Pareto efficiency criterion assumes that a voter's satisfaction with a result can be measured purely by the number of elected candidates they have approved. And going by that, any result containing any A or B candidates is Pareto dominated by CD.

This can be seen as a two-dimensional voting space with an AB axis and a CD axis. PAV would ignore the AB axis. Arguably though COWPEA makes better use of the voting space.

That was an example where any proportions are allowed, but here is another where 2 candidates are to be elected.

150: AC
100: AD
140: BC
110: BD
1: C
1: D

Basically any deterministic would elect either AB or CD. CD Pareto dominates AB in this multi-winner sense as all 502 voters have an approved candidate under CD, compared with 500 under AB (no-one has two approved candidates). However, AB is more proportional. 250 have approved A and 250 have approved B. 291 have approved C and 211 D. So under the CD result, the 211 D voters would wield a disproportionate amount of power. So what do you think? Does that matter or is it all about number of approved candidates?

If you go for AB, then you are rejecting the multi-winner Pareto efficiency criterion, but also consistency as a by-product. This is because you can have a set of ballots where the C and D approvals are just swapped round. So it's 211 for C and 291 for D but otherwise the same. Then combining the two sets of ballots together you get:

250: AC
250: AD
250: BC
250: BD
2: C
2: D

If there are 2 to elect here, it must be CD. So if you went for AB in the other examples, then you are rejecting the consistency criterion in multi-winner elections.

@toby-pereira Yes, and if only extremists show up, they deserve the power.

If this was extended to score voting, I think it should elect the Condorcet winner in the single-winner case, if there is one. Otherwise, obviously it would have to choose between the candidates in some way, like other Condorcet methods do so it's not a big problem.

When there's more than one winner, what happens depends on how you interpret the scores. You could measure a voter's satisfaction by adding up the scores the voters have given to the elected candidates, but I think that might be unsatisfactory in a few ways. There's always debate about how to interpret scores and what they mean, and whether absolute numerical values should really be used in their raw form.

Instead, the scores could be used as layers of approval. This basically means that a voter's satisfaction with a candidate set is determined by the single highest score they've given to a candidate in the set, next best used as a tie-break, and so on. So for scores out of 5, a single 5 is better than multiple 4s etc.

This should keep it relatively simple. Also if candidates are elected sequentially, it should be simple enough to calculate the results.

I think this should be a decent enough method and I think I'd prefer it to things like Allocated Score and Sequentially Spent Score.

Obviously COWPEA Lottery using scores as layers of approval is God-tier in terms of criterion compliance, and very simple to implement, but it is non-deterministic, which might be too much for some people, so this method could be a good compromise.

Edit - You'd have to work out exactly how to measure the stability of a candidate set though. Let's say the first 2 candidates elected are AB. Then you need to test e.g. ABC, ABD, ABE etc. to find the 3rd candidate. But I think you might be able to test them against each individual candidate not in the set. So test ABC against, D, E, F etc. separately.

Edit 2 - You'd probably have to test each potential set against all the other subsets. So ABC would go against ABD, ABE etc., plus AD, AE, BD, BE, as well as D, E etc. Still not that many in the general scheme of things.

Here is one by Piotr Skowron as well titled "Consistent Approval-Based Multi-Winner Rules".

And here's one called "Proportionality and the Limits of Welfarism" (comparing Phragmén and Thiele). I didn't actually catch the guy's name. It didn't sound like the names in the paper - Dominik Peters or Piotr Skowron (and it doesn't look like Piotr Skowron) - and the subtitles have it as Scott Anand, which is a name I'm not familiar with.

Thanks for the replies everyone. My confusion was because I wasn't aware that the load could be spread arbitrarily. Interestingly I think this is equivalent to the idea of partially or wholly removing approvals that are detrimental to a candidate set's "score", which I suggested years ago, but seems to be a case of reinventing the wheel. There's quite a good discussion of Phragmén's voting methods in this paper by Svante Janson.

@toby-pereira Quite amusing.

@toby-pereira said in Lottery PR methods compared:

COWPEA Lottery with layers of approval - Voters score or grade candidates. The actual values are irrelevant, but when a ballot is picked at random only the top layer of (relevant) candidates is looked at. A pro is that it gives voters more distinguishing power between candidates. The cons are that it becomes more complex and to vote optimally a voter would have to grade basically all of the candidates, which could be quite a lot of them.

This one could be used nicely with the 0 to 5 "star" ballot. So to clarify the process to elect a candidate - pick a ballot at random and eliminate all but the top or joint top rated candidates on that ballot. Pick another ballot and retain only the candidate(s) that are the highest rated among those still in contention. Continue until one candidate remains. Elect that candidate.

I discussed in this thread how COWPEA can fail a multiwinner version of Pareto efficiency, which you could define as follows:

If every voter has approved at least as many candidates in set X as set Y, and at least one voter has approved more candidates in set X than set Y, then set Y should not be the winning set.

But I then countered in this thread that it might not actually be such a desirable property.

However, it does leave me wondering how optimal candidate Thiele voting would behave. Thiele's biggest failing is that it fails ULC, but with different candidate weighting allowed, a universally liked candidate would get all the power, so the problem might go away. But there might be residual problems when a candidate isn't universally liked, but is by certain factions, or if factions mix in a certain way. So I'm not sure if the problem would remain. Thiele would automatically pass the Pareto criterion above, though as said, it's debatable how desirable it is.

I think in general it would lead to more majoritarian but less purely proportional results than COWPEA.