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@toby-pereira you know much more about this subject than I do, so whatever my 2 cents is worth, I think PRIL is a reasonable necessary criterion for PR. As you indicated, it can be a weak criterion to satisfy. Something like order (and type) of convergence to PR with N also seems important to me. I probably made a similar comment previously as well, because I know I’ve thought about this before.

Another reasonable criterion seems to be that whenever exact PR is possible (and under other reasonable assumptions like sincerity, etc.), it is guaranteed. And stronger, if it isn’t possible, then the result should be guaranteed to be “as close as possible” in some way.

But I wager nondeterministic methods tend not to make those kinds of guarantees except probabilistically, despite probably having reasonable properties in expectation and variance. Which I’m fine with personally.

Are there issues with those properties above? Other than maybe being difficult to satisfy with other criteria?

Something to add here is the discussion of what it means to be proportional. I said above that I used the Perfect Representation In the Limit (PRIL) criterion. As discussed in this thread, most of the proportionality criteria that have been used for approval methods rule out methods that reduce to Sainte-Laguë in the case of party voting. A former member here went off the deep end insisting that those methods must then not be proportional in this thread.

But really it just shows that the criteria are not up to scratch. I acknowledge that PRIL is not perfect either, because it's not at all demanding in how a method reaches perfect representation, only that it happens in the limit.

For example, if you had the following ballots for party voting:

200 voters: A
100 voters: B

A method could pass by awarding the first 200 seats to A and then the next 100 to B. So in a 200-seat scenario, A would get all the seats. But obviously a method would have to be very contrived to do that, and it would likely cause it to break other symmetries and fail other criteria.

Another criterion I would demand for deterministic methods is that they would pass perfect representation when the number of elected candidates is a multiple of the number of voters (so 300, 600 etc. in this case).

But if you required that a method also only looked at voter ratios, rather than absolute numbers, we could massively reduce that problem.

2 voters: A
1 voter: B

This ballot would have to give perfect representation whenever the number of elected candidates is a multiple of 3, so the previous example would have to as well. But you could have:

201 voters: A
100 voters: B

This doesn't divide down further. But then if you also demanded monotonicity, it would have to give B at least as many seats as it would in cases where the B vote is lower. So:

3 voters: A
1 voter: B

B gets at least every fourth seat right from the start so the above case would do that as a minimum. And you could go further with the demands.

But PRIL works quite well with non-deterministic methods with fairly minimal demands. You probably only need to specify that each one is an independent pick.

There has been another update to the paper. The paper discusses how COWPEA Lottery is the only ABC (approval-based committee) method that is proportional and passes, monotonicity, strong candidate Pareto efficiency, participation, Independence of Irrelevant Ballots (IIB), Independence of Irrelevant Alternatives (IIA), Independence of Universally Approved Candidates (IUAC) and independence of clones.

Optimal PAV Lottery passes all of these except monotonicity. Both of these methods are obviously non-deterministic, but I think it's probably impossible for a deterministic method to pass all the criteria.

@marcus_sir I would agree with @cfrank that some examples would be good to get a feel for how it works. I've always tended to think that quota removal methods are fairly unsophisticated, and I believe MES falls into this category. See the thread here.

Hi @marcus_sir! Thanks for posting. Do you think you could provide a streamlined version of this, any links to supplementary information, and perhaps an example or two?

My understanding is that this is about a modification of the Method of Equal Shares, also discussed by @Toby-Pereira in this thread: https://www.votingtheory.org/forum/topic/553/general-stuff-about-approval-cardinal-pr/4, and who is (in my estimation) one of the most well-versed members we have on PR methods.

I plan to read in more detail, just busy!

@masiarek you're right, the "97/90/81/74" sequence was from a different experiment. The plots I shared have the accurate rates for the relevant experiments.

Your point about information is also important, because PSRO involves repeated optimization against mixed opponent strategies with effectively complete information about the modeled strategic environment, and a real election is definitely not like that. Present elections are probably better represented by the bounded-strategy region than by the PSRO endpoint, which does strengthen the practical case for Smith compliance. One could still speculate that declining computational and informational costs may move real strategic behavior toward that endpoint.

Also to clarify, the initial post was based entirely on the PSRO adaptive-strategy results. The later "human-like" bounded-strategy experiments were prompted by curiosity based on your feedback, and they changed the interpretation in the opposite direction. Unless we assume extremely capable strategic agents with very strong coordination and information, the PSRO results probably aren't descriptively realistic except as an approximation to a highly sophisticated strategic limit. It is interesting and somewhat unfortunate, though, that most formally Smith-compliant method seem basically unable to maintain a comparative advantage over non-Smith methods under the kind of strategic pressure PSRO simulates, at least in terms of preserving the sincere Smith structure.

Your last point about the fresh runoff aspect is also correct. I did also try several Smith-compliant methods with fresh runoffs during exploration, but those should be compared again systematically. In my exploratory simulations, most fresh runoffs showed only marginal improvements under PSRO, except for IRV vs runner up. There may be some interesting theoretical reasons for this, see Durand et al. 2026, "Super Condorcet Winners and Limit Coalitional Manipulability of IRV"--this uses Impartial Culture. I did not try fresh-runoff Smith methods in the later "human-like" analysis, so that would definitely be interesting to test as well.

And thank you for sharing your code base! I want to dig into this kind of thing more.