Seat capping is a technique I've been working on developing for a little while now. It allows various multiwinner voting methods, including many proportional methods, to be made precinctsummable by limiting the number of seats that the voting method can fill. Many specific instances of seat capping have been discovered by others, but to my knowledge I am the first person to generalize this insight to a wide variety of voting methods, including allocated score (the current STARPR proposal), sequentially spent score, reweighted range voting, harmonic score voting, and Bucklin transferable vote.
Best posts made by BTernaryTau

Precinctsummability through seat capping

RE: New Thieletype proportional voting method
@marylander I see, thank you for catching this! I'm honestly quite surprised that this sort of behavior arises from what to me feels like a pretty natural extension of SPAV. Are there any other methods known to behave this way? I need to think more about what exactly went wrong here. If there aren't any other examples, I guess that would at least mean I discovered a voting method with a novel failure mode!
@KeithEdmonds SDV should not have this issue. If you give all candidates from your party the same initial score S and all others 0s, then your ballot will always contribute a score of S²/(S + K · SUM) = S²/nS = S/n, where n is a positive integer (assuming K=1 or K=2) that depends on how many candidates from your party have been elected. This means that a greater initial score will maximize your ballot power in this case.
More generally, the limit of S²/(S + K · SUM) as SUM goes to infinity is 0, so no party should be able to win an arbitrarily large number of seats with a fixed number of voters.

Hello!
In online voting theory circles I am known as BTernaryTau, or just Tau for short. I'm a supporter of cardinal methods like approval, STAR, and allocated score. I write a blog where I often discuss the subject of voting methods and voting method criteria, sometimes with a theory focus and sometimes with an activism focus. I developed the mathematical cancellation criterion as a formalization of Mark Frohnmayer's equality criterion concept, and I am currently working on the sequential cancellation criterion, which extends the cancellation criterion to sequential multiwinner methods in a manner compatible with proportional representation.
Latest posts made by BTernaryTau

Precinctsummability through seat capping
Seat capping is a technique I've been working on developing for a little while now. It allows various multiwinner voting methods, including many proportional methods, to be made precinctsummable by limiting the number of seats that the voting method can fill. Many specific instances of seat capping have been discovered by others, but to my knowledge I am the first person to generalize this insight to a wide variety of voting methods, including allocated score (the current STARPR proposal), sequentially spent score, reweighted range voting, harmonic score voting, and Bucklin transferable vote.

RE: New Thieletype proportional voting method
@keithedmonds Don't worry, I'm not planning to stick with Thieletype methods forever. For now I'm just trying to get a better feel for exactly what went wrong and what I need to look out for in the future.

Does participation imply monotonicity?
Looking at the criteria tables on Wikipedia, there don't seem to be any voting methods listed that pass participation but fail monotonicity. And given how similar participation failures are to monotonicity failures, it seems plausible that these two criteria are linked in some way. However, when I investigated this further I found what I believe is a counterexample to the claim that participation implies monotonicity.
Consider an election with candidates A, B, C, and D, where the winner is chosen by plurality voting with ranked ballots and alphabetical tiebreaking. Now modify the voting method slightly such that when D>C>B>A is the only ballot cast, B wins, and when D>B>C>A is the only ballot cast, C wins. This creates an instance of nonmonotonicity where ranking B higher changes the winner from B to C. However, it does not create any instances of participation failure. In a singlevoter election, a voter who honestly casts either of these ballots improves their result (B or C is elected instead of A). In a twovoter election, if one voter casts either of these ballots, the other voter is guaranteed to have their top candidate win (since D never wins when there's a tie). And all other cases consist of elections identical to those under plurality voting.
Assuming I didn't make any mistakes, my next question is whether there are any restrictions that can be placed on the class of voting methods being considered to change this result by eliminating contrived examples like the one above. For example, maybe the combination of unanimity, anonymity, neutrality, and participation implies monotonicity, and as a result pretty much any "reasonable" voting method that passes participation will also pass monotonicity.

RE: New Thieletype proportional voting method
Thought a bit about the results of applying other means in place of the arithmetic mean. The geometric mean seems to remove the issue where a party can win an arbitrarily large number of seats with a fixed number of voters (the relevant limit is 0), but it doesn't eliminate the more general problem of being able to win more seats by giving all preferred candidates lower scores. The harmonic mean doesn't have either problem, and in fact using it in place of the arithmetic mean yields RRV.

RE: New Thieletype proportional voting method
@marylander I see, thank you for catching this! I'm honestly quite surprised that this sort of behavior arises from what to me feels like a pretty natural extension of SPAV. Are there any other methods known to behave this way? I need to think more about what exactly went wrong here. If there aren't any other examples, I guess that would at least mean I discovered a voting method with a novel failure mode!
@KeithEdmonds SDV should not have this issue. If you give all candidates from your party the same initial score S and all others 0s, then your ballot will always contribute a score of S²/(S + K · SUM) = S²/nS = S/n, where n is a positive integer (assuming K=1 or K=2) that depends on how many candidates from your party have been elected. This means that a greater initial score will maximize your ballot power in this case.
More generally, the limit of S²/(S + K · SUM) as SUM goes to infinity is 0, so no party should be able to win an arbitrarily large number of seats with a fixed number of voters.

New Thieletype proportional voting method
While thinking about how to keep some of the advantages of SPSV while ditching the KP transform, I came up with a new voting method that I'm currently calling sequential threshold average score voting, or STAS voting for short. It essentially operates the same way RRV does, but reweights ballots using a different formula (see the link above) that was inspired by the KP transform. This formula seems to preserve (and sometimes even strengthen) SPSV's tendency to avoid heavy deweighting when candidates are elected that were given low ratings on the ballot in question, while simultaneously using a simple "one weight per ballot" system instead of splitting ballots up like SPSV does. In my opinion this makes it a candidate for best Thieletype proportional voting method (though I doubt it's the best rated partyagnostic proportional voting method), but I'd like to see if anyone has any major objections or other comments.

RE: Handling nondeterministic tiebreaking in voting criteria
Ok, I think I figured out how to get around the keybased version's issue. The problem is that the key is essentially exposing the voting method's internal randomness mechanism to the criterion rather than treating it as a black box. So to get around that issue, we can require that, for some nondeterministic method m, there exist a function f(k, e) that passes the keybased version of the criterion such that for all elections e and candidates c, P(f(k, e) = c) = P(m(e) = c). That way we can isolate the randomness of f to one variable just as before while still treating the voting method itself as a black box.
@Marylander do you see anything I'm missing here?

RE: Handling nondeterministic tiebreaking in voting criteria
Alright, I've determined that this approach would not behave exactly like Marylander's version. Specifically, it's possible to have two voting methods with the same probability of electing each list of candidates in every election and yet have only one of them pass the random key version. As a concrete example, here are two implementations of breaking approval voting ties uniformly at random:
 break an nway tie by picking the ith candidate, where i = key mod n
 break an nway tie by picking the ith candidate, where i = key + (total number of approvals) mod n
Here is a simple example election:
1: approves A and B
And here is a pair of cancelling ballots:
1: approves A and C
1: approves BFor the first tiebreaking implementation, i stays the same when the cancelling pair is added, so the tie is broken the same way and no violation of sequential cancellation occurs. For the second tiebreaking implementation, i is flipped by the addition of the cancelling ballots because the total number of approvals increases by 3. Thus, the tie is broken in favor of the opposite candidate and sequential cancellation is violated. This behavior generalizes such that the first implementation passes keybased sequential cancellation while the second does not.
This is unfortunate because one of the intended advantages of the sequential cancellation criterion is that if two voting methods always produce the same results, either both will pass it or both will fail it. This is already weakened a little by requiring the order in which candidates are elected to remain the same, and under the random key implementation it would need to be weakened more by requiring the candidates elected to remain the same for every key.
As of now, I still prefer the keybased version to the fully probabilistic version, but this seems like a major downside. I'm not sure if there is any way to combine the advantages of both approaches, but if there is a way it would be very helpful.

RE: Handling nondeterministic tiebreaking in voting criteria
Coming back to this, I'm wondering if a better approach would be to model nondeterministic voting methods as functions that take a random key as input in addition to the ballots, similar to the way that pseudorandom functions are handled. Then if the deterministic sequential cancellation criterion is passed for every possible key, we can say that the nondeterministic method passes sequential cancellation.
If I'm correct, this version of the criterion is equivalent to Marylander's version but isolates the randomness to a single variable (the key) in a way that allows it to be ignored for the most part. I believe this makes the criterion easier to reason about, and this feels like the approach that I was searching for when I started this thread. Is there any reason to avoid it?

RE: Handling nondeterministic tiebreaking in voting criteria
@Marylander Thank you for this! This is exactly the criterion I was picturing for handling nondeterministic methods. I have decided I'm going to keep the deterministic version of the criterion as the default, but this is definitely worth mentioning as an extension.