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.
Best posts made by BTernaryTau

Hello!
Latest posts made by BTernaryTau

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.

Handling nondeterministic tiebreaking in voting criteria
I'm trying to figure out what the best way to handle ties would be for the sequential cancellation criterion. For the cancellation criterion, I was able to get around this issue by treating voting methods as functions without worrying about the specifics of what those functions output. Thus, voting methods that break ties randomly could simply output something like a weighted collection of candidates, and the cancellation criterion would be satisfied if that output stayed the same, even if the nondeterministic outcome did not. However, the sequential cancellation criterion has to set a specific output format since it cares about whether an individual candidate is elected at the same time across multiple elections, so that approach isn't an option.
My initial thought is to have the voting methods output a weighted collection of lists, then require that the individual lists satisfy a modified version of the sequential cancellation requirements. But this adds a lot of complexity to an already complex criterion. Another option would be to just restrict the criterion to deterministic methods and have those methods handle ties by violating neutrality (e.g. breaking ties by alphabetical order). Are there any better ways to handle this issue?

RE: Majority Judgment
It's easy to show that MJ fails the opposite cancellation criterion.
2: A/Excellent, B/Very Good, C/Reject
1: A/Good, B/Very Good, C/RejectMedians: A/Excellent, B/Very Good, C/Reject
A is electedWe can add a pair of opposite ballots to change this result.
2: A/Excellent, B/Very Good, C/Reject
1: A/Good, B/Very Good, C/Reject
1: A/Good, B/Excellent, C/Poor
1: A/Acceptable, B/Reject, C/Very GoodMedians: A/Good, B/Very Good, C/Reject
B is electedDemonstrating that MJ fails the cancellation criterion is a bit more difficult since we must consider all possible cancelling ballots for A/Good, B/Excellent, C/Poor.
2: A/Excellent, B/Very Good, C/Reject
1: A/Good, B/Very Good, C/Reject
1: A/Good, B/Excellent, C/Poor
1: A/?, B/?, C/?Medians: A/?, B/Very Good, C/Reject
Luckily it's possible to construct an election where B and C's medians are independent of the final ballot (as I did here), so we only need to consider the possible ratings for A. An Excellent rating will lead to A keeping their median rating of Excellent and winning, but no other rating will. A Very Good rating creates a tie between A and B which is broken in B's favor, and anything lower leads to B winning as well.
Now all that remains is to find another election in which A/Good, B/Excellent, C/Poor cannot possibly be cancelled out by a ballot with an Excellent rating for A.
2: A/Poor, B/Acceptable, C/Reject
1: A/Good, B/Acceptable, C/RejectMedians: A/Poor, B/Acceptable, C/Reject
B is elected2: A/Poor, B/Acceptable, C/Reject
1: A/Good, B/Acceptable, C/Reject
1: A/Good, B/Excellent, C/Poor
1: A/Excellent, B/?, C/?Medians: A/Good, B/Acceptable, C/Reject
A is electedSo MJ fails both of these formalizations of Frohnmayer balance.

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.

RE: Too Dependent on One Person
@JackWaugh How involved would that be? I'm interested but I may not have the time, depending on how long it would take to learn everything and how urgent it is.

RE: Too Dependent on One Person
I'm familiar with basic SSHing if you still need someone.