Recent paper about voting in non-deterministic voting systems: https://www.mdpi.com/2409-9287/9/4/107 (open access)

@toby-pereira Yes, this was about what "ensures a win" aka what guarantees it. This post didn't really go into the details on which threshold is necessary since it was more about comparing step functions and non-step functions, and what that entails for voters. However, since you're curious I'll try to explain why you need ⅔:
Assume you control X% of the votes, want to make sure that option A wins, you do not know how the other 100–X% will vote, and there are K many options overall. No matter how you vote, the average rank of A within your X% of votes will be at least 1, and there will be some option B other than A that has an average rank of at most K/2 + 1 within your X% votes. The other 100–X% of voters might provide A with a rank of K and B with a rank of 1, in which case the overall average rank of A will be at least X% · 1 + (100 − X%)·K = K − X% · (K − 1) and that of B will be at most X% · (K/2 + 1) + (100 − X%) · 1 = 1 + X% · K/2. So, to guarantee that the latter is larger than the former, 1 + X% · K/2 > K − X% · (K − 1), you need to have X% > (K − 1)/(3K/2 − 1) ≈ K/(3K/2) = ⅔.

@cfrank Yeah, I don't see nondeterministic systems being used for national elections anytime soon, but luckily these arguments also work for smaller/local procedures and there they can be implemented much more easily. I think we could make these systems robust against rigging; what will be much harder is making them robust against accusations of rigging. In this polarized political landscape where conspiracies are flying left, right and center, it would be trivially easy for a demagogue to claim the result of a nondeterministic election was rigged. While nondeterministic systems may be more fair, they could appear to be less fair, which is unfortunate since we need social trust to keep a democracy in place.

@toby-pereira See the "effective power" more as something over time. So for the random ballot, with a guaranteed 51% of the vote you'd be in power 51% of the time, but with a guaranteed 51% of the vote in a majoritarian system, you'd be in power 100% of the time. I'd gestured at this by saying "if you have 51% of the vote you are in power 100% of the time", but I'll make this all a bit clearer if I use this graph again in the future.

@toby-pereira See the "effective power" more as something over time. So for the random ballot, with a guaranteed 51% of the vote you'd be in power 51% of the time, but with a guaranteed 51% of the vote in a majoritarian system, you'd be in power 100% of the time. I'd gestured at this by saying "if you have 51% of the vote you are in power 100% of the time", but I'll make this all a bit clearer if I use this graph again in the future.

@toby-pereira Yes, this was about what "ensures a win" aka what guarantees it. This post didn't really go into the details on which threshold is necessary since it was more about comparing step functions and non-step functions, and what that entails for voters. However, since you're curious I'll try to explain why you need ⅔:
Assume you control X% of the votes, want to make sure that option A wins, you do not know how the other 100–X% will vote, and there are K many options overall. No matter how you vote, the average rank of A within your X% of votes will be at least 1, and there will be some option B other than A that has an average rank of at most K/2 + 1 within your X% votes. The other 100–X% of voters might provide A with a rank of K and B with a rank of 1, in which case the overall average rank of A will be at least X% · 1 + (100 − X%)·K = K − X% · (K − 1) and that of B will be at most X% · (K/2 + 1) + (100 − X%) · 1 = 1 + X% · K/2. So, to guarantee that the latter is larger than the former, 1 + X% · K/2 > K − X% · (K − 1), you need to have X% > (K − 1)/(3K/2 − 1) ≈ K/(3K/2) = ⅔.

Looking at Jason Brennan's "Polluting the polls" argument in the context of probabilistic voting
https://bobjacobs.substack.com/p/should-we-abstain-from-voting-in

@cfrank Yeah, I don't see nondeterministic systems being used for national elections anytime soon, but luckily these arguments also work for smaller/local procedures and there they can be implemented much more easily. I think we could make these systems robust against rigging; what will be much harder is making them robust against accusations of rigging. In this polarized political landscape where conspiracies are flying left, right and center, it would be trivially easy for a demagogue to claim the result of a nondeterministic election was rigged. While nondeterministic systems may be more fair, they could appear to be less fair, which is unfortunate since we need social trust to keep a democracy in place.

Recent paper about voting in non-deterministic voting systems: https://www.mdpi.com/2409-9287/9/4/107 (open access)