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This post does not prove Maximal Lottery Lotteries exist. Instead, it redefines MLL to be equivalent to the Nash bargaining solution (in a way that is obscured by using the same language as the MLL proposal), and then claims that under the new definition MLL exist (because the Nash bargaining solution exists).

I like Nash bargaining, and I don't like majoritarianism, but the MLL proposal is supposed to be a steelman of majoritarianism, and Nash bargaining is not only not MLL, but it is not even majoritarian. (If a majority of voters have the same favorite candidate, this is not sufficient to make this candidate win in the Nash bargaining solution.)

Dang. I wasn't entirely sure whether you were firm on the definition of lottery-lottery dominance or if that was more speculative. I guess I wasn't clear that MLLs were specifically meant to be "majoritarianism but better"? Given that you meant for it to be, this post sure doesn't prove that they exist. You're absolutely right that you can cook up electorates where the majority-favored candidate isn't the Nash bargaining/Geometric MLL favored candidate.

Hello! I'm glad to read more material on this subject.

First I want to note that it took me some time to understand the setup, since you're working with a modified notion of maximal lottery-lotteries than the one Scott wrote about. And this made it unclear to me what was going on until I'd read a bunch through and put it together, and changes the meaning of the post's title as well.

For that reason I'd like to recommend adding something like "Geometric" in your title. Perhaps we can then talk about this construction as "Geometric Maximal Lottery-Lotteries", or "Maximal Geometric Lottery-Lotteries"? Whichever seems better!

It seems especially important to distinguish names because these seem to behave distinctly than the linear version. (As they have different properties in how they treat the voters, and perhaps fewer or different difficulties in existence, stability, and effective computation.)

With that out of the way, I'm a tentative fan of the geometric version, though I have more to unpack about what it means. I'll divide my thoughts & questions into a few sections below. I am likely confused on several points. And my apologies if my writing is unclear, please ask followup questions where interesting!

Underlying models of power for majoritarian vs geometric

When reading the earlier sequence I was struck by how unwieldy the linear/majoritarian formulation ends up being! Specifically, it seemed that the full maximal-lottery-lottery would need to encode all of the competing coordination cliques in the outer lottery, but then these are unstable to small perturbations that shift coordination options from below-majority to above-majority. And this seemed like a real obstacle in effectively computing approximations, and if I undertand correctly is causing the discontinuity that breaks the Nash-equilibria-based existence proof.

My thought then about what might more sense was a model of "war"/"power" in which votes against directly cancel out votes for. So in the case of an even split we get zero utility rather than whatever the majority's utility would be. My hope was that this was both a more realistic model of how power should work, which would also be stable to small perturbations and lend more weight to outcomes preferred by supermajorities. I never cached this out fully though, since I didn't find an elegant justification and lost interest.

So I haven't thought this part through much (yet), but your model here in which we are taking a geometric expectation, seems like we are in a bargaining regime that's downstream of each voter having the ability to torpedo the whole process in favor of some zero point. And I'd conjecture that if power works like this, then thinking through fairness considerations and such we end up with the bargaining approach. I'm interested if you have a take here.

Utility specifications and zero points

I was also a big fan of the full personal utility information being relevant, since it seems that choosing the "right" outcome should take full preferences about tradeoffs into account, not just the ordering of the outcomes. It was also important to the majoritarian model of power that the scheme was invariant to (affine) changes in utility descriptions (since all that matters to it is where the votes come down).

Thinking about what's happened with the geometric expectation, I'm wondering how I should view the input utilities. Specifically, the geometric expectation is very sensitive to points assigned zero-utility by any part of the voting measure. So we will never see probability 1 assigned to an outcome that has any voting-measure on zero utility (assuming said voting-measure assigns non-zero utility to another option).

We can at least offer say  probability on the most preferred options across the voting measure, which ameloriates this.

But then I still have some questions about how I should think about the input utilities, how sensitive the scheme is to those, can I imagine it being gameable if voters are making the utility specifications, and etc.

Why lottery-lotteries rather than just lotteries

The original sequence justified lottery-lotteries with a (compelling-to-me) example about leadership vs anarchy, in which the maximal lottery cannot encode the necessary negotiating structure to find the decent outcome, but the maximal lottery-lottery could!

This coupled with the full preference-spec being relevant (i.e. taking into account what probabilistic tradeoffs each voter would be interested in) sold me pretty well on lottery-lotteries being the thing.

It seemed important then that there was something different happening on the outer and inner levels of lottery. Specifically when checking for dominance with , we would check . This is doing a majority check on the outside, and compares lotteries via an average (i.e. expected utility) on the inside.

Is there a similar two-level structure going on in this post? It seemed that your updated dominance criterion is taking an outer geometric expectation but then double-sampling through both layers of the lottery-lottery, so I'm unclear that this adds any strength beyond a single-layer "geometric maximal lottery".

(And I haven't tried to work through e.g. the anarchy example yet, to check if the two layers are still doing work, but perhaps you have and could illustrate?)

So yeah I was expecting to see something different in the geometric version of the condition that would still look "two-layer", and perhaps I'm failing to parse it properly. (Or indeed I might be missing something you already wrote later in the post!) In any case I'd appreciate a natural language description of the process of comparing two lottery-lotteries.

Thinking about what's happened with the geometric expectation, I'm wondering how I should view the input utilities. Specifically, the geometric expectation is very sensitive to points assigned zero-utility by any part of the voting measure.

This comment may be relevant here.

I gave a short and unpolished response privately.

I haven't engaged with this in enough detail, but some people who engaged with Scott's sequence who I can imagine being interested in this: @Scott Garrabrant , @James Payor, @Quinn, @Nathan Helm-Burger, @paulfchristiano

dedicated to a very dear cat;


Fraktur is only ever used for the candidate set  and the dregs set . I would also have used it for the Smith set , but \frak{S} is famously bad. I thought it was a G for years until grad school because it used to be standard for the symmetry group on n letters. Seriously, just look at it: .

Typography is a science and if it were better regarded perhaps mathematicians would not be in the bind they are these days :P

It suffices to show that the Smith lotteries that the above result establishes are the only lotteries that can be part of maximal lottery-lotteries are also subject to the partition-of-unity condition.

I fail to understand this sentence. Here are some questions about this sentence:

  • what are Smith lotteries? Ctrl+f only finds lottery-Smith lottery-lotteries, do you mean these? Or do you mean lotteries that are smith?

  • which result do you mean by "above result"?

  • What does it mean for a lottery to be part of maximal lottery-lotteries?

  • does "also subject to the partition-of-unity" refer to the smith lotteries or to the lotteries that are part of maximal lottery-lotteries? (it also feels like there is a word missing somewhere)

  • Why would this suffice?

  • Is this part also supposed to imply the existence of maximal lottery-lotteries? If so, why?

To avoid confusion: this post and my reply to it were also on a past version of this post; that version lacked any investigation of dominance criterion desiderata for lottery-lotteries.

what are Smith lotteries?

Lotteries over the Smith set. That definitely wasn't clear - I'll fix that.


which result do you mean by "above result"?

  • Proposition: (Lottery-lotteries are strongly characterized by their selectivity of partitions of unity) 

 This one. "You can tell whether a lottery-lottery is maximal or not by how good the partitions of unity it admits are." Sorry, didn't really know a good way to link to myself internally and I forgot to number the various statements.


What does it mean for a lottery to be part of maximal lottery-lotteries?

Just that some maximal lottery-lottery gives it nonzero probability.


does "also subject to the partition-of-unity" refer to the smith lotteries or to the lotteries that are part of maximal lottery-lotteries? (it also feels like there is a word missing somewhere)

Oh no! I thought I caught all the typos! That should be "also subject to the partition-of-unity condition", that is, you look at all the lotteries (which we know are over the Smith set, btw) that some arbitrary maximal lottery-lottery gives any nonzero probability to, and you should expect to be able to sort them into groups by what final probability over candidates they induce; those final probabilities over candidates should themselves result in identical geometric-expected utility for the voterbase.


Why would this suffice?

Consider: at this point we know that a maximal lottery-lottery would not just have to be comprised of lottery-Smith lotteries, i.e., lotteries that are in the lottery-Smith set  - but also that they have to be comprised entirely of lotteries over the Smith set of the candidate set. Then on top of that, we know that you can tell which lottery-lotteries are maximal by which partitions of unity they admit (that's the "above result"). Note that by "admit" we mean "some subset of the lotteries this lottery-lottery has support over corresponds to it" (this partition of unity).

This is the slightly complicated part. The game I described has a mixed strategy equilibrium; this will take the form of some probability distribution over . In fact it won't just have one, it'll likely have whole families of them. Much of the time, the lotteries randomized over won't be disjoint - they'll both assign positive probability to some candidate. The key is, the voter doesn't care. As far as a voter's expected utility is concerned, the only thing that matters is the final probability of each candidate.

That's where your collapse of different possible maximal lottery-lotteries to the same partition of unity comes in. Because we know that equivalent candidate-lotteries give voters the same expected utility, the only two ways you get a voter who's indifferent between two candidate-lotteries are 1) they're the same lottery or 2) the voter's utility function is just right to have two very different lotteries tie. Likewise, the only two ways you get a voterbase to be indifferent between two lottery-lotteries is 1) they reduce to the same lottery or 2) the geometric expectation of a voter's utility over candidates sampled from the samples of the lottery-lottery Just Plain Ties.

So: if we can show that whatever equilibrium set of candidate-lotteries Alice and Bob pick always collapses to some convex combination of the Best partitions of unity...? Yeah, I don't think that the second half of the proof holds up as is.

I think I've slightly messed up the definition of lottery-Smith, though not in a fatal way nor (thankfully) in a way that looks to threaten the existence result. The set condition is too strong, in requiring that a lottery-Smith lottery contain all lotteries which correspond to any of the admissible partitions. I'm just going to cut it; it's not actually necessary.


Is this part also supposed to imply the existence of maximal lottery-lotteries? If so, why?


Yes, and in particular, it implies the existence of maximal lottery-lotteries before it even tries to prove that they're also lottery-Smith in the sense I define.

Scott's proof doesn't quite work (as he says there) - it almost works, except for the part where the reward functions for Alice and Bob can quite reasonably be discontinuous. My proof is intended as a patch - the reward functions for Alice and Bob should now be extremely continuous in a way that also corresponds well to "how much better did Alice do at picking a candidate-lottery that V will like than Bob did?".


Hopefully this helped? Reading this is confusing even for me sometimes - the word "lottery/lotteries", which appears 59 times in this comment alone, no longer looks like a real word to me and hasn't since late Wednesday. Your comment was really helpful - I have some editing to do! (update - editing is done.)

strictly weaker

add "than Condorcet" in this sentence since its only implied but not said