Forum Council

@gregw definitely. I think the main persuasive route is in framing the argument. I tried to present the most persuasive argument I could think of for the principle. There might be more persuasive arguments, and there might be persuasive arguments to the contrary that should be considered.

Still, if the principle is accepted, then the technical details required to actually implement it should be acceptable as long as they’re presented well—that gets difficult, because it really is a technical problem whose solution is not immediately obvious to people outside relevant fields (math, computer science, cryptography, economics, etc.).

@gregw I think the most practical way to proceed would be to just compute the full majority margin matrix. The final Mij would need to be the legal object outputted from the counting process.

If there’s a Condorcet winner, you’re done. Otherwise, the randomization process would need to follow a “commit” then “reveal,” where a secret seed is generated and committed to before the resulting lottery is known from multiple independently verifiable sources. The selection can be audited by comparing the seed’s output with the inverse CDF of the published maximal lottery distribution (and proof that it is actually a maximal lottery).

To determine the maximal lotteries from the Smith/Landau restricted majority margin matrix, the maximal lotteries are the mixed strategy Nash equilibria of the game with payoff matrix equal to the majority margin matrix. Since this is a zero-sum game, it follows that the Nash equilibrium is the minimax solution by von Neumann's minimax theorem. This is a feasibility problem that is polynomial-time computable via linear programming as a convex optimization problem. I’ll probably put a small script together that works soon, there are almost certainly existing ones online.

One caveat mentioned is that maximal lotteries aren’t always unique. One could compute the Jeffrey’s prior over the maximal lottery set and sample one accordingly, equivalently that yields a “canonical” choice of maximal lottery.

But basically, it goes (1) pre-commit to an auditable distributed seed with information asymmetry, (2) compute Mij, (3) reveal the winner with an auditable certificate of validity (either by showing they are the Condorcet winner, or that they were genuinely produced by the maximal lottery procedure via the distributed seed in (1)).

@gregw with the caveat that I strongly encourage anyone to correct me: a maximal lottery is the unique kind of single-winner method that satisfies the following properties:

• Condorcet-consistency;

• Reinforcement: if two electorates independently select the same outcome, combining them does not change that outcome; and

• Participation.

That’s the main selling point. If we want reinforcement, Condorcet-consistency, and participation all together in a single-winner method, then maximal lotteries are forced. Without randomization, Condorcet-consistency and participation are already incompatible. But if we allow randomization, then once we require Condorcet-consistency, reinforcement, and participation, not only is it possible, but we actually have no other choice but to use a maximal lottery.

Why that’s true is because of how maximal lotteries are defined—they are exactly the undominated mixed strategies of the majority margin game. That’s the technical/mathematical machinery behind the result, which may not itself be easy to sell per se. But the result is pretty compelling—trust aside, acceptance or rejection becomes primarily a question of whether we need to satisfy those three fairly intuitive properties. If you demand all three, you're forced to reject determinism and to accept maximal lotteries.

Maximal lotteries can’t solve the fact that Condorcet cycles exist, but they do guarantee the strongest possible form of stability compatible with majority rule: no alternative decision rule can be majority-preferred on procedure. Stable preference by majority on outcome is not nominally possible when Condorcet cycles exist. If you want ex-post stability of majority preference when cycles exist, you need supplementary structure that actually changes the decision problem (compensation, bargaining, agenda constraints, etc.).

As a bonus, they also satisfy independence of clones. In fact, if you require independence of clones instead of participation in the list above, the same uniqueness result holds. (With a slight caveat—you need to consider all maximal lotteries over candidates, so you could choose one at random). Importantly as well, in the generic case, the set of maximal lotteries from the majority margin matrix Mij is continuous in its entries. There are abrupt boundaries that can be crossed, but those boundaries have measure zero in the space of all majority margin matrices (they are almost guaranteed not to occur in any real election with many voters).

Lastly, they satisfy the Smith criterion. Even the Landau criterion. Actually, they induce a slightly stronger criterion called the “bipartisan” criterion—the “bipartisan set” is exactly the set of candidates that can attain nonzero probability under a maximal lottery, and it is a (sometimes strict) subset of the Landau set, which itself is a (sometimes strict) subset of the Smith set.

I stress single-winner because designing principled multi-winner extensions of maximal lotteries under comparable axioms remains an active research problem.

This is unbelievably ridiculous on one hand. Although, on the other hand, it also means that plurality rules can be contested in court for failing fairness criteria such as independence of clones.

https://newrepublic.com/post/205290/supreme-court-major-blow-mail-in-voting?utm_sf_cserv_ref=27532535073004573&utm_campaign=SF_TNR&utm_sf_post_ref=655090698&utm_source=Threads&utm_medium=social&media_id=3813670661894575203_63371295673&media_author_id=63371295673&source_quote_media_id=3813869635691710276&utm_source=ig_text_post_permalink

If voting reformers are strategic, they can systematically sue for specific criteria failures, leaving the only options for legal voting rules to fall in a narrow, ideally more preferable category.

One of the most interesting suits could be failure of the Smith or bipartisan criterion. Imagine if that case were won. Imagine also if the Supreme Court, ignorant of voting theory, established contradictory laws Wouldn’t that be wonderful.

For any Smith compliant method, detecting a Condorcet cycle is easy given the method’s winner—you check if the winner is beaten head to head by some other candidate.

BTR is Smith compliant and the winner is fast to compute, therefore it yields very fast cycle detection. Specifically, we can determine existence of a cycle in worst-case linear time O(n) in n the number of candidates. Specifying the cycle is more complex, at worst O(n^2). I don’t think either can be improved. But just in case, what are some other comparably efficient methods to detect cycles?

This forum has gone through ups and downs in activity, but over the years it has generated a lot of content that can be intimidating to parse. @Jack-Waugh (and others? don’t let me leave them out) has done a fantastic job of putting this forum website together, and the council and moderators contribute to keep the site up and active while many users generate great content.

With today’s LLMs, we have an opportunity to consolidate and organize a lot of the information here into a more navigable, unsupervised and context driven resource. This is something I would like to work on at some point in the future, but I am also quite busy with my own responsibilities outside of this forum.

I’m just putting this out there to get other minds thinking on the subject. I may simply not have spent enough time with the forum interface to make a clear assessment of navigability, but my impression is that as it stands, one mostly has to know what they’re looking for already to find it, and even the content within our broad categories has become fairly diverse. Is this something others agree with? Otherwise, how do you navigate content? Do you simply keep up with the latest topics in the forum and recall connections to prior discussions by memory? Or is there a system you use? Lastly, is this a non-issue?

I’m noticing that tags are underused (I am definitely guilty of this), and are only visible through mobile in landscape mode. That may be part of the problem.

I’d be pleased to hear any thoughts on this. Thank you!

@gregw BTR/score (or BTR/approval) is an excellent system, although it is not stable ex ante under majority preference—only a maximal lottery is. Maximal lotteries also satisfy participation and Condorcet (they can do that because they are inherently non-deterministic in the case of Condorcet cycles—those properties are incompatible for deterministic methods).

BTR was invented by Nicolaus Tideman. He is still around, I don’t know how accessible he is but he is certainly involved in voting theory. Sorting/tie-breaking by score or approval in BTR is an obvious extension.

For primaries, a specified multi-winner method is needed, which could be a “natural” extension of a single-winner method. Peeling winners of BTR off recursively is one option, although it would be more stable to prioritize candidates in the Smith/bipartisan set (the naive recursion can lead to results that violate multi-winner Smith compliance). A PR/multi-winner method would probably be theoretically preferable, maybe some others more versed in multi-winner methods can comment on options for simplicity.

Something on my mind, for any Condorcet method, even a maximal lottery, being intrinsically stable under majority preference after a winner is chosen is simply impossible with Condorcet cycles. My thinking lately is, this implies that stability requires a supplementary mechanism that compensates dissatisfied majorities in the event of Condorcet cycles, specifically to the extent that majority grievances are sufficiently reduced. However, I don’t know what that mechanism ought to be or how it ought to be enforced, and serious consideration of that enters the interface between technical voting theory, real politics, and law. I mused about that here: https://www.votingtheory.org/forum/topic/591/maximal-lotteries/8

Just food for thought. I’m glad your reform efforts are picking up steam!

P.S.: While I do like Condorcet methods, my opinion is that realistic and highly impactful reforms would be easiest to implement by pushing for approval voting. We discussed that point here as well and it seems to have broad agreement, but obviously that’s just my personal interpretation and some disagree for their own reasons: https://www.votingtheory.org/forum/topic/495/approval-voting-as-a-workable-compromise/20?_=1768708850097

Ultimate approval would still require a multi-winner primary system.

@toby-pereira yes and that’s interesting in itself. I thought it would be about compensating disaffected majorities since that’s what the Chatbot said lol

@cfrank I've had a short look at it. The main conclusion seems to be that you can approximate maximal lotteries with balls in urns!

@cfrank I’m bringing this topic up again, because it seems necessary to consider the implications of how alternative systems at lower levels of government translate effects upward, especially when higher levels maintain a winner-take-all style.

For instance, in a presidential election, say we adopt maximal lotteries (Condorcet compliant) to generate social choice rankings per state. For this to translate nicely to the federal level and accommodate the electoral college, the natural extension seems to be another maximal lottery where voters are states casting ranked choice ballots with electoral college weightings.

I think this could be a viable system in principle, but the question is about how feasible the structural and institutional changes would be to make.

The same sort of question comes up with approval voting. Essentially, my worry is that without upper-level criteria met, lower-level changes, while still locally impactful and potentially inducing long run changes, would not translate effectively upward in the short term, and may even destabilize upper levels of government.