Posts made by cfrank

@toby-pereira your interpretation is correct, and yes with an even number of voters and non-strict rankings, ties can occur and that can induce non-uniqueness. For example there may be two separate, disconnected Condorcet cycles of different sizes in the Smith set for instance. The proof of uniqueness in general is probably less straightforward than just the intuition, I haven’t dug into it.

With ties in other methods, the resolution of ties is typically standard because the set over which the ties occur is discrete—uniformly sample one from among the tied candidates. But for maximal lotteries, when non-uniqueness holds there is a continuum of admissible lotteries as you indicated. The analog of a uniform distribution in this case would be using Jeffrey’s prior, which is why I think that’s the “right” way to go.

But yes it ultimately doesn’t really matter how the maximal lottery used is chosen since they are all maximal, but that’s also kind of the issue, because a choice has to be made.

Maximal lotteries are guaranteed to be unique when rankings are strict and there is an odd number of voters. Uniqueness can hold in other cases, and generally non-uniqueness only occurs when the majority margin matrix exhibits "pathological" and non-generic symmetries. It follows that any random perturbation of the majority margin matrix almost certainly yields uniqueness.

This post is just a space to discuss how this non-uniqueness should be handled in a fair way when it arises. Here is an example of one idea (not novel):

1. If the maximal lottery is unique, great.
2. Otherwise, if we allowed non-strict rankings, induce strict rankings on each ballot's indifferences independently at random. (This raises questions about ballot format and implementation).
3. If the maximal lottery is still not unique, then the number of counted ballots is even. Produce one additional ballot, such as by randomly sampling one late ballot that was not yet counted (if one exists), by constructing a ballot from those already submitted (randomly sampled or a distributed construction), or by blatant authority (or even just a completely random ranking).

This will guarantee uniqueness of the maximal lottery. However, it may no longer satisfy the formal properties of maximal lotteries---as in, the achieved unique lottery may not actually be maximal for the ex-ante majority margin matrix (unless we use the latecomer ballot).

Alternatively, we can use a rule to select a maximal lottery from the admissible set, such as the maximum entropy maximal lottery, or sampling the maximal lottery from the Jeffrey's prior over maximal lotteries. My personal opinion is that the Jeffrey's prior makes the most sense.

As always, any thoughts are welcome.

@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’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.

@toby-pereira hopefully it’s interesting. I had another thought which might be something actually workable: Say there is a first provisional winner by lottery. That candidate then has the opportunity to secure their victory by offering concessions to the offended majorities within a fixed time horizon. After that time horizon, if an agreement is not reached, then another lottery is run, and that second candidate will be declared the final winner. To ratify concessions from candidate A, each majority bloc preferring some B to A with margin M(B,A) would need to provide approval of size at least ceil(M(B,A)/2) after the concession contract is made public. Or something like that.

@Toby-Pereira this seems in line with compensation, I haven’t read in full yet; it seems complex: https://econtheory.org/ojs/index.php/te/article/viewFile/5380/39121/1195

EDIT: It isn't quite what I thought it was, but is still interesting. And thinking more about the idea of compensation, I think it’s probably impossible while also preserving fair majoritarianism, since compensation in the form of increased future electoral influence becomes strategically relevant information. Probably, compensation would have to be outside of the electoral process, and that gets into problems of money and power in politics.

@toby-pereira yes monotonicity fails, but I think pairwise probability margins are apparently monotonic.

I think maximal lotteries are interesting too, and I wonder if it makes sense to compensate the dissatisfied majorities with electoral credit to spend in subsequent election(s).

For example, say the maximal lottery (or maybe any other Condorcet method) elects A when there is no Condorcet winner. Let beat(X) be the set of candidates that beat X in a head-to-head majority.

Then for each B in beat(A), it seems like the majority preferring B to A should be compensated in some way. Coming up with any way to compensate majorities for electoral grievances that generalizes across time and is game-theoretically stable might get complicated. There may already be literature on this kind of thing.

The ideal would be for the electoral process to be history dependent in cases of Condorcet cycles, but still preserve long-run maximal lottery distribution properties (somehow). Maybe the maximal lottery is the only way to go… the worry is that majorities with instantaneous grievances might not internalize future lotteries as granting them sufficient compensation.

I just learned about this now. It’s a Smith compliant method, and is participation compliant up until necessary randomization.

https://en.wikipedia.org/wiki/Maximal_lotteries

Basically, you form the majority margin matrix Mij over candidates. Then consider the two player game where each player chooses one candidate: if player 1 chooses i, and player 2 chooses j, then player 1 gets Mij and player 2 gets -Mij.

This game has at least one mixed strategy Nash equilibrium, which is a distribution over candidates. A maximal lottery selects a candidate at random according to one such mixed strategy Nash equilibrium. In this sense, a maximal lottery is a mixed “candidate” that cannot be challenged by a majority.

Occasionally maximal lotteries are non-unique, but they often are. A trivial case is when a Condorcet winner exists, in which case the maximal lottery selects the Condorcet winner as a pure strategy. Orthogonally, if there is an odd number of voters, and if preferences are strict, the maximal lottery is always unique. Otherwise, symmetries like multiple dominance components in the Smith set can induce degeneracies.

Still, any maximal lottery satisfies probabilistic notions of participation, where participation cannot reduce the chances of a preferred outcome for a voter.

I think this is nearly the “right” notion of compliance. There is only slight room in choosing which maximal lottery to implement. For example, in case of degeneracy, one could choose a maximal lottery that optimizes expected score. Or, maybe compute Jeffrey’s prior over maximal lotteries, and then sample a maximal lottery accordingly.

What thoughts do others have?

@cfrank I revisited this concept, and I'm quite sure now that participation will not hold even with careful adjustments to the method. Several simple counterexamples were discovered involving the generation of a top Condorcet cycle. C'est la vie.

@kaptain5 said in I'm designing an experiment on voting systems, what would you like to see?:

My method will have human participants make the decisions for the voting blocks instead of attempting to simulate what a selfish imperfect-utility-maximizing person might do. Participants will be screened for competitive people trying to win.

Interesting for sure. It makes sense you would not want to share the full methodology if this is original research. I do have questions, I'm curious primarily about the scope of the simulation, because my observation is that often the routes to exploitation of systems are discovered by operating outside of or on the boundaries of their intended framing. For example, some questions would be, how are participants encouraged to be competitive? Will they engage in (quasi) long-run participation so that they can learn from the strategies of others? Which voting systems are you considering to test, and how large scale would this study be in terms of the number of participants?

I personally think key systems to include would be vote-for-one (obviously), approval, score, STAR, IRV, and a Condorcet method (I would suggest bottom-2-runoff). I would say it's important to keep in mind limitations of these, especially in terms of independence of clones.

I am not wholly familiar with the findings in this field, but I think it would be very helpful to see how coalitions emerge in different voting settings.
This is probably well outside the scope of what you want to investigate, but I personally would be happy to see this kind of system simulated:
https://www.votingtheory.org/forum/topic/299/pr-with-ambassador-quotas-and-cake-cutting-incentives/6

So IMO some basic questions would be about the emergence of coalitions, investigating the stability of multi-party coalitions versus duopolistic structures. Resistance to outsized influence of radical minority factions would also be good to test if possible.

In this realm, a question myself and @SaraWolk found interesting is whether "over time the two opposing factions [naturally] become more and more polarized." (see https://www.votingtheory.org/forum/topic/586/is-duopoly-more-resistant-to-fascism)

@SaraWolk I agree with you. When I say majoritarian, I only mean in the sense of Lijphart’s ontology, because you’re right that ultimately it isn’t even actually majoritarian but largely an illusion of it.

I’m curious about examples of this: “the center-squeeze effect ensures that over time the two opposing factions become more and more polarized.” I feel you know more than I do in this area, but it seems possible to me that there could be a stable polarization that doesn’t necessarily explode. In principle, as long as there is a large enough population of centrists, if one party leans too far in one direction, naively I would imagine the other could gain more power by appealing to those centrists than by appealing to the fringes.

Naivety aside, I think you’re probably right. The problem is that even if there is a population of centrists, if the representatives aren’t held accountable to them, then the parties themselves seem to have no good reason not to polarize once they’ve duopolized the political market. I’m really just curious about what causes that—is polarization actually steadily preferred? Or is it just a matter of time and drift before one party tips over the edge? It seems like the opposite of Hotelling’s law.

I also don’t think multiparty systems or PR for instance alone would solve the problem, there are examples of both systems falling to authoritarianism, and resistance depends on many contingent factors that ultimately bring about your main point, which is preventing extremists from gaining leverage or control.

Hopefully we can get some technical voting reform and see whether things change. Approval would be great.

This might be kind of anathema to our movement… but I think it’s a really important question to ask. While a duopoly exploits voters, it also establishes a pre-assembled, entrenched, large opposition faction to fascism, which might be diluted in a multi-party government.

I think this is something to investigate in terms of historical evidence, and also something to keep in mind during reform efforts. I wonder whether, theoretically, it would be better to keep a duopolistic structure of government in the House and Senate, but also enable a third house with multi-party deliberations. This is obviously just day-dreaming.

I used to consider that multiparty democracy would dilute authoritarian movements from the bottom, but right now the answer isn’t clear to me. I guess we’re seeing one “case study” play out in real time…

Any thoughts welcome. Thanks!

FOLLOW UP:

Based on my research, important factors for resisting fascism external to the style of democracy are horizontal and vertical separations of power (ex: checks and balances and federalism), the strength of democratic norms, and possibly also the priming of democratic resistance by salient failures of other states.

Commonly, authoritarians will do away with multiparty consensus structures and replace them with majoritarian systems (which are more efficient, especially when they’re only for show). In terms of fascism, the main route in majoritarian democracies with duopolistic structure is capture of one major party by a radical faction (as we see in the U.S. now), and while the opposition faction is likely to be large and organized, there’s also the concerning fact that there is essentially no other horizontal obstacle to the fascist movement. When this resistance fails, what remains is vertical opposition, e.g. federalism, or external opposition (other states).

The issue with multiparty consensus democracies is that they can be too fluid and fail to offer resistance to organized radicals unless coalitions are primed and strong, and can be less efficient than majoritarian governments (which don’t have to deliberate as much).

I think a “dual-phase” system would be most resilient if set up properly, because strong, fast, pre-existing opposition could be paired with a more slowly deliberating but broader coalition—basically, the large opposition party can act as a stopper to give time while a more overwhelming consensus forms.

@toby-pereira I agree with this. Something in that spirit I am considering is that the power allocations can still be traced back to ballots. For example, if the seated representatives and powers were {A:45, B:35, C:20}, in principle, those single seats could be subdivided into multiple seats of roughly equal power, depending on the candidate pool (i.e. how many candidates are available).

Possibly, a sub-election could be run to determine the representatives within the A:45 group, etc. Maybe they could be given 4 seats, the B:35 group 3 seats, and C:20 2 seats. That could refine representation, some candidates might pick up multiple seats. It’s probably getting messy and complicated, it essentially becomes a hierarchical partitioning of ballots. I’m not sure what to make of that prospect, it starts looking like a network/phylogenetic tree or forest architecture, and that can become arbitrary fast.

I do see what you mean. An individual voter may actually prefer a particular coalition of candidates, rather than just want to get their top guy in. I wonder if non-strict rankings and distributed power would mitigate this issue, or for instance, if the A:45 group's sub-election guaranteed a seat for A, and ran the election on the remaining candidates, that might align with the spirit of preference for whole coalitions.

You definitely are more familiar with this space than I am, so I wager some of my objections may be non-issues when one considers alternative PR methods. But it seems to me that in this case, pushing for coalitions that respect single individual preferences for whole coalitions can lean toward reduced diversity and reduced minority representation. Is that inaccurate? Or is that a common tradeoff issue in PR systems?

“Would the weighting purely count towards their voting power in the elected body, or does it have other effects such as more time to speak?”

Yeah, it does beg some questions.

EDIT: After multiple adjustments made to guard against clone dependence and tactical voting, there is a non-monotonicity issue in my latest version (/branch, it is not my original concept so I don’t claim ownership in any way), where a minority faction can gain strictly preferred representation in the form of a seated candidate by merely withholding approval for that candidate. I have a concrete example of this, and may try to see how to address it. It may be due to something unnecessary.