I think there are many of us here who prefer some voting system or another over approval voting. I also think there is room for improvement. However, approval voting has a huge advantage in its simplicity and potential for integration into existing infrastructure. This is totally besides the comparisons to make in terms of game theoretical stability with Condorcet methods and expressivity with Score or others.

My thought is that, if we are really going to make progress by consolidating our support behind a single voting system, then realistically, Approval voting fits the bill. That isn’t to say that it should be the final destination for voting reform, but it would absolutely be a major step forward. While IRV is something of a tokenism, Approval would be an actual game changer.

Any thoughts about this are welcome.

This is a minor attempt to modify Condorcet methods in a simple way to become more responsive to broader consensus and supermajority power. It’s sort of like the reverse of STAR and may already be a system that I don’t know the name of. In my opinion, the majority criterion is not necessarily a good thing in itself, since it enables tyrannical majorities to force highly divisive candidates to win elections, which is why I’ve been trying pretty actively to find some way to escape it.

For the moment I will assume that a Condorcet winner exists in every relevant case, and otherwise defer the replacement to another system.

First, find the Condorcet winner, which will be called the “primary” Condorcet winner. Next, find the “secondary” Condorcet winner, which is the Condorcet winner from the same ballots where the primary Condorcet winner is removed everywhere.

Define the Borda difference from B to A on a ballot as the signed difference in their ranks. For example, the Borda difference from B to A on the ballot A>B>C>D is +1, and on C>B>D>A is -2.

If A and B are the primary and secondary Condorcet winners, respectively, then we tally all of the Borda differences from B to A. If the difference is positive (or above some threshold), then A wins, and if it is negative or zero (or not above the threshold), then B wins.

For example, consider the following election:

A>B>C>D [30%]
A>B>D>C [21%]
C>B>D>A [40%]
D>B>C>A [9%]

In this case, A is a highly divisive majoritarian candidate and is the primary Condorcet winner. B is easily seen to be the secondary Condorcet winner. The net Borda difference from B to A is

(0.3+0.21)-2(0.4+0.09)<0

Therefore B would be chosen as the winner in this case.

Some notes about this method:
It certainly does not satisfy the Condorcet criterion, nor does it satisfy the majority criterion. These are both necessarily sacrificed in an attempt to prevent highly divisive candidates from winning the election. It does reduce to majority rule in the case of two candidates, and it does satisfy the Condorcet loser criterion, as well as monotonicity and is clearly polynomial time. It can also be modified to use some other metric in the runoff based on the ballot-wise Borda differences.

Continuing with the above example, suppose that the divisive majority attempts to bury B, which is the top competitor to A.
This will change the ballots to something like

A>C>D>B [30%]
A>D>C>B [21%]
C>B>D>A [40%]
D>B>C>A [9%]

And if the described mechanism is used in this case, we will find instead that C is elected. So burial has backfired if B is "honestly" preferred over C by the divisive majority, and they would have been better off indicating their honest preference and electing B.

And again, suppose that the divisive majority decides to bury the top two competitors to A, namely B and C, below D, keeping the order of honest preference between them. We will find

A>D>B>C [30%]
A>D>B>C [21%]
C>B>D>A [40%]
D>B>C>A [9%]

In this case, the secondary Condorcet winner is D, and the mechanism will in fact elect D, again a worse outcome for the tactical voters.

Finally, suppose that they swap the order of honest preference and vote as

A>D>C>B [30%]
A>D>C>B [21%]
C>B>D>A [40%]
D>B>C>A [9%]

Still this elects D.

As a general description, this method will elect the Condorcet winner unless they are too divisive, in which case it will elect the secondary Condorcet winner, which will necessarily be less divisive. I believe that choosing the runoff to be between the primary and secondary Condorcet winners should maintain much of the stability of Condorcet methods, while the Borda runoff punishes burial and simultaneously addresses highly divisive candidates.

This is a concept I had in mind which may already have been described, although not all of the logistics are necessarily hashed out and there may be issues with it. The idea is described below, but first I want to make a connection to “cake-cutting.” The standard cake-cutting problem is when two greedy agents are going to try to share a cake fairly without an external arbiter. An elegant solution is a simple procedure where one agent is allowed to cut the cake into two pieces, and the other agent is allowed to choose which piece to take for themselves. The first agent will have incentive to cut the cake as evenly as discernible, since the second agent will try to take whichever piece is larger. In the end, neither agent should have any misgivings about their piece of cake.

So this is my attempt to apply that kind of procedure to political parties and representatives. Forgive my lack of education regarding how political parties work:

• There should be a government body that registers political parties and demands the compliance of all political parties to its procedures in order for them to acquire seats for representation;
• (Eyebrow raising, but you might see why...) Every voter must register as a member of exactly one political party in order to cast a ballot (?);
• Each political party A is initially reserved a number of seats in proportion to the number of voters with membership in A; the fraction of seats reserved for A is P(A). however
• For each pair of political parties A and B (where possibly B=A), a fraction of seats totaling P(A~B):=P(A)P(B) will be reserved for candidates nominated by A, and elected by B; these seats will be called ambassador seats from A to B when B is different from A, and otherwise will be called the main platform seats for A;
• Let there be a support quota Q(A~B) for the number of votes needed to elect ambassadors from A to B, and call P(A~B) the ambassador quota of party A for B. If E(A~B) is the fraction of filled A-to-B ambassador seats (as a fraction of all seats), I.e. nominees from A who are actually elected by members of B, then A will only be allowed to elect P(A~A)*min{min{E(A~B)/P(A~B), E(B~A)/P(B~A)}: B not equal to A} of its own nominees. That is, the proportion of reserved main-platform seats that A will be allowed to fill is the least fraction of reserved ambassador seats it fills in relation to every other party, including both the ambassadors from A to other parties, and the ambassadors from other parties to A.

This procedure forces parties to also nominate candidates that compromise between different party platforms in order to obtain seats for any main-platform representatives. If a party fails to meet its quota for interparty compromises, it will lose representation. On the flip side, this set up will also establish high incentives for other parties to compromise with them in order to secure their own main-platform representation. In total, this system would give parties high incentives to compromise with each other and find candidates in the middle ground, which will serve as intermediaries between their main platforms.

Basically, here the outlines indicate seats open to be filled by candidates who are nominated by the corresponding party, and the fill color indicates seats open for election by the corresponding party:

Seats with outlines and fills of non-matching color are ambassador seats, and seats with matching outline and color are main platform seats. In terms of party A, by failing to nominate sufficiently-many candidates who would meet the support quota Q(A~B) to become elected as ambassadors from A to B, or by failing to elect enough ambassadors from B to A, party A restricts its own main platform representation and that of B simultaneously. By symmetry the reciprocal relationship holds from B to A. Therefore all parties are entangled in a dilemma: to secure main-platform representation, parties must nominate a proportional number of candidates who are acceptable enough to other parties to be elected as ambassadors.

To see that all needed seats are filled in the case of a stalemate, where parties refuse to nominate acceptable candidates to other parties and/or refuse to elect ambassadors, the election can be redone with the proportions being recalculated according to the party seats that were actually filled.

The support quotas collectively serve as a non-compensatory threshold to indicate sufficient levels of inter-party compromise. Ordinary PR is identical to PR with ambassador quotas but with all support quotas set to zero, whereby there is no incentive to nominate compromise candidates.

The purpose of this kind of procedure is twofold: firstly, it should significantly enhance the cognitive diversity of representatives, and secondly, it should significantly strengthen more moderate platforms (namely those of the ambassadors) that can serve as intermediaries for compromises between the main platforms of parties. Every party A has a natural “smooth route” from its main platform to the main platform of every other party: The main platform of A should naturally be in communication with ambassadors from A to B, who should naturally communicate with ambassadors from B to A, who should naturally communicate with the main platform of B.

Also, this procedure gives small parties significant bargaining power in securing representation. Large parties will have much more representation to lose than the small parties that are able to secure seats if the small parties refuse to elect any ambassadors, so rationally speaking, large parties should naturally concede to nominating sufficiently many potential ambassadors whose platforms are closer to the main platforms of those small parties. The same rationale holds for the potential ambassadors nominated by small parties, who also should tend to have platforms closer to the main platform of the small party.

Finally, this system creates significant incentives for voters to learn about the platforms of candidates from other parties who stand to reserve seats for representatives.

@rob especially if the state is a swing state, making it more difficult for the large parties to secure voters for their platform I think would be a significant influence forcing large parties and their candidates to more scrutinizingly determine the real interests of voters in those states. It may dilute the interests of less competitive states, but since the competitive states are crucial to obtaining the presidency, the large parties will still have to invest strongly in the interests of voters in those states in order to compete with alternatives (and obviously each other) for the crucial swing points. This may lead to something like an arms race of concessions, which happened in New Zealand in 1996 and led to the national adoption of a PR system, according to Arend Lijphart. Obviously that's quite a leap for the U.S., but maybe a less extreme analogue is not so far-fetched.

Maine is one of the thirteen most competitive states for elections according to a 2016 analysis (Wikipedia: Swing state), so I’m not sure their recent establishment is actually strategically foolish, although it’s possible that it wasn’t fully thought through. I agree it isn't clear.

I think it will definitely be interesting to observe how the current political apparatus responds to Maine--and apparently, more recently, and strangely, Alaska:

https://news.yahoo.com/alaska-is-about-to-try-something-completely-new-in-the-fall-election-193615285.html

Since Alaska is far from competitive, I do think this transition was in fact foolish for the reasoning you stated, but it remains to be seen. If we saw a state like Florida transition to a system like Maine's, it would be very interesting to study the relative differences between federal treatments of Florida, Maine, and Alaska as a case study for how "swingy-ness" might influence the effect of such voting system transitions. If Maine experiences an increase in federal power, it would be a good case for the remaining swing states to make a similar transition. If that occurred, the swing states would become a platform foothold for alternative parties to grow.

@k98kurz I don’t think there should be any uncertainty in the default for a voter’s ballot.

@lime the point of this post isn’t to argue that approval voting is superior to other methods or that modifications wouldn’t improve approval voting, it’s to point out that despite other methods being potentially superior, standard approval voting is probably the most realistic target for near future steps toward substantially reformed voting.

Unfortunately, more choices does mean the system is more complicated. You can observe that the addition of even a very simple, marginal modification as you suggest already raises questions. Every question about a method is an opportunity for distrust to be exploited, even if the method is ultimately better. Plurality is terrible, but almost nobody had questions about it, and that’s why it’s stuck around for so long. Do you see what I mean? I may be a bit jaded, but I’m hoping to be realistic.

I don’t mean to be a downer, but my point is a bit sad: in terms of what people would prefer, such as more choices or buttons, what we have to deal with is exactly the fact that people are having a hard time getting what they prefer. The political status quo is strongly opposed to voting reform, it will have to relinquish substantial power and accountability to the people under an effective voting system. There’s a reason only flawed tokenisms like IRV have passed through legislature in recent times. In fact, there is a history of voting reforms being enacted and then reversed.

@democrates I meant a Condorcet winner only among the front runners (for example, the candidates with the top K scores, here we are taking K=2). If there is no Condorcet winner among them, then we can choose the top scoring candidate.

In your case, if two candidates have the same second-greatest score, and the three front runners form a Condorcet cycle, then you can use the scores to break the tie. If this was used, then Jill Stein would have won the election.

That isn’t “the correct” solution (there is no such thing), but it is somewhat less arbitrary than flipping a coin or operating by alphabetical order, neither of which has anything to do with relevant information that is readily available on the ballots.

If we were being engineers about choosing a high quality candidate to win the election, we could even compute the distribution of scores, take the candidates whose scores exceed some elbow point, and find the Condorcet winner among those candidates with the top scoring candidate as the backup if no Condorcet winner exists. That’s basically a generalization of STAR with a dynamic front-runner selection method.

There are other ways to proceed. For example, we could remove Condorcet losers, then try to find the Condorcet winner of all remaining candidates, iteratively eliminating the lowest scoring candidate until a Condorcet winner emerges.

@toby-pereira yes you’re right, it was just a thought that occurred to me when I was thinking about how to discourage bullet approvals, but it has irreconcilable flaws that are now apparent.

@k98kurz mirroring @Lime, I think any advantage conferred to one candidate over any other in an election should be granted on an opt in basis. A voter shouldn’t have to opt out from conferring an advantage to a candidate.

@wolftune this is a well-known Condorcet method due in spirit to Tideman and called “Bottom N Runoff” where N=2 (hence “Bottom Two Runoff,” I.e. BTR or B2R). Generally speaking, these methods use some kind of absolute criterion (like least number of first place votes, lowest score, lowest approval, etc.) to decide which “bottom” candidates to subject to an elimination round, eliminates a Condorcet loser among them, and iterates until the desired number of winners remain. You are right, they’re pretty good methods. I like them.

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 small perturbations 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.