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.

@toby-pereira apparently so, because they left. But honestly in terms of the purpose of a forum, that doesn’t really subtract from anything.

Anyway, this original post was made well before RFK Jr.’s (imo reluctant) alignment with Trump. At least one of RFK Jr.’s predictions was correct, namely that Biden and/or Harris would not beat Trump. His “no spoiler” pledge would have given beating Trump the greatest possible chance, but Democrats refused to cooperate because they are power hungry, greedy, and benefit too much from the duopoly to concede to a third party candidate, even at the cost of Trump winning.

IMO, that’s primarily why RFK Jr. angled against them, in game theory terms it was as a punishment. It was a textbook failed prisoner’s dilemma, and they got a taste of their own medicine in a way that hurt everybody and could have been avoided. But I digress.

@Isocratia I mean, maybe. But if you bail from a conversation just because people are discussing ideas that don’t neatly align with your views, I think that kind of runs counter to productive discussion. Engagement is the whole point of a forum. Why not take the opportunity to make your case? On that point, I don’t think I was being dogmatic, I was just putting a moderate, measured perspective out here. In particular, that if a candidate has comparable support to what Nader did, he should also be on the debate stage.

As for Kennedy, I’m not sure if you followed his campaign directly, but from what I saw, his platform had some surprisingly rational moments. Whatever mistaken views he holds about healthcare, his core message was about dismantling corporate capture of government—which, let’s be honest, is exactly the route that’s brought us to the brink of fascism today. Frankly, he seemed more committed to stopping Trump than the Democrats did.

Like him or not, he was a third-party candidate who genuinely threatened to shake up the duopoly—something we haven’t really seen since Nader. And given how deeply dysfunctional the two-party system has become, that’s not nothing. The political landscape is a real-time disaster, and reform doesn’t just happen on its own. While it wasn’t his main agenda, one thing I appreciated about his run is that he was literally the only candidate to talk about ranked-choice voting and other technical fixes.

That said, I completely checked out when he aligned himself with Trump. At that point, his platform basically collapsed. His current sellout stance disillusioned a lot of his supporters—and honestly, he should’ve just bowed out once it was clear he couldn’t win.

If you see it differently, I’d be interested to hear why. That’s why I brought this topic up in the first place.

@toby-pereira yes definitely. I started trying to actually prove participation last evening, and it got much hairier than I would have liked... lots of branching edge-case conditions. I think an actual proof (or counterexample) of participation for this system would require some nice insights, and/or a larger scale planning and brute-force organized accounting of every relevant case.

For example, I’m quite certain the new participant V can never change the top sorted candidate to somebody they prefer less. So it would have to be an upset via the B2R survivor, who would have to become the new winner, and be preferred less than the old winner by V. But that situation gets complicated in terms of the sorting and the rank tie-breaking authority… Maybe some day!

@toby-pereira yes it’s a bit particular, that’s the part that’s designed to preserve participation. The +1 advantage plus tie break conferred to the adversary is essentially to prevent any single vote from changing the outcome of the participation criterion-satisfying method. It still needs proof or more auditing and adjustment. But it was motivated empirically by finding examples of participation failure without introducing the advantage and some other aspects.

I think voters could have an anonymous ID given to them upon voting, it would have to be done with encryption. You’re right that in this case we would have to preclude latecomers, which I think would be fine. I think it could be done securely without an extra trip. This whole situation really makes “recounts” potential difficult though.

Having the “sincere” rank be attached to the original ballot might also be an option, but voters would somehow need to know that the second ballot would not be used in the first election, for instance. The only way they can know for sure is if they don’t provide it until after the first election winners are revealed. That could also be done with encryption.

In terms of preserving participation, the final runoff may not even be necessary. I’m trying to combine two things that can be looked at separately.

Also thanks for reading and your thoughts! I’m starting to wonder about how to guarantee the Condorcet loser criterion while still preserving participation. As of now though I think the method is essentially approval but with significantly stronger Condorcet-like guarantees.

EDIT: I just learned that there is an impossibility theorem about participation, independence of clones, and Condorcet loser. My guess is that the context and proof are similar to Arrow’s theorem, but the details I don’t know.

DISCLAIMER: This is conjectural and needs adjustments, but at least participation compliance can be improved. I have had LLMs run thousands of examples of this iteration, and have found no examples of participation failure. Still, it needs to be audited closely.

To understand this method, you should also know about Bottom Two Runoff (B2R), which you can learn about here for example:

https://www.votingtheory.org/forum/topic/564/bottom-n-and-bottom-2-runoffs-are-equivalent

The method is a development of this concept:

https://www.votingtheory.org/forum/topic/563/direct-independent-condorcet-validation/20

For those perhaps not familiar with some voting theory, there are often desirable criteria of a voting system that are, unfortunately, mathematically impossible to reconcile with each other. An important example of this is the contention between the Condorcet criterion and the participation criterion. It turns out that satisfying the Condorcet criterion in every case guarantees some instances where participation fails, and vice versa.

Consider the following method, which I’m just calling Approval vs. B2R, similar to Approval-seeded Llull as per @Jack-Waugh ---it's a bit technical, but there's a reason for that, so bear with me:

DEFINITION

(1) Voters submit rank ballots with approval cutoffs.
(2) Candidates are sorted by approval rates, with rank-based head-to-head breaking ties if possible—by this, I mean that within a tied score group, we recursively pull Condorcet winners and losers to the top and bottom of the ranking until none remain, and then defer to an authoritative and consistent rank-order over the candidates to break Condorcet cycles and ties in the final sorting.
(3) According to this sorting order, run B2R and identify the B2R survivor, with head-to-head ties broken by the sorting order.
(4) Consider the top-sorted candidate. If this candidate is the same as the B2R survivor, elect them.
(5) Otherwise, the top-sorted candidate will be an adversary to the B2R survivor. Run an independent head-to-head election between the B2R survivor and their adversary, with the following caveats:

--> Voters are not tied down in any way to their original preference between the B2R survivor and the adversary, and can freely vote for either in the independent head-to-head. Also, barring logistical prohibitions, voters who did not participate in the first round should fully be allowed to participate in the final round. By default, voters' original ballots will be used to determine the preference between the B2R survivor and their adversary, but voters may opt in to swap their preference either 0 or 1 times, whichever is necessary for their final ballot to confer an advantage that they wish to disclose.
--> However, based on these swaps, we can count the net number of swaps that are advantageous to the adversary over the B2R survivor compared with the original ballots. If this number is positive, the election proceeds as you would expect, with ties broken by the sort order. However, if the number is not positive, if the original head-to-head was in favor of the B2R survivor, and if a material difference would be incurred, then the adversary will be conferred an automatic +1 head-to-head advantage, and will also automatically win ties.

I conjecture that these caveats about the runoff—including setting aside the B2R survivor, increasing the adversary’s advantage by +1, and giving them ties—together restore the participation criterion under fully sincere ballots, at the expense of the full Condorcet criterion. However, when the margins are not close, the Condorcet criterion is still satisfied if ballots are sincere.

Here’s what else is fascinating about this: if participation does hold, then we can directly control the tradeoff between participation and Condorcet—in particular, when applicable, we can choose to increase the B2R adversary's margin over the B2R survivor by +1 with probability P. Then the method satisfies participation under sincere ballots with probability at least P, and simultaneously satisfies the Condorcet criterion under sincere ballots with probability at least 1-P.

IN SUMMARY:

This method I believe is participation compliant, which it is supposed to be by intentional design. This still needs to be proved. But as a consequence of this intention, it was also designed to fail Condorcet compliant in a controlled way. As per my comprehension, it will fail Condorcet compliance under these exact conditions:

(1) The Condorcet winner C exists (and will therefore be the B2R survivor);
(2) The approval winner Y is different from C;
(3) The head-to-head rank-based margin of C over Y is either 0 or +1; and
(4) The +1 boost to Y is applied, with ties going to Y.

More generally, it will fail the Smith criterion under these exact conditions:

(1) The approval winner Y is not in the Smith set;
(2) The head-to-head rank-based margin of the B2R survivor over Y is either 0 or +1; and
(3) The +1 boost to Y is applied, with ties going to Y.

In all other cases, it satisfies the Smith criterion. Thus in a sense, this method tries to be as close to Smith compliant as possible while enforcing participation compliance as non-negotiable. It unconditionally satisfies a weakened version of the Condorcet criterion: If the Condorcet winner exists and its weakest margin of victory is at least 2, then the Condorcet winner is elected. It also unconditionally satisfies a more significantly weakened version of the Smith criterion: If every member of the Smith set has a weakest margin of victory against non-Smith set members of at least 2, then the election winner will belong to the Smith set.

However, when the method fails the Smith criterion, it will necessarily fail the Condorcet loser criterion in the following circumstance:

(1) The Condorcet loser is the only available adversary to the B2R survivor, which is only possible if the Condorcet loser alone attains the maximum approval rate.
(2) The B2R survivor beats the Condorcet loser with a margin <=+1.

While this is unlikely to occur in realistic elections, it is possible, and pathological examples can be generated. Addressing this is hard, because independence of clones, participation, and the Condorcet loser criterion are a “cursed triangle,” where any two are incompatible with the third. In our case, we satisfy independence of clones and participation, so examples of failing the Condorcet loser criterion inevitably must exist.

Not that it's necessary to frame the properties in terms of Smith or Condorcet criterion-sounding conditions, but possibly those weak conditions can be strengthened. It is what it is.

Future work will be to prove participation and to refine the mechanisms for guaranteeing participation.

@jack-waugh but that’s always the tradeoff with score systems, which is why people bullet vote and/or min-max (translate to approval). I totally understand what you’re saying though.

I’m not saying the “Condorcet adversary” should be the score winner, just that they should be a strong alternative. Your approval-seeded Llull ballot could accommodate an approval winner as the Condorcet/Smith-method’s adversary, for example.

I think a “rank with approval cutoff” ballot makes sense. Then there could be the approval winner, and, say, the B2R (Smith compliant) winner, followed by an independent head-to-head.

@jack-waugh I see to an extent, but I would argue that your collapse of rankings is incompatible with the distinction by score. You prefer one to the other, even if they are both horrible, right?

@jack-waugh I’m not sure, I think coupling encourages consistency, which is a prerequisite to honesty. The coupled structure is also simpler and more efficient. The score winner is not necessarily the Condorcet winner for instance, and need not be the winner of a Condorcet compliant method when the Condorcet winner doesn’t exist (e.g. B2R).

The same validation logic also works in a homologous sense for Smith compliance. I think it’s less flexible if for instance two divergent Smith compliant methods were pitted against each other in the absence of a Condorcet winner. In that instance though, the Condorcet winner could not be tested with the final head-to-head. I’m not sure about extending to particular special subsets of the Smith set in the same way, depending on the subset.

@toby-pereira I would say they are declared the winner if the methods coincide. Otherwise there may be potential clone issues. It’s also simpler that way. Although if we did have an unambiguous runner up, that would be preferred to validate any potential Condorcet winner status. For example, the second highest scorer.

But if we wanted the decision process to be consistent, that has the potential to cause a recursion of successive head to heads, say, if the two methods repeatedly coincide. Unlikely but still. That would probably be the “right” way to do it in this context.

@jack-waugh exactly, that's the kind of thing I'm considering. Now I'm kind of puzzled though about whether a head-to-head between two winners of clone-independent methods can fail independence of clones. I actually don't think it's possible that the composite system fails independence of clones. FYI my preference for a Condorcet method would be Tideman's Bottom 2 Runoff (B2R), which I'm now certain is equivalent to BNR. For example, we could even pit the Approval winner against the B2R winner head-to-head. That would be a form of Approval-seeded Llull with a final independent head-to-head.