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.

@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.

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?

@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.

@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.

@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.

@toby-pereira I’m sure the courts will conjure up whatever question-begging definition of vote they need to for whatever ruling they decided on beforehand. I think getting the reasonable notions settled on to facilitate progress without constitutional amendments will require more lawyering than only appeals to reason. It’s unfortunate that Maine’s constitution encoded plurality into its state voting law, I think it’s important to know what other states might have this same kind of language issue in their constitutions.

@masiarek you're right, the "97/90/81/74" sequence was from a different experiment. The plots I shared have the accurate rates for the relevant experiments.

Your point about information is also important, because PSRO involves repeated optimization against mixed opponent strategies with effectively complete information about the modeled strategic environment, and a real election is definitely not like that. Present elections are probably better represented by the bounded-strategy region than by the PSRO endpoint, which does strengthen the practical case for Smith compliance. One could still speculate that declining computational and informational costs may move real strategic behavior toward that endpoint.

Also to clarify, the initial post was based entirely on the PSRO adaptive-strategy results. The later "human-like" bounded-strategy experiments were prompted by curiosity based on your feedback, and they changed the interpretation in the opposite direction. Unless we assume extremely capable strategic agents with very strong coordination and information, the PSRO results probably aren't descriptively realistic except as an approximation to a highly sophisticated strategic limit. It is interesting and somewhat unfortunate, though, that most formally Smith-compliant method seem basically unable to maintain a comparative advantage over non-Smith methods under the kind of strategic pressure PSRO simulates, at least in terms of preserving the sincere Smith structure.

Your last point about the fresh runoff aspect is also correct. I did also try several Smith-compliant methods with fresh runoffs during exploration, but those should be compared again systematically. In my exploratory simulations, most fresh runoffs showed only marginal improvements under PSRO, except for IRV vs runner up. There may be some interesting theoretical reasons for this, see Durand et al. 2026, "Super Condorcet Winners and Limit Coalitional Manipulability of IRV"--this uses Impartial Culture. I did not try fresh-runoff Smith methods in the later "human-like" analysis, so that would definitely be interesting to test as well.

And thank you for sharing your code base! I want to dig into this kind of thing more.

@masiarek thank you, I appreciate your feedback.

First, to clarify the simulation: ChatGPT used lightweight PSRO-like strategic agents representing coordinated factions. Each faction’s objective was to maximize its members’ mean underlying utility for the elected candidate, not to defeat the sincere Condorcet winner. An evolutionary best-response procedure searched over strategically available ballot policies against the current mixture of opposing strategies. This was done in spare time with ChatGPT, so I don’t have the code base yet, but can share once I dig it out.

The three largest sincere-favorite factions were strategic agents. Their policies controlled strategic participation/sincerity and candidate-specific rank and score offsets. I used 9 candidates and 71 voters per election.

Also, the benchmark was not impartial culture. I was using synthetic stress profiles including center-squeeze, polarized, clone-heavy, and ring-type electorates. So your broader point about generator dependence still applies, but the ~70% IRV result wasn’t coming from IC.

So, in that respect, a manipulation that displaced the sincere Condorcet winner but produced a worse outcome for the manipulating faction would count against that strategy. That said, model dependence is definitely a serious limitation.

Second, I did test a fairly broad range of completion rules for Smith-compliant methods rather than treating “Smith” as a single method. These included Ranked Pairs, B2R, IRV, Approval, Score, Benham, and several other standard or experimental completions. I also tested variants using fresh second-round ballots, including cases where the second-round completion was itself Smith-restricted. I may certainly have missed some more obscure possibilities, but the convergence did not seem specific to one particular Smith completion.

The main failure mode was often burial or burial-like pairwise distortion: the sincere Condorcet winner had already been pushed out of the Smith set before the completion rule was applied, so at that point changing the completion rule could not rescue it, although different completion rules might change some strategic incentives and definitely impact results once the sincere CW makes it into the Smith set.

I agree with your assessment in an important respect. These simulations assume unusually capable, coordinated strategic factions. They are basically searches for strategies that sophisticated actors could discover, rather than models or predictions of what ordinary voters would actually do, discover, or successfully coordinate.

There is also a normative question here. A system can perform relatively well once voters behave ruthlessly and strategically while performing worse under sincerity, but it is not obvious that this is desirable, and it is unlikely to be descriptively realistic. Obviously, effective strategy has informational, computational, coordination, and cognitive costs that many voters will not pay—or may not be able to pay equally.

I did some follow-up experiments after reading your response where I restricted behavior to be more “human-like” in the loose sense that strategic agents used simpler heuristic strategies, and I varied the fraction of strategic voters. The results were quite different: conditioning on elections with a unique sincere Condorcet winner, Ranked Pairs elected that winner about 97%, 90%, 81%, and 74% of the time when 25%, 50%, 75%, and 90% of voters, respectively, were strategic. IRV remained around 63–65% across those conditions.

You can see these plotted here.

So I think the distinction you’re drawing is important, and I largely agree. The earlier convergence between IRV and Condorcet methods seems to be a result about sufficiently powerful adaptive strategic behavior, not about strategic voting in general, or in human circumstances. Under simpler and more heterogeneous strategic behavior, the sincere advantage of Condorcet methods can remain very large.

However, I don’t think the highly adaptive case is therefore unimportant. As AI tools become increasingly capable and accessible, the informational and computational costs of identifying sophisticated election strategies may decline substantially. So even if the adaptive simulations are poor descriptive models of present-day individual voter behavior, they may still be useful as stress tests of how a voting method behaves when strategic optimization becomes easier.

Over the past few weeks, I’ve been thinking a lot about sincere Smith compliance under what seems to be its main strategic adversary: burial.

I’m increasingly coming to the conclusion that formal Smith compliance may be much less informative about strategic behavior than I previously assumed. A Smith-compliant method will, of course, continue to elect from the Smith set of the ballots actually cast. But strategic burial can change that reported Smith set. So a method can remain perfectly Smith-compliant while nevertheless failing to elect a candidate who would have been the Condorcet winner (or more generally Smith compliant) under sincere preferences.

Put differently, there seems to be an important distinction between formal Smith compliance and strategic preservation of the sincere Smith set.

I’ve been exploring this through a large number of LLM-assisted simulations of adaptive strategic behavior. In the simulations, strategic blocs are allowed to alter their ballots freely and adapt in response to the strategies of other blocs. These are obviously exploratory tests and should be taken with a grain of salt, as I may be out of my element.

In any case, conditioning specifically on elections in which a unique sincere Condorcet winner exists, I’ve found that even fairly sophisticated two-round mechanisms designed specifically to counter burial do not seem to improve the probability that the sincere CW actually wins by very much. The best variants I’ve tested have been around 72% under adaptive strategy.

What surprised me is that this is very close to the performance of much simpler one-shot methods under the same strategic conditions. IRV, for example, elected the sincere Condorcet winner about 70% of the time. Stable Voting and Ranked Pairs were roughly in the high-60% range, and differences between the better-performing methods were often pretty small.

So I don’t mean that IRV is somehow “more Condorcet compliant” than Ranked Pairs or Stable Voting. Obviously it isn’t in the formal criterion sense. Rather, it seems that formal Condorcet/Smith compliance did not translate into dramatically greater preservation of the sincere Condorcet winner once strategic behavior was introduced.

This seems important, because we often talk about Smith compliance as though it provides strong protection against electing the “wrong” candidate. But the guarantee of Smith compliance applies only to the Smith set induced by the submitted ballots. If strategic voting substantially changes the pairwise structure, that guarantee can diverge from the thing we may actually care about: whether a candidate who would beat everyone else under sincere preferences wins the election.

That has made me wonder whether formal Smith compliance is less useful as a practical discriminator between methods than I had thought. Maybe a more relevant question is: how difficult, risky, or dependent on coordination is it for strategic voting to displace the sincere Condorcet winner? And that’s a different property from formal Smith compliance itself.

In fact, a formally non-Smith-compliant method might preserve the sincere CW about as often as a formally Smith-compliant method under strategic conditions, since a Smith-compliant method might remain perfectly compliant with its reported ballots even after burial has manipulated the pairwise structure away from sincerity.

I’m still testing this, and I’m not an expert in this area, so I wouldn’t put too much weight on the values yet. But the general pattern has been pretty consistent: once sufficiently adaptive strategic behavior is allowed, a number of very different methods seem to converge toward fairly similar rates of sincere-CW preservation—namely, in this case, ~65-75%.

What do you think of this? Personally, it makes me more interested in IRV, since it consistently outperformed many other methods under strategic pressure. I haven’t looked into the sincere bottom-Smith situation yet, but that’s another thing to consider.

At the same time, it makes me feel that prioritizing some criteria that are incompatible with formal Smith may be a wise decision, such as participation.

Shout out to this post: https://www.votingtheory.org/forum/topic/620/score-voting-is-king-condorcet-not-so-much/2

I’ve been considering generalizations of SSVV designed to improve robustness against strategic voting, and landed on the following design outline for a voting system, which can be called “MARE,” standing for “Majoritarian vs. Alternative with Rescue Election.”

I tested this framework using limited simulations of strategic voter behavior, and found significantly improved outcomes against benchmarks. For example, successful burial rates dropped from ~15% to ~7%, with >50% of buried Condorcet winners being rescued. This needs further investigation and more rigorous analysis.

The idea is as follows: in the first round, voters submit rank-score ballots, which determine two sets of candidates: a Majoritarian set M, and an Alternative set A.

For example, M could be the Smith set or Bipartisan set. A could be the Approval winner set or the IRV winner set.

Next, voters are given information about M and A. For example, M may be made public. A may also be made public, or some relevant statistics about A may be made public. For example, maybe instead of revealing the score winner, only the score is shown.

The key is, voters can use the information about M and A to do two things at once: (1) vote for whether to proceed by electing a candidate from A, or a candidate from M. And (2) possibly submit a “Rescue” signal, designed to counter burial tactics that may have manipulated M.

For example, the Rescue signal may be implicit, as the M-outsider with the highest top-rank support among outsiders. The design goal for the Rescue signal is to re-introduce a buried candidate into M with high efficiency, without ruining a sincere election. The cost is strict Smith compliance (although, if the Smith set is already manipulated and insincere, that may not be a bad thing).

If the electorate chooses M, then the pre-specified completion rule is run on M with the Rescue set R. For example, Benham could be run on M, and the winner could go head-to-head against a winner in R.

Otherwise, if they choose A, then the pre-specified completion rule is run on A.

The goal of MARE is to enable information/incentive engineering by selecting the rules for M, A, and R. In some experiments, I had M be Smith, A be IRV, and R be top outsider. There are various other considerations to explore within this framework. I just wanted to share it to see if others wanted to experiment with simulations.

@toby-pereira makes sense, it seems Condorcet/Smith methods consistently rise to the top in multiple ways.

I think Smith//STAR (implicit) as mentioned above has some problems, but I think Smith//Approval (explicit) as denoted in the electorama vote is a very good method despite taking two rounds. I’m half surprised Smith//Approval (implicit) didn’t do better.

I understand the general aversion to two-stage methods, but in the USA, we often effectively do that with primaries anyway, and it seems commonly used elsewhere. I think introducing alternative methods into party primaries could be pretty significant change in itself.

@masiarek I think this is a useful way of looking at it: sincere, consistent voting should lead to Smith//Approval, as long as the Approval (or score) round is conducted after the Smith set is already known. So I would push back somewhat on “same winner, no second election,” because in SSVV, the Smith set (as well as the Score winner set) is known before the scoring of candidates in step (2.2). That sequencing can matter, even if sincere voting ultimately produces the same outcome as Smith//Approval, unless voters have genuine independent approvals per candidate.

In fact, it may be that most strategic benefits of SSVV are caused by the score/approval in the Smith set being informed by the Smith set itself, which would mean Smith//Smith-informed-Approval is a reasonable comparator, and it may be superior.

I also agree that in the singleton case with sincere voters, the validation step is redundant. If the Score set is ({A}) and the Smith set is ({B}), then a majority vote consistently with sincere preferences should choose the Smith set, since that same majority sincerely prefers (B) to (A). Probably, often the Score winner and Condorcet winner will coincide anyway.

Where I think SSVV differs most substantially from Smith//Approval is in strategic behavior rather than sincere outcomes, particularly when strategic voting changes the composition of the reported Smith set. Smith methods like Smith//Approval automatically commit the electorate to choosing from the reported Smith set, whereas SSVV does not—since voters first see the Score and Smith sets, the majority can decide whether they actually want to validate a Smith set that may have been distorted by tactical voting.

That may matter specifically for burial. Suppose burial changes the Smith set from something relatively compelling into a larger or stranger set. The burial strategy has succeeded in manipulating the pairwise structure, but has also changed the object voters are being asked to validate. Voters who prefer the Score winner to every member of the Smith set obviously choose Score, but even some voters who prefer one Smith candidate to the Score winner might choose Score rather than risk one of the other Smith candidates winning.

So I think your observation points toward an interesting strategic question about the method. Manufacturing a cycle can indeed get a candidate into the Smith set or remove another candidate from it, but unlike under Smith//Approval, that does not automatically get the preferred candidate into the decisive election, nor does it necessarily eliminate the buried candidate from consideration altogether, since the manipulated Smith set has to survive validation by majority vote.

I don’t know yet whether that additional hurdle is strong enough to make enough of a difference to warrant it, but I think that’s the main distinction from Smith//Approval.

My simulations so far have been exploratory and conducted interactively through ChatGPT 5.6-Sol, using spatial and nonspatial electorates with coordinated factions iteratively trying various score-compression, compromise, and burial strategies, so I would not treat them as an equilibrium analysis. They do suggest that the validation stage can sometimes neutralize successful manipulation of the Smith set (primarily accomplished through burial), but that result needs more systematic investigation.

I also just experimented with public B and hidden A, or revealing only the mean score of A rather than candidate identity. This mostly improved behavior even more, because manipulating the Smith set becomes more risky in validation if the Score winner set is kept hidden. But it opens a can of worms in terms of “information engineering” to improve incentives. Even if the Score “escape hatch” isn’t used, its mere existence changes strategic incentives.

This is a single-winner system with two voting stages. I did some cursory simulations against related alternatives, and this looked like the most promising of them.

Definition

(1.1) Voters submit a rank-score ballot: candidates are scored, with the induced score ordering determining the ranking; candidates receiving equal scores may be explicitly ranked to break the ordinal tie.

From (1.1), compute:

• A, the Score winner set; and
• B, the Smith set induced by the rankings.

Both sets are then made public.

In the second voting stage, knowing A and B, voters answer two independent questions:

(2.1) Vote either to elect from the Score winner set A, or to elect from the Smith set B.

This is deliberately an ex ante judgment. The voter knows both candidate sets, but—when either contains multiple candidates—does not necessarily know which particular candidate will ultimately be selected from the chosen set.

(2.2) Cast a fresh ballot over the candidates needed to resolve the chosen branch. In the version I tested, this was a fresh Approval ballot over the Smith set; if the Score winner set contains multiple candidates, the ballot can analogously include the members of A needed to break that tie.

Finally:

• if a majority in (2.1) chooses the Score branch, elect the fresh-ballot score winner from A (ordinarily A will contain only one original Score winner);
• otherwise, elect the fresh-ballot score winner from B.

Thus the first ballot produces two competing claims about the appropriate winner: an unrestricted cardinal winner and a winner drawn from the reported majority-dominant set. The second stage asks the electorate, after seeing those sets, which claim it wishes to validate.

The structure can be generalized, for instance, using the IRV winner rather than Score/Approval, or using a different completion method in the Smith set.

I wanted to bring up something from the past that never came to fruition, which was the idea to vote on voting systems. The idea was to have users include a ranked score-carrying ballot in their forum signature (you can see mine below as an example), and to intermittently compare them to see what users thought. All that was ~discussed here:

https://www.votingtheory.org/forum/topic/237/proposed-options-for-voting-on-voting-methods/25?_=1785984453512

Anyway, after digging through the old forum threads, I tried to reconstruct a “metaelection” from the score ballots that various users have put in their signatures or explicitly posted.

So far I’ve recovered commensurate ballots from only a small collection of users:

• cfrank (myself)
• Jack Waugh
• multi_system_fan
• Marylander
• Andy Dienes (former user)
• Keith Edmonds

(I also found evidence that rob had a ballot in his signature, but I haven’t yet recovered the actual scores.)

While this is not necessarily representative of all users or even voting theory enthusiasts in general, and does not include all voting methods one might in principle prefer, I wanted to see what the results were from the limited scope and “turnout.”

To make the ballots comparable, I grouped together the various rated/cardinal Condorcet methods (e.g. Score Sorted Margins, generic cardinal Condorcet, rated Condorcet), rescaled Marylander’s 0–100 ballot to 0–10, and then compared the common methods.

The aggregate score totals are:

• Cardinal Condorcet: 55.0
• STAR: 48.0
• Approval: 42.9
• Score: 39.1
• IRV: 12.8
• Choose-one/FPTP: 4.0

Pairwise, cardinal Condorcet defeated STAR, Approval, and Score by 4–2, making it the Condorcet winner of the metaelection as well.

It also won under Score, STAR, Borda, Plurality, and IRV. Under Approval, the outcome depended on the approval threshold: using the suggested threshold of 5 gave a three-way tie between cardinal Condorcet, Approval, and STAR, while a stricter threshold (>5) elected cardinal Condorcet outright.

At least from the ballots I could recover, the collective ordering was:

1. Cardinal/rated Condorcet
2. STAR
3. Approval
4. Score
5. IRV
6. Choose-one/FPTP

If anyone knows of additional signature ballots (or can recover rob’s old ballot), it would be interesting to incorporate them.

We can compare with an external poll on Reddit (this poll is not a vote, but asks which single-winner system among alternatives is most preferred):

https://www.reddit.com/r/EndFPTP/comments/1cukxrz/what_singlewinner_method_do_you_support_the_most/

Interestingly (or perhaps uninterestingly?), if we pool together all Condorcet methods, our small toy election matches quite well with the top preferences in the Reddit poll.

I would also like to hold that kind of election not in terms of the most preferred voting method outright, but in terms of what the most feasible reform target to implement would be. That’s a bit vague—for example, would that mean implement state-based/federal approval voting? Or to push for proportional representation in districts?

Based on this preliminary outcome, a cardinal-Smith method with a STAR-style fallback or completion rule might be worth considering, although that specific composite was not itself included uniformly on the ballots. I was also surprised to find that, unless I overlooked something, this specific hybrid had not been proposed before now on this forum. It has been proposed elsewhere, including within various Reddit threads.

Finally, on the subject of voting for voting methods, I wonder if it would make sense to have the electorate vote on their preferred method along with their ballots, or to vote for their preferred method in advance. That’s probably a different topic.

@clay, this appears to undermine itself at several important points. You claim to have completely debunked Kant, but then concede that your definitional move yields only a stalemate, that the genealogical critique cuts equally against your own foundation, that “free and fully informed” may smuggle in normativity, and that Kantian constitutivism remains a live dispute. Those concessions are incompatible with the claimed total refutation. I recommend writing the argument as explicit premises and implications; the problems should become apparent.

Even if we admit preference sovereignty, its own formulation is severely self-limiting. It applies only where all affected parties would freely and unanimously choose an outcome while fully informed, which is already immediately divorced from any dilemma that involves conflicting interests. Consequently, that principle in itself does not imply aggregate utility maximization in cases of conflicting interests.

To derive aggregate utility maximization, Harsanyi introduces additional assumptions through an identity-lottery construction. That construction can make aggregate utility numerically equivalent to the expected utility of a hypothetical identity-uncertain chooser, but it does not make that quantity identical to each actual person’s utility, nor does it establish why taking Harsanyi’s hypothetical standpoint is normatively authoritative.

What you have, in effect, is a selected collection of normative operations—idealized preference under conditions of rationality, freedom, and full information; identity uncertainty; expected-utility reasoning; and interpersonal aggregation—that define your ethical framework. Choosing different operations can and does yield competing ethical theories that you have not eliminated.

To be clear, some of those operations are at least superficially appealing. The preference-sovereignty principle is almost a tautological no-brainer within its stated scope, and the arguments that follow once we accept the normative premises built into Harsanyi’s construction may be perfectly valid. But those premises are exactly where justification is lacking, where competing alternatives are abundant, and where legitimate disagreement obviously exists.

We are clearly not getting anywhere, so I’ll offer my ultimate relevant point on this subject and stop. I think the election-by-jury concept is an interesting and reasonable proposal with compelling arguments in its support, and that while there are also completely valid concerns, it’s also plausible that those concerns could be effectively addressed with more collaborative examination and careful approaches to implementation. It has controversial aspects, but it isn’t a wholly radical idea. I think there are also legitimate generalizations that are equally interesting and reasonable topics of discussion.

@clay you can plug your ears and say La La La all you like. Kantian ethics is a perfectly legitimate alternative to utilitarian ethics.