Forum Council

@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.

@cfrank There was a vote on electorama not that long ago Ranked Pairs with winning votes (as opposed to margins, which wasn't on the ballot) won the poll. It was the Condorcet and approval winner. In that poll I put approval as my top method.

@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.