On the technical properties there are no issues here.

On the practical considerations there are a number.

It requires explaining the Smith Set It requires explaining Condorcet winners It requires explaining Approval voting. It requires introducing two new voting methods , with two different ballot types, each with different instructions, at the same time.

I recommend having one voting method only. If you want to have a primary and general then use the same good method for both and don't narrow it down too much before the general. That's actually simpler and will get much broader support from stakeholder factions and minor parties who want real choices on the general election ballot.

Condorcet is accurate enough to just skip the primary all together unless there are too many candidates. If you want Condorcet with a simple tiebreaker why not just make the tiebreaker between the finalists (tied candidates) automatic in the event that it's even needed? In the tiebreaker, each voters ballot counts as a vote for their favorite(s) still in the running. If a voter ranked two candidates equally that would count as a vote for both. The finalist with the most votes wins.

The tiebreaker never even has to be explained. It's literally just a built in tiebreaker like a Plurality provision might call for a drawing of lots or a coin toss, but the ballots already have the tiebreaker info they need so it can be done instantly.

@gregw I think Condorcet is great when a Condorcet winner exists, but when one doesn’t exist it’s really troublesome. Ideally, we would have a method to check whether one exists without unearthing the Condorcet cycle that reveals the jilted majority upon the choosing of any winner, but that is essentially impossible.

I think it makes sense to do Condorcet//Approval, in two separate rounds, the approval round restricted to the Smith set. But two round voting outright is a difficult sell in the USA (even though two-round voting is pretty common all over the rest of the world…).

People try to put the two together in a single round vote, but the strategic incentives of casting rank and approval/score indications on the same ballot cause issues.

My personal belief is that this system of two-round voting for single winner elections, I.e. approval conditional on already knowing the Smith set, would be most ideal. I think actually implementing the approval aspect first however is an easier sell than implementing the Condorcet aspect first. I can envision a natural progression as: (1) implement straight approval, (2) eventually indicate the shortcomings of approval in guaranteeing election of Smith set candidates, (3) reform to include a ranked primaries to restrict to the Smith set before the final approval vote.

I feel even having rank/Smith-based primaries makes way more sense than what we have if the subsequent system is approval. There’s no issue with vote splitting in that instance, and it fits at least partially into the political system we already have (although this would also require substantial changes).

@toby-pereira Anything can be argued and counter-argued in court, but my assessment above is based on what a fair and likely ruling would be, in my professional opinion.

I'm not a lawyer, but have been studying these things for years and we did win the case in the Oregon Supreme Court against the Oregon Legislative Council which found that RCV's ballot title in Oregon was misleading and inaccurate as it pertains to RCV's majority winner claims, though there is apparently no mechanism anymore to enforce such a ruling.

Two confounding data points:

RCV has spent years making the argument that RCV guarantees majority winners. That's directly at odds with an argument that it guarantees plurality winners. Ironically, there's a solid case to be made that due to exhausted ballots it guarantees neither. The Constitution of Maine Article IV, Section 5 requires that winners be elected "by a plurality of ALL votes returned.” In cases like these where the wording is explicit that it's "all" votes "returned" or all votes "cast" RCV is eliminated from compliance by the existence of exhausted ballots alone.
Screenshot 2026-04-15 at 6.15.24 PM.png

@jack-waugh I’m imagining the decisions could be automated by making assumptions about the group utility and computing the mixed strategy Nash equilibria of the game.

The deepest problem there is the utility function, since it would be an aggregate utility of some kind that we would have to presume emerged somehow from the individual utility functions of the voters that constitute the group.

All of that is definitely questionable, I don’t believe in utility functions, but I’m unsure how else to proceed in putting this kind of situation into an analytical framework.

@toby-pereira I see, it seems that the properties of maximal lotteries related to participation aren’t what I believed they were. Looking into this, I misunderstood the definition of an (x,y)-improvement, this is swapping the adjacent positions of x and y in a ballot that already exists in the election, not introducing a new ballot where x>y.

That’s unfortunate, but also makes sense.

@cse4129 no problem, that’s what forums are for! Your idea is good, it just happens it was already invented. It also does have some problems/failed properties, though (like any method).

For example, it isn’t Condorcet compliant or Condorcet loser compliant, and it fails participation. It doesn’t satisfy independence of clones either, and it fails various other binary criteria. That doesn’t mean it’s a bad method, but it means it can occasionally produce pathological results. It satisfies the majority winner and majority loser criterion, and later-no-help.

There’s a “mind map” I made some years ago of some of the best-known/characterized voting systems in terms of the binary criteria they satisfy and fail in that discussion above. Probably nowadays I would make a better one (I might at some point). I should have done something like PCA or a graph embedding, but I tried to make that map before I knew about those analysis methods.

@jack-waugh Oh, okay! Thanks for the shoutout! Here's a more stable link to use: https://electorama.com/discord .

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

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

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

Hi @cfrank - as @Toby-Pereira mentioned, there is electowiki for consolidating ideas from over here. At this point, I'm already relying on some pretty substantial help from LLMs for the tools I use to put together the electorama news. The latest draft edition is here:

https://electowiki.org/wiki/ElectoramaNews/2026-January

The prose in the ElectoramaNews is pretty much all mine, but the code I've "written" to provide the summaries of each of the forums (like my summary of this forum) was mostly generated from a vibecoded script I run periodically. My hope is that I'll be able to provide a more-and-more sophisticated summary of the various forums every month, and maybe I can have my tools call the LLMs to provide the summaries of the discussions. Right now, the "summary" in many sections is basically just a link to every forum post or email or whatever.

Hopefully, by regularly publishing ElectoramaNews, we'll be able to drive some traffic over here to the Voting Theory forum.

Opps – Correction

I made a mistake in an earlier post on this thread:

I stated that in a BTR-Score election with a 3-way cycle, the highest seed in the cycle would have a 50% chance of winning, and the two lower seeds would each have a 25% chance of winning.

This is incorrect. The highest seed in the cycle would always win.

Consider this BTR-Score tournament with a 3-way cycle:

Candidate A is the highest seed in the cycle.
Candidate B is the 2nd highest seed in the cycle.
Candidate C is the 3rd highest seed in the cycle.

In every such election, B and C will face off in the tournament; the winner will then face A in the deciding contest.

In a 3-way cycle, each of the candidates wins one match and loses one match to the other two candidates. The winner of the B vs C contest uses its only win to prevail; therefore, that winner has no chance of defeating A.

2 Possibilities:
A > B > C > A - B beats C then B loses to A
C > B > A > C - C beats B then C loses to A

A BTR-Score election with a Condorcet winner rewards majority.

A BTR-Score without a Condorcet winner rewards utility.

It would be possible, but extremely weird, for a BTR-Score election to have a 3-way cycle that does not include the top seed, the candidate with the highest total score.

@cfrank I’m bringing this topic up again, because it seems necessary to consider the implications of how alternative systems at lower levels of government translate effects upward, especially when higher levels maintain a winner-take-all style.

For instance, in a presidential election, say we adopt maximal lotteries (Condorcet compliant) to generate social choice rankings per state. For this to translate nicely to the federal level and accommodate the electoral college, the natural extension seems to be another maximal lottery where voters are states casting ranked choice ballots with electoral college weightings.

I think this could be a viable system in principle, but the question is about how feasible the structural and institutional changes would be to make.

The same sort of question comes up with approval voting. Essentially, my worry is that without upper-level criteria met, lower-level changes, while still locally impactful and potentially inducing long run changes, would not translate effectively upward in the short term, and may even destabilize upper levels of government.

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

@kaptain5 said in Professional Politicians skew towards altrusism? [study to read]:

In this study an interesting finding was that on average politicians we behaving altruistically (giving more to the other party than they would receive). On average over all sampled politicians were giving 49% of the prize to political opponents and giving 57% to in-group members. Compared to most other studies of the ultimatum game this is far above most people and very far above the rational expectation. The result is also remarkably consistent between countries. When the out-group and in-group results are combined this gives the result that the politicians surveyed are slightly altruistic in the game.
This one surprised me, I expected the politicians surveyed to be much more greedy.

I haven't properly read the paper but this bit is interesting, and I've been considering whether we should be surprised or not. Politicians certainly have a reputation for being greedy and corruptible. But a lot of people who go into politics probably do so for the right reasons - that they actually want to make things better for people and remove unfairness etc.

I think there is more corruption at the top, but this isn't most politicians. Plus I also think that higher-up politicians might still have some good principles, but these conflict with other things that get in the way of that - wanting to get in with the right people, make the right deals etc. So removed from politics, they still might make fair decisions. They just happen to be morally weak in certain ways.

If you have an allocation game dividing a budget, a pizza, etc... this game is zero-sum in the sense that a you getting a slice of pizza is a slice that isn't going to me.

If you are negotiating with an agent who you are unsure is altruistic, rational, or spiteful there is a strategic incentive to misrepresent as a spiteful agent. An altruistic agent gets less than their fair share and a spiteful agent gets more than the rational agent if they can credibly convince other agents they are spiteful. It is worth it to pay off a spiteful agent rather than provoke it.

So if you are trying to optimize for the social choice there's a big problem with spiteful agents. With limited information all agents are incentivized to misrepresent as spiteful and over-represent how much they care about minor concerns. And for making social choice with limited resources there is a genuine zero-sum nature to the problem. Rational agents operating in a zero-sum environment will behave spitefully (you having it is negatively correlated with me having it.)

So with that irrationality baked in, if we could magically find what the social choice was, with spiteful agents the social choice isn't very sociable. And spiteful agents should be common under resource constraints. This correlation between individual's utility functions will also heavily restrain the types of group choice sets which exist.

@toby-pereira Thanks for the feedback!

National elections generally take place several years apart and a lot can change in between, and so voters wouldn't be able to apply game theory in the same way as they would with multiple elections close together under the same conditions. And also I imagine people in the study are more likely to be "clued up" than the average member of the public.

One of my goals is looking towards what the long-run equalibria might be after people have figured out the system. So in this way I see that as a plus.

So that raises a question - with multiple elections, will it just be the same conditions each time just to see how the methods behave in these ideal conditions, or will certain variables change to make it more "realistic", or possibly you'd model both? Both would be interesting in their own ways.

In trying to set up the models one of my conclusions is that a "correct" set of parameters doesn't exist. I think instead of trying to make a study "correct" or "accurate" it would be more useful to search for robustness. A common result of repeated games is multiple Nash Equalibria. Another common result is sensitive Nash Equalibria. So if one number changes slightly a whole new set of equalibria might be optimal now. So I think it is important to vary conditions to see if an equilibrium is stable.

Reading on different strategies and failure conditions of voting systems you come across some really weird ones where it is technically optimal to do make some really weird decision. But that often requires very specific information. An important concept I came across while brushing up on my game theory is imperfect and incomplete information. When you change the information available it can sometimes change what the Nash Equilibrium strategy is. There's also an idea that a strategy might not be optimal over any specific set of game parameters but is optimal when you are unsure of the exact parameters. The more limited the information the simpler a game sometimes becomes.

Yes, for PR elections. Candidates from the voter's district could be optionally identified by name in bold font.

@cfrank If others have examples, simulations, or citations for my claim that "over time the two opposing factions become more and more polarized” I'd love to have those on hand too. I'm not sure I'm referencing any one thing I've read or heard in particular, but more putting multiple things together to get the big picture.

On the math alone I think you're right that the center is an important block for the two parties to court as well, but in practice I think that that incentive is outweighed by the other perverse social incentives to demonize the other side, to punish "traitors" considering switching, and so on.

Cancelling people who question the party line costs center voters, but it also discourages others from following. I think this was the subliminal Democratic Party tactic over the last decade that paid dividends for a while but then ultimately lost them the "big-tent" advantage and the presidency. I'm not saying it was an intentional strategy. There are big cultural forces at play here. That's obviously my own personal opinion.

Back on topic: As the narrative gets dominated by two polarized factions and the middle is silenced, the real middle (the center of public opinion) almost ceases to be a part of the political spectrum because it's doesn't actually map on to the left, right, and swing voter boxes.

Identifying and presenting consensus win-win policy and then getting it passed is the goal. We need to incentivise and empower that one way or another.