A number of (old and) recent blog posts and manifold markets have come out debating the efficacy of Futarchy.
This post will address none of these! Instead, I'll cover some thoughts (originating mostly in Manifold discord discussions dating a few years back) about what I think is likely a strict improvement to futarchy as it is almost always presented. Namely: Rather than using prediction markets that replace the "called-off"/"reverted"/"N/A" bets with a combinatorial market over the decision and outcomes.
Futarchy as it is usually explained
Here is an explanation of futarchy (really it's an explanation of "decision markets", which I think is the right term for the decision making mechanism behind futarchy).
We have a decision to make, let us suppose there are two choices and each choice can either lead to a good or bad outcome (you could imagine more choices and a larger spectrum of outcomes, but this is simpler to explain. All the ideas in this post can be generalized to other settings). We want to maximize the chances of getting a good outcome.
We make two prediction markets
The first market will:
Resolve YES if option A is chosen and we get a good outcome.
Resolve NO if option A is chosen and we get a bad outcome.
Resolve N/A if option A is not chosen, meaning all trades made in the market will be undone, as if they had never happened.
The second market will:
Resolve YES if option B is chosen and we get a good outcome.
Resolve NO if option B is chosen and we get a bad outcome.
Resolve N/A if option B is not chosen.
Thus the first market price reflects the probability of getting a good outcome if A is chosen and the second market reflects the probability of getting a good outcome if B is chosen. We therefore choose according to which market has a higher price/probability (either averaged over some window, or at a specific time)
The issue with N/A resolutions
The issue with this design is the part where we undo all trades in the market for the option that isn't chosen.
To see why this is a problem, or at least more complicated than it appears, consider the following scenario:
You think option A is underpriced, so you put €50 on the platform to buy 100 shares of YES at 50%.
The rest of the market comes around to your point of view. The probability of a good outcome for option A goes up to 75%. Yay!
You sell your 100 shares for €75.
You withdraw the money from your account, and buy yourself something nice with the €25 you profited.
Option B turns out to be priced even higher, so option A resolves N/A.
At this point, the market has a problem. It's supposed to undo all the trades made on market A, which should mean making sure that the profit you made is reverted. But you have already left the building! Now the prediction market has to find someone to go out and break your kneecaps to get you to return the money.
More seriously, the upshot of all this is that the platform can't let you cash out your profits, or use them to bet on other markets, until the decision is made. This restricts the usefulness of your holdings and makes you less likely to trade on the markets in the first place.
The better option
But this is unnecessary. Here's an alternative market design for futarchy based on a single market with four linked share types.
The first share type pays out if decision A is taken and we get a good outcome
The second share type pays out if decision A is taken and we get a bad outcome
The third share type pays out if decision B is taken and we get a good outcome
The fourth share type pays out if decision B is taken and we get a bad outcome
But now that we don't revert the markets, how will we know what the conditional probabilities are?
Using the law of conditional probability!
$$ P(Good | A) = P(Good & A)/P(A) $$
Really this is just how conditional probability is defined in the first place: The probability that we get a good outcome given that we choose A is just the probability of choosing A and getting a good outcome, divided by the chance that we choose A at all. The probability that we choose A and get a good outcome is the probability given by the first share, and the probability that we choose A at all is the sum of the probabilities of the first two shares. Therefore, we can caluclate the conditional probability and use it just as we would in normal futarchy.
But what if I want to make a trade that pays out like the conditional markets do?
You can still do that!
Let's say that that we're in the situation from step 1 above: The conditional probability of a good outcome given A is 50% according to the market, and you want to spend €50 making it higher, but you want to do this in such a way that if option B is chosen, you don't have any profit or loss. The first thing to do is to buy 50 shares each of share types 3 and 4. This costs less than €50, since the probabilities of outcomes 3 and 4 can't total 100% or more, and it guarantees you'll be left with what you started with if option B is chosen. Then, you simply spend what remains of your €50 on shares of the first type.
Because all of this is done on a computer, the computer can even simulate for you what the outcome will be, so that you can get exactly the same user experience as if the market were conditional.
What advantage does this have that we don't get by conditional markets with locked-in profits?
Let's continue the thought experiment above.
After your initial buy of "Good conditional on A" You have 50 shares each of (B, Good) and (B, Bad), and 100 Shares of (A, Good).
Again, the rest of the market comes around to your point of view. The probability of a good outcome for option A goes up to 75%.
You again sell shares as if you were using conditional markets.
In the scenario above, you sold 100 YES shares in A at 75%, which is equivalent to buying 100 NO shares at €0.25 per share.
So the analogous thing to do is to spend €25 to buy 25 more shares each of (B, Good) and (B, Bad), and also 100 shares of (A, Bad)
Thus you now have 75 shares each of (B, Good) and (B, Bad), and 100 shares each of (A, Good) and (A, Bad)
As is usual in prediction markets, you can redeem a complete set of shares for €1. So you do this x75, and so you get back all of the €75 you put into the market, and you still have 25 shares each of (A, Good) and (A, Bad)
So there's a few benefits over the locked-in profits approach: The first is that you are able to cash in at least some of your shares (though technically the locked-in profits model could be implemented smartly enough to let you withdraw the minimum of all amount you would be able to withdraw under all future scenarios).
But more tellingly, the benefit is that your long position in A shares is now tradeable. If you think the probability of A being chosen is overpriced, you can sell that position without further need to add money to the system. Or if you don't have a strong opinion about what decision will be made, but you trust that the market is efficient, you can sell your position at market price to reduce your risk.
What are other ways to frame the benefit of this approach?
Another way you could think about this is that it allows more general flexibility in how you express your needs to the market.
For example, it's not often talked about, but one source of liquidity for futarchic markets could be individuals who have a more direct stake in the decision and its outcome than the typical trader. Indeed, many government policies are important to groups that they affect directly. you could imagine that there's a trader who would be especially distraught to see the market choose option B and simultaneously get a bad outcome (perhaps they are part of the group it is speculated policy B would most hurt, if it goes wrong (we could even make separate arbitraged markets on the parts of welfare sustained by different groups to better capture this)). That trader could hedge their risk by buying shares in the possibility that this would come to pass. But why should we force them to implicitly take a long position in outcomes in A in order to be able to?
Open question: Details on how liquidity changes
My examples above tries to be as clean as possible in showing the analogies between the two different market structures. The main difference with the combinatorial market is that it allows trades directly on the likehood of decisions.
Could this be a bad thing in some cases? If one option is obviously much better than the other, perhaps it is also obviously more likely to be chosen, which could lead to less available liquidity in the market shares for the alternative. I think this is hopefully not too much of an issue if the liquidity provision is designed carefully - in the worst case we could partially simulate the liquidity structure of the conditional markets by setting up separate automated market making mechanisms trading the conditionals and then arbitraging them against the main markets. It's worth noting that even if the prediction platofrm itself doesn't allow trades on decisions, a black market could spring up to take this liquidity anyway.
Maybe it's actually actively bad to spread information about how low the likelihood of one outcome is, precisely because it will cause people to see that correcting prices on that conditional will not be profitable. But yet again, maybe it's better to have this happen than to have only traders who are not sophisticated enough to estimate this chance accurately for themselves be the dominant forces in that market.
In any case, I would like to see more experimentation with futarchy structured this way.
@BoltonBailey Perhaps jumping the gun on this, but after my experience on the market below and thinking about this some more, I'm confident that combinatorial futarchy is more manipulation resistant than standard futarchy due to the lack of N/As, as I think you are alluding to.
As well as greater risk to manipulators due to none of their bets getting rolled back, anti-manipulators can sometimes make extremely safe counter-bets by betting no on some outcomes (e.g. decisions that are unlikely to be or shouldn't be taken), vs in regular futarchy where all anti-manipulation bets will either get N/A'd or be evaluated according to how well you've actually predicted the outcomes of the chosen action. That said, I'm not sure if this advantage becomes less relevant as the manipulation becomes more strategic (in the linked market the manipulation was very chaotic).
Fun combinatorial futarchy experiment in progress over the next 48 hours: https://manifold.markets/Jasonb/who-should-i-send-1k-mana-to-combin
Can this help you make profit if you know for example that A > B but aren't sure if A & Good is actually under or over priced? Not being able to leverage this type of information without significant risk / it only being profitable in expectation over many decisions seems like a somewhat reasonable flaw of standard futarchy
@Jasonb Good question! I have thought about this some and unfortunately I think there is just no way to profit from this knowledge state consistently: Basically, P(Good | A) > P(Good | B) is consistent with any of the four outcomes (Good & A), (Good & B), (Bad & A), (Bad & B) being close to certain (e.g. it could be the case that P(Good & B) is close 1 as long as P(Bad & A) is much closer to 0). And so any bet you make in the four markets will leave you net negative in at least one outcome, which you can't know isn't likely, unless you also introduce knowledge of the relative likelihood of A and B. This is true no matter whether you use combinatorial or conditional markets.
@BoltonBailey This is maybe another argument in favor of combinatorial markets though: If there is enough liquidity on the market for A vs B, you might be more confident that that price is correct. If you are confident that the market is pricing P(A) and P(B) correctly, you can make a trade that you know is positive in expectation - i.e. buy (Good given A) and (Bad given B) in proportion to the inverse of the probabilities that A and B are chosen respectively.
@BoltonBailey Nice, these are some good points. I'm also starting to feel that even though you can't guarantee profit you can definitely reduce risk under a variety of mild conditions (e.g. if it was binding you could bet up A&G and A&NG a little less, or you could bet so you only lose money in B&G outcome and even in non-binding case this feels much safer than a bet on a normal futarchy market). I might try and get round to just crunching the numbers, would be good if there was some property (in the most hopeful case) that for all market and knowledge states, your expected profit in log-space could always be higher for combinatorial markets by virtue of it having more hedging opportunities and whilst still being able to encode standard futarchy bets in it.
Right now I wonder if part of the problem is as simple as a UI problem: it's hard to bet on any market structure like this and be sure you're getting efficient trades!
None of the experiments done so far are sufficiently well funded to pay for any software development by participants. Maybe the answer is to try something with some free software that helps users interpret probabilities and make trades?
Anyway I've vibecoded a few pieces of this. I might try publishing them in the next few weeks if no one else gets to that first. If there are medium-term attempts to run more experiments I would be happy to provide liquidity, arb bot services, and/or my best attempts at an improved UI relative to the Manifold default.
@EvanDaniel LMK if you open source anything related to this, I have done some bot trading on manifold using CVXPY in the past but I have never gotten around to integrating the patten for arbitraging a conditional market agains its two underlyings.
@BoltonBailey Will do. Right now it's all vibecode slop and my excitement about attaching my name to that in the wild is... limited. But Claude and I will get there :)
The pattern I used for conditional markets was fairly straightforward: define exhaustive sets, trade on them when they're priced < $1. Config file holds lists of such sets. A 2x2 conditional market + underlyings then ends up with 4 sets of 3 outcomes each. The config file is a little verbose, but not complicated or hard to write and modern LLMs can do so with a little coaxing. That format is also flexible enough to trade on things like A -> B or your 2x2x2 market but needs extensions for cases where things might N/A I think.
So far the hard part has actually been re-implementing all the cost calculations so I can run the arb optimizer without hammering the API (which isn't even possible when using the multi-bet endpoint).
Anyway, for now feel free to tag me if there's a conditional you want me to set the arb bot on, I'm happy to add to the config file and dump some liquidity in any such market.
I like your ideas! I didn't know what futarchy actually was until reading through some of the links at the top of your post.
This isn't directly related to this post, but wouldn't the wealthy be able to manipulate the values of decision markets by purchasing many shares to achieve policies that benefit them but wouldn't benefit the society, profiting even while they lose money in the decision market?
@MaxE Haha I'm glad you asked that because I actually have another post in this pipeline about this question.
To summarize my thoughts: I think a proponent of futarchy (and the originator Robin Hanson in particular) would say something along the lines of that this is not a concern, because even if some of the wealthy choose to do this, all it takes is for a few wealthy defectors who could make the decision to (and make massive profits from) betting probabilities that reflect the truth.
But I actually think that there is still danger along these lines due to the inherent desire to hedge bad outcomes really scaling up the activity of the wealthy people to do this, and the fact that in some permutations, they don't lose money at all because those trades are reverted!
@BoltonBailey I've thought about this with the added idea of wealthy defectors.
If the wealthy know that the policies they want will result in worse outcomes than the current decision market rates indicate, they will want to bet in favor of the policies they don't want, but just little enough so that the policies they want still go into effect. Each individual corporation or rich person wont want to pay to sway the policy if they know others will do it for them. Therefore, the market will stabilize at a point where the probability of a good outcome given the wealthy benefitting policy is very slightly higher than the probability of a good outcome given the alternative policy, kept there by automated systems.
In the case where one policy is obviously better than the alternative, it will be easier for those who only care about profiting off of the market to force a policy through, but as the merits of the options converge, there will be less difference between the manipulated market probability and the actual probability and therefore there will be less profit to make by those who don't care about the policy as much. So as the options become less different in effect, they become easier to manipulate. But, as they become more different in effect past a certain point, they also become easier to manipulate
For example: a policy that would implement a 100% wealth tax over 100 million dollars, adjusted for inflation. Assume this policy would be effective at making the nation better because this is taking place in the future US and and would raise 50 trillion dollars after loopholes, and it has been determined that this wont drive business away to a detrimental extent. Rational actors with less than 100 million dollars of net worth will support this policy.
No very wealthy person will be defecting here; even if you profit hugely from the market, most of that money will instantly be taxed. If my intuition is correct, everyone else in the US would have to collectively raise 50 trillion dollars to counter the 50 trillion dollars that the wealthy stand to lose and would rather spend on preventing this policy, assuming they wouldn't spend more because of future profits. We would have to raise more if there was a nonzero chance that this policy would work out worse than if it wasn't implemented and there are actual market forces at play.
This comes out to $125 per person given 400 million people in the minimum cost case, but most people realistically won't contribute to this. I don't know how hard it would actually be to raise that much money, but it doesn't seem likely.
Is this correct or did I make a mistake in my logic somewhere?
You don't need to give a long response if you don't want since you are making a post along these lines anyway
@MaxE I think that your logic works. I also think the "Wealth tax" is a clever example because it's basically forcing a situation where there is literally no way for the wealthy to benefit from the tax policy. Maybe it's too clever - a futarchy proponent might say something like that it's unreasonable to expect futarchy to work in situations where you are literally directly changing the payout of traders, and so we should obviously not use Futarchy to set taxes, but maybe should use it for other goods. But OTOH, part of the capitalist mindset is that money is fungible with other goods, so perhaps you can't really get away from this.
@BoltonBailey Another point that I think I want to make in my upcoming post is - if we are treating social classes as aggregations, we could maybe also make a symmetric argument where we put a 100% marginal tax on the poor (defining poor broadly enough to contain the majority of total wealth), and then argue that futarchy will benefit the poor. So there has to be some kind of external influence or coalition factor here.
@BoltonBailey it is true that the wealth tax case is a very specific case designed to break futarchy. I suppose I didnt examine whether the failure would extend to less clear-cut policies that also greatly hurt the wealthy but to a lesser extent.
If you define "the poor" to be the poorest 51% of the wealth, that will include a supermajority of the population. This group is generally not currently advantaged by policy in most democratic countries in my opinion, even though it gets far more votes. This group is advantaged in theory in the modern system but not in practice. I expect this could apply in futarchy as well. Also, the longterm benefit of the majority of people in a society is almost the same as the longterm benefit of the society, which is what futarchy wants.