I’ve seen similar derivations before, but it’s been a few years since I looked at AMMs in detail. I’ve spent some time recently looking at mechanism design for prediction markets, so this is a timely reminder!
Three questions— Would you agree that this captures your main conclusion for a binary prediction market: CPMM ⟺“price = probability”?
I seem to recall that CPMMs easily generalize to multiple assets. Instead of ab=a′b′ you have abc=a′b′c′ and so on. Do you happen to know if that matches the generalization of your prediction market with binary outcome to one with categorical outcome?
You mention in your 2nd footnote that there are “different probability functions matching these desiderata”. Care to say more on this?
If you could link me to these similar derivations I’d be interested to read them, I mostly wrote and worked through this because I couldn’t find any existing ones from first principles and was sure it would be possible.
price = probability is a general rule for prediction markets, it’s more that CPMMs can be derived from the probability function we described (no/(no+yes)).
The generalization I’m using in my current implementation is https://manifoldmarkets.notion.site/Multi-CPMM-62fe5b99013c4d5a87dfa84e0b8fa642, although I believe Manifold currently uses some bizarre auto arbitrage system between linked binary predictions for exclusive categories. Similarly, you can also extend CPMM by adding a parameter to allow market initialization at different probabilities, and also to allow users to inject liquidity without pushing probability towards 50%.
Regarding other probability functions, there are of course a whole family of constant function market makers that CPMM is a member of. As a trivial example, (no2/(no2+yes^2)) should also match our desiderata, I believe.
Additionally, starting from the angle of “We have a market maker with pools of shares, how do they calculate probability from these pools” is just one approach you can take.
There’s also the LMSR (Logarithmic Market Scoring Rule) also developed by Robin Hanson, which approaches it from an entirely different angle, starting from asking how you can score predictors based on how well they performed, and then applying this to rewards and incentive alignment in a prediction market. This is actually more reflective of the Bayesian structure of the market than CPMM is, I was largely joking when I made that claim.
There’s also DPM (Dynamic Parimutuel), which adapts existing parimutuel betting systems to prediction markets. It does have the disadvantage of not being able to know ahead of time how much money you’ll receive from your bet, only how much money you’ll receive in expectation from your bet, but it has some advantages of its own.
I have this paper saved to read through and think about, I don’t really understand it yet but it also proposes a unique solution to this problem.
Largely, CPMM is the one I understand most intuitively out of all of these, which is part of why I’m using it in my personal prediction market implementation.
Thanks for the questions!
Edit: After encountering some problems I have since done more research. Multi-CPMM is dumb and bad and Multi-Binary (manifold’s current implementation) is superior in every regard actually. I am signing the manifold markets apology form (reason for behavior: thought the decision was for architectural reasons, was repelled by the auto arbitrage nature). I will hereby respect Manifold Markets and I will NOT talk down on the greatest prediction markets platform of all time.
I will try to dig up some references for you. Sorry it really was a small side project and has been several years.
Ah so I can’t imagine a probability function for that market that isn’t yy+n. y2y2+n2 is a fine pricing function that doesn’t appear to adhere to the rules of probability theory. If I try to compose two y2y2+n2markets, one conditional on the other, then can I multiply their prices to find the joint probability? Does this violate “price=probability”?
”price=probability is a general rule for prediction markets” is a very interesting claim. Seems obvious, but then you have to ask yourself whose probability?
I’m familiar with some of those operations (I’ve skimmed the Boyd paper). Unfortunately, there are a lot of different ways of expressing the same constraints, so I can’t immediately tell whether Manifold’s implementation is equivalent to what I had imagined.
Thanks for your answers, I’ll look into some of the other ideas you referenced.
What I mean is that the if the AMM estimates the probability at .75, it should charge .75 for a marginal YES share, by law of expected utility. I don’t think a different probability function should alter the probabulity theory, just change the pricing curve.
If “price=probability”, then changing the pricing curve is equivalent to changing how the AMM updates its probability estimates (on evidence of buy/sell orders).
Yes, but it just affects how liquidity is allocated, and it doesn’t just affect how the AMM updates, it affects how users trade as well since they respond to that, either way they’d want to bet to their true probability. So changing the pricing curve is largely a matter of market dynamics and incentives, rather than actually affecting the probabilistic structure.
I’ve seen similar derivations before, but it’s been a few years since I looked at AMMs in detail. I’ve spent some time recently looking at mechanism design for prediction markets, so this is a timely reminder!
Three questions—
Would you agree that this captures your main conclusion for a binary prediction market:
CPMM ⟺“price = probability”?
I seem to recall that CPMMs easily generalize to multiple assets. Instead of ab=a′b′ you have abc=a′b′c′ and so on. Do you happen to know if that matches the generalization of your prediction market with binary outcome to one with categorical outcome?
You mention in your 2nd footnote that there are “different probability functions matching these desiderata”. Care to say more on this?
If you could link me to these similar derivations I’d be interested to read them, I mostly wrote and worked through this because I couldn’t find any existing ones from first principles and was sure it would be possible.
price = probability is a general rule for prediction markets, it’s more that CPMMs can be derived from the probability function we described (no/(no+yes)).
The generalization I’m using in my current implementation is https://manifoldmarkets.notion.site/Multi-CPMM-62fe5b99013c4d5a87dfa84e0b8fa642, although I believe Manifold currently uses some bizarre auto arbitrage system between linked binary predictions for exclusive categories. Similarly, you can also extend CPMM by adding a parameter to allow market initialization at different probabilities, and also to allow users to inject liquidity without pushing probability towards 50%.
Regarding other probability functions, there are of course a whole family of constant function market makers that CPMM is a member of. As a trivial example, (no2/(no2+yes^2)) should also match our desiderata, I believe.
Additionally, starting from the angle of “We have a market maker with pools of shares, how do they calculate probability from these pools” is just one approach you can take.
There’s also the LMSR (Logarithmic Market Scoring Rule) also developed by Robin Hanson, which approaches it from an entirely different angle, starting from asking how you can score predictors based on how well they performed, and then applying this to rewards and incentive alignment in a prediction market. This is actually more reflective of the Bayesian structure of the market than CPMM is, I was largely joking when I made that claim.
There’s also DPM (Dynamic Parimutuel), which adapts existing parimutuel betting systems to prediction markets. It does have the disadvantage of not being able to know ahead of time how much money you’ll receive from your bet, only how much money you’ll receive in expectation from your bet, but it has some advantages of its own.
I have this paper saved to read through and think about, I don’t really understand it yet but it also proposes a unique solution to this problem.
Largely, CPMM is the one I understand most intuitively out of all of these, which is part of why I’m using it in my personal prediction market implementation.
Thanks for the questions!
Edit: After encountering some problems I have since done more research. Multi-CPMM is dumb and bad and Multi-Binary (manifold’s current implementation) is superior in every regard actually. I am signing the manifold markets apology form (reason for behavior: thought the decision was for architectural reasons, was repelled by the auto arbitrage nature). I will hereby respect Manifold Markets and I will NOT talk down on the greatest prediction markets platform of all time.
I will try to dig up some references for you. Sorry it really was a small side project and has been several years.
Ah so I can’t imagine a probability function for that market that isn’t yy+n. y2y2+n2 is a fine pricing function that doesn’t appear to adhere to the rules of probability theory. If I try to compose two y2y2+n2markets, one conditional on the other, then can I multiply their prices to find the joint probability? Does this violate “price=probability”?
”price=probability is a general rule for prediction markets” is a very interesting claim. Seems obvious, but then you have to ask yourself whose probability?
I’m familiar with some of those operations (I’ve skimmed the Boyd paper). Unfortunately, there are a lot of different ways of expressing the same constraints, so I can’t immediately tell whether Manifold’s implementation is equivalent to what I had imagined.
Thanks for your answers, I’ll look into some of the other ideas you referenced.
What I mean is that the if the AMM estimates the probability at .75, it should charge .75 for a marginal YES share, by law of expected utility. I don’t think a different probability function should alter the probabulity theory, just change the pricing curve.
If “price=probability”, then changing the pricing curve is equivalent to changing how the AMM updates its probability estimates (on evidence of buy/sell orders).
Yes, but it just affects how liquidity is allocated, and it doesn’t just affect how the AMM updates, it affects how users trade as well since they respond to that, either way they’d want to bet to their true probability. So changing the pricing curve is largely a matter of market dynamics and incentives, rather than actually affecting the probabilistic structure.