A new paper, "An Axiomatic Characterization of CFMMs and Equivalence to Prediction Markets", just came out (h/t @dreev). Skimming it, it seems kinda cool and potentially relevant to Manifold; but it uses a lot of math notation and cites prior literature.
I'm posting a M$500 bounty to anyone who writes up a summary of the paper & it's implications for Manifold, in a couple paragraphs! This market resolves to YES if this bounty is claimed before market close in a week.
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My ELI5 attempt:
This paper is the Rosetta Stone of prediction market AMMs. It turns out that constant function market makers (like Uniswap), cost function market makers (like Hanson's LMSR), and market mechanisms derived from proper scoring rules (like the Brier score), are all equivalent*. That is, you can turn turn one type of AMM into the other.** Which in turn means the a lot of findings from the defi world have implications for forecasting and vice versa.
The main thing Manifold should take away is more confidence that our Uniswap-based AMM is both good and fundamentally the same sort of thing as other more popular AMMs in the prediction market world.*** From this paper and earlier work by Chen & Pennock, we know that Maniswap**** is equivalent to a constant log-utility AMM that starts with a uniform prior, which it "updates" based upon subsidies it receives.
*assuming they satisfy basic properties you'd expect from a functional AMM like not being able to rob the AMM; that buying more of an asset doesn't decrease the price; etc.
** For instance, Uniswap is usually stated as a constraint x*y = k, where x and y are the units the AMM holds in reserve for two assets X and Y. But you can also re-write it in terms of a cost function C(x, y). A cost function means the cost of buying one unit of X is C(x+1,y) - C(x,y).
*** Our DPM free-response AMM does not meet the requirements of a good AMM, but we already knew that.
**** At least viewed statically, at a single point in time.
@SG Yes! And apparently that cost function is
C(x,y) = (x + y + Sqrt[4*k + (x-y)^2]) / 2
I want to read this paper so maybe producing this summary will be a good way to make sure I've gotten my head around it. In the meantime, from talking to Dave Pennock, the relevance to Manifold is for killing DPM for free-response markets. We now know how to generalize constant-function market makers to multiple outcomes!
@dreev Yes, the constant log utility market maker is naturally generalized to 3,4,5, or any number of outcomes, even countably infinite outcomes, even adding outcomes on the fly by, for example, dividing an "other" outcome into a new outcome plus a smaller "other" outcome. The cost function will not have a nice closed form like for 2 outcomes, but the cost function is well behaved and can be computed numerically using binary search. So perhaps this will be better than DPM for free response markets. As much as I like the idea of DPM being used in practice!!