Thread by @tarunchitra

⚠️ Γ Alert ⚠️

What does part of Paul Milgrom's 2020 Nobel Memorial Prize have to do with 🦍-ing into pool 2?

2nd part of our series on CFMM shape looks at:

💹 How do you compare LP return from different pools?
🤼 Quantifying adverse selection in CFMMs

medium.com/gauntlet-netwo…

Post 1 tl;dr: Curvature controls pool price stability

Post 2: Curvature *directly* controls:
- LP profits when asset pairs are mean reverting
- ∃ a magic formula relating LP profit to adverse selection (probability α of LP realizing IL), curvature, and fees for *any* CFMM!

These results generalize Glosten & Milgrom (1984), Kyle (1985) to arbitrary CFMMs

This seminal work shows the shape of the order book represents the amount of adverse selection a market maker feels, leading to strategies where they remove liquidity to avoid adverse selection

Inspired by Jun Aoyogi's work on Uniswap adverse selection, we generalized his results to @CurveFinance & more

Similar to Jun, we construct a game between LPs 🐳, informed traders 🦍, & arbs 🧙 to find an equilibrium condition

But our game is different in a couple key ways 🔮

1. We model roundtrip trades: This is when an 🦍 makes a trade but ends up being wrong, this leads to 🐳, 🧙 profits

2. Our game more carefully connects impermanent loss to the shape of the CFMM as we explicitly consider the effects of changes in marginal price on LP share value

This shows that protective CFMMs like @BreederDodo and any CFMM @hosseeb says needs an oracle simply by reducing LP adverse selection by adjusting curvature

But this also intimates you can do this endogenously *without* an oracle if you are willing to store more on-chain state

So what does this mean for the future of CFMM design?
1. Curvature is dual to adverse selection (e.g. stablecoin-stablecoin trades aren't v. adverse, so you don't need much curvature)
2. Dynamic updates to a CFMM, either by an oracle or endogenously, need to account for shape

Designing dynamic update rules will inevitably be tied to designing the correct gradient flow that adjusts a CFMM's curvature

Given the centuries of work in differential geometry on such flows, it is inevitable that we will see them (and control theory) interacting with CFMMs

What's next? 🔮

The final post (and the release of the paper) will show how curvature and yield farming incentives (especially pool 2!) interact

We'll illustrate the optimization problem for *optimally* choosing farming rewards for pool 2

Thanks go out to:

Collaborators (@GuilleAngeris, @alexhevans)
Feedback: @_prestwich and @ciamac
Protocols that inspired this: @NotionalFinance @yield @BreederDodo @MonteCarloDEX @ShellProtocol

✌️🏿✌️🏿✌️🏿

P.S. If you're designing options AMMs (@0mllwntrmt3, @anisimov_andrei, @AndreCronjeTech, @opyn_) all of this advice about curvature applies doubly so — next week's post will go more into curvature + Greeks

*simply works