Hey all!

We just put out a new paper showing some interesting properties of Uniswap and its generalizations. The paper can be found on arXiv.

Essentially, we show a few things:

- The instantaneous Uniswap arbitrage problem (along with the Balancer arbitrage problem) is convex, even when adding in most reasonable (read: convex, monotonic) models of the market. This implies that the arbitrage conditions are often easy to compute and are extremely likely to hold in practice.
- There doesn’t seem to be a way of depleting Uniswap markets of their reserves by only trading the pair of coins found in the market (i.e., the only possible way to truly deplete a market is by burning UNI coins).
- As one might expect, the larger the fees, the larger the no-arbitrage bounds are, which means that the Uniswap price may stray further from the true market price.
- The paper also gives an explicit formula for Uniswap returns in the no-fee case by constructing an equivalent portfolio.

We also leave some questions which may be of interest to the Eth research group:

- What is the right view of convexity for AMMs? We mention that both Balancer and Uniswap are also log-log convex (as is any AMM which uses concave, nonincreasing functions for their bonding curves), but don’t explore this topic further.
- Under what conditions can we guarantee that liquidity providers have positive expected value relative to a portfolio with equivalent weights to the UNI coins minted?
- Are there even more natural generalizations (or classes) of AMMs which have better properties than Uniswap/Balancer? Can we characterize what conditions are necessary/sufficient for a “good” AMM? What does this even mean in practice?

Anyways, please feel free to ask us any questions (or pose any problems that come to mind!) about/from this paper