Paradigm is the largest investor of Uniswap, which makes this article worth reading. The theme of this article is to regard V3 as a more generic AMM: any static AMM can be approximated by a custom liquidity provision strategy involving multiple positions on Uniswap V3.

We can visualize the liquidity provided by any AMM as a curve in “tick space”. Applying this method to existing AMMs like Curve, Balancer, and the logarithmic market scoring rule (LMSR) shows how those AMMs concentrate their liquidity across different prices, revealing the unique “liquidity fingerprint” corresponding to each of those AMMs.

Uniswap V3:

In Uniswap V3, anyone can create a position to provide a certain amount of liquidity – L – within a price range between two ticks. Tick indexes (ti) are logarithmic in price and specify the lower and upper prices at which that position provides liquidity.

Liquidity (L) can also be defined as the rate of change of y for a given change in the square root (P):

L3’s curve and distribution of liquidity:

Uniswap V2:

V2’s liquidity can be described with the following equation: , where L=s, and liquidity is a constant.

Curve:

The liquidity can be described with the following equation:

Balancer

The liquidity can be described with the following equation: 

The exponential function of liquidity: 

Logarithmic market scoring rules

The liquidity can be described with the following equation: 

L: 

This paper shows how several popular AMMs could be simulated using Uniswap V3 and demonstrates how graphing curves in liquidity space provides insight into their unique “liquidity fingerprint”.  However, several limitations need to be overcome before Uniswap V3 can be used to simulate most of these AMMs:

• Ticks are not infinitely divisible, and liquidity providers have to approximate this curve using the available intervals of certain granular ticks.
• The gas cost of minting and burning is proportional to the number of ticks updated, so providing liquidity at custom levels across all of tick space would be insurmountably inefficient.
• Finally, not every useful AMMs can be easily represented as a function in liquidity space.

Future work will demonstrate some techniques for getting around these problems using numerical approximation, as well as some smart contract tricks to make custom liquidity provision much more gas-efficient.

Link to original article: https://www.paradigm.xyz/2021/06/uniswap-v3-the-universal-amm/