Popular automated market makers (AMMs) use constant function markets (CFMs) to clear the demand and supply in the pool of liquidity. A key drawback in the implementation of CFMs is that liquidity providers (LPs) are currently providing liquidity at a loss, on average. 

In this paper, we propose a new design for decentralised trading venues, the arithmetic liquidity pool (ALP). In the ALP, the protocol chooses (i) impact functions that determine how liquidity taking orders impact the marginal exchange rate of the pool, and (ii) sets the price of liquidity in the form of dynamic quotes around the marginal rate. The impact functions and the quotes determine the dynamics of the marginal rate and the price of liquidity. We show that CFMs are a subset of ALP; specifically, given a trading function of a CFM, there are impact functions and quotes in the ALP that replicate the marginal rate dynamics and the execution costs in the CFM. We show that the price of liquidity in CFMs is suboptimal in the ALP for LPs that maximise wealth. Also, we give conditions on the impact functions and the liquidity provision strategy to prevent arbitrages from roundtrip trades. 

The ALP design accommodates any dynamic LP strategy that depends on the ALP state. In this paper, we derive the optimal strategy where the price of liquidity maximises the expected profit of LPs. The strategy depends on the (i) tolerance to inventory risk and (ii) views on the demand for liquidity. Our strategy admits closed-form solutions and is computationally efficient.   Finally, we use transaction data from Binance and Uniswap v3 to show that liquidity provision is not a loss-leading activity in an ALP that implements our proposed strategy.