The Cost of Permissionless Liquidity Provision in Automated Market Makers

Automated market makers (AMMs) allocate fee revenue proportional to the amount of liquidity investors deposit. In this paper, we study the economic consequences of the competition between passive liquidity providers (LPs) caused by this allocation rule. We employ a game-theoretic model in which $N$ strategic agents optimally provide liquidity. In this setting, we find that competition drives LPs to provide excess liquidity. In the limit, the excess liquidity converges to a constant that linearly increases with the amount of base demand, demand that is insensitive to trading costs. Providing excess liquidity is costly as more capital is exposed to adverse selection costs, leading to a loss in welfare. Our main result is that the price of anarchy, defined over the liquidity provider performance, is $O(N)$, implying that the welfare loss scales linearly with the number of liquidity providers. We show that this result is still observable when using richer aggregate demand models.