  Automated market makers (AMMs) allocate fee revenue \textit{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 and two types of liquidity traders
trade. In this setting, we find that competition drives LPs to provide excess
liquidity. Excess liquidity is costly as more capital is exposed to adverse
selection costs. One of our main results 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. This inefficient
capital allocation is masked when considering the welfare of elastic liquidity
traders as the total price of anarchy is $O(1)$. Since this result is driven by
elastic liquidity traders benefiting from the liquidity provided because of
inelastic liquidity traders, we show that different types of liquidity traders
complement each other. Finally, we show that AMM designs that reduce the
arbitrage intensity per unit of liquidity do increase utility for liquidity
traders but importantly not for LPs nor do they necessarily decrease total
arbitrage volume.
