  Constant Function Market Makers (CFMMs) are a tool for creating exchange
markets, have been deployed effectively in prediction markets, and are now
especially prominent in the Decentralized Finance ecosystem. We show that for
any set of beliefs about future asset prices, an optimal CFMM trading function
exists that maximizes the fraction of trades that a CFMM can settle. We
formulate a convex program to compute this optimal trading function. This
program, therefore, gives a tractable framework for market-makers to compile
their belief function on the future prices of the underlying assets into the
trading function of a maximally capital-efficient CFMM.
  Our convex optimization framework further extends to capture the tradeoffs
between fee revenue, arbitrage loss, and opportunity costs of liquidity
providers. Analyzing the program shows how the consideration of profit and loss
leads to a qualitatively different optimal trading function. Our model
additionally explains the diversity of CFMM designs that appear in practice. We
show that careful analysis of our convex program enables inference of a
market-maker's beliefs about future asset prices, and show that these beliefs
mirror the folklore intuition for several widely used CFMMs. Developing the
program requires a new notion of the liquidity of a CFMM, and the core
technical challenge is in the analysis of the KKT conditions of an optimization
over an infinite-dimensional Banach space.
