Augmenting Batch Exchanges with Constant Function Market Makers
Geoffrey Ramseyer, Mohak Goyal, Ashish Goel, David Mazières
TL;DR
The paper develops an axiomatic framework to integrate CFMMs into multi-asset batch exchanges, revealing fundamental trade-offs that prevent simultaneously achieving all desirable properties. It defines minimal axioms and several target properties (PO, PC, PPC, JPD, LCR, and path independence) and proves key impossibility results, guiding design choices. It then offers two natural equilibrium-formation algorithms (Trading Rule U and Trading Rule S), analyzes which properties they satisfy (notably that U yields JPD and S yields PPC under the CLCP class), and provides a convex program that computes equilibria for two-asset, WGS-demand settings. Finally, it discusses beyond-equilibrium ideas like post-processing by CFMMs to reach JPD, highlighting practical trade-offs and open problems, including the role of fees and multi-asset generalizations.
Abstract
Batch auctions are a classical market microstructure, acclaimed for their fairness properties, and have received renewed interest in the context of blockchain-based financial systems. Constant function market makers (CFMMs) are another market design innovation praised for their computational simplicity and applicability to liquidity provision via smart contracts. Liquidity provision in batch exchanges is an important problem, and CFMMs have recently shown promise in being useful within batch exchanges. Different real-world implementations have used fundamentally different approaches towards integrating CFMMs in batch exchanges, and there is a lack of formal understanding of different design tradeoffs. We first provide a minimal set of axioms that are well-accepted rules of batch exchanges and CFMMs. These are asset conservation, uniform valuations, a best response for limit orders, and non-decreasing CFMM trading function. In general, many market solutions may satisfy all our axioms. We then describe several economically useful properties of market solutions. These include Pareto optimality for limit orders, price coherence of CFMMs (as a defence against cyclic arbitrage), joint price discovery for CFMMs (as a defence against parallel running), path independence for simple instances, and a locally computable response of the CFMMs in equilibrium (to provide them predictability on trade size given a market price). We show fundamental conflicts between some pairs of these properties. We then provide two ways of integrating CFMMs in batch exchanges, which attain different subsets of these properties. We further provide a convex program for computing Arrow-Debreu exchange market equilibria when all agents have weak gross substitute (WGS) demand functions on two assets -- this program extends the literature on Arrow-Debreu exchange markets and may be of independent interest.