Defensive Rebalancing in Networks of Automated Market Makers

Abstract

This paper introduces and analyzes defensive rebalancing, a novel mechanism for preventing arbitrageurs from extracting surplus from networks of constant-function market makers (CFMMs).
A rebalancing transfers assets directly from one CFMM’s pool to another’s, bypassing the CFMMs' standard trading protocols. In any arbitrage-prone configuration, we prove there exists a rebalancing to an arbitrage-free configuration that strictly increases the liquidity of at least one participating CFMM without decreasing the liquidity of any other participating CFMM. Moreover, we prove that a configuration is arbitrage-free if and only if it is Pareto efficient under rebalancing.
We prove that for any log-concave trading function, including the ubiquitous constant-product market maker, finding an optimal arbitrage-free rebalancing that maximizes the aggregate liquidity of the participating CFMMs while preserving each participant’s liquidity can be formulated as a convex optimization problem with a unique, computationally tractable solution.
We extend this framework to mixed rebalancing, where a subset of participating CFMMs use a combination of direct transfers and standard trades to transition to an arbitrage-free configuration while capturing arbitrage opportunities from non-participating CFMMs and external price-setting oracles such as centralized exchanges.
Our results provide a rigorous foundation for future CFMM protocols that coordinate defensive rebalancing to internalize arbitrage surplus within networks of CFMMs.