This paper introduces and analyzes \emph{defensive rebalancing}, a novel mechanism for protecting constant-function market makers (CFMMs) from value leakage due to arbitrage. A \emph{rebalancing} transfers assets directly from one CFMM's pool to another's, bypassing the CFMMs' standard trading protocols. In any \emph{arbitrage-prone} configuration, we prove there exists a rebalancing to an \textit{arbitrage-free} configuration that strictly increases some CFMMs' liquidities without reducing the liquidities of the others. Moreover, we prove that a configuration is arbitrage-free if and only if it is \emph{Pareto efficient} under rebalancing, meaning that any further direct asset transfers must decrease some CFMM's liquidity. We prove that for any log-concave trading function, including the ubiquitous constant product market maker, the search for an optimal, arbitrage-free rebalancing that maximizes global liquidity while ensuring no participant is worse off can be cast as a convex optimization problem with a unique, computationally tractable solution. We extend this framework to \emph{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 harvesting arbitrage profits from non-participating CFMMs, and from price oracle market makers such as centralized exchanges. Our results provide a rigorous foundation for future AMM protocols that proactively defend liquidity providers against arbitrage.

翻译：本文提出并分析了一种名为"防御性再平衡"的新机制，旨在保护常函数做市商（CFMMs）免受套利导致的价值流失。所谓"再平衡"操作，是指直接将资产从一个CFMM资金池转移至另一个资金池，从而绕过CFMM的标准交易协议。我们证明，在任何存在套利风险的配置状态下，总存在一种再平衡方案可以转化为无套利配置，该方案能在不降低其他CFMM流动性的前提下，显著提升部分CFMM的流动性。此外，我们证明配置状态具有无套利性的充要条件是该状态在再平衡操作下达到帕累托最优，这意味着任何进一步的直接资产转移都必然导致某些CFMM的流动性下降。针对包括广泛使用的恒定乘积做市商在内的所有对数凹交易函数，我们证明寻找最优无套利再平衡方案（在确保所有参与者利益不受损的前提下实现全局流动性最大化）可转化为具有唯一解且计算可行的凸优化问题。我们将此框架扩展至"混合再平衡"场景，其中参与再平衡的CFMM子集通过结合直接资产转移与标准交易的方式，在向无套利配置过渡的同时，还能从非参与CFMM以及中心化交易所等价格预言机做市商处获取套利收益。本研究为未来主动保护流动性提供者免受套利侵害的AMM协议奠定了严格的理论基础。