Defensive Rebalancing for Automated Market Makers

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.

💡 Research Summary

The paper tackles a fundamental vulnerability of constant‑function market makers (CFMMs): when multiple pools trade the same assets at divergent prices, arbitrageurs can extract risk‑free profit, which is ultimately borne by liquidity providers (LPs). To eliminate this leakage, the authors introduce “defensive rebalancing”, a mechanism that moves tokens directly between the pools of different CFMMs, bypassing the usual trade‑by‑trade interaction. A rebalancing is self‑funding – it conserves the total amount of each token – but can raise the liquidity of one or more pools while never decreasing any other’s liquidity.

The authors first prove that any arbitrage‑prone configuration admits a rebalancing that both removes arbitrage opportunities and strictly increases the liquidity of at least one pool without harming the rest. They then define Pareto efficiency under rebalancing: a state where any feasible rebalancing would necessarily lower the liquidity of some pool. The central theoretical result (Theorem 3.7) shows that a configuration is arbitrage‑free if and only if it is Pareto‑efficient. Consequently, achieving one property automatically guarantees the other.

For practical deployment, the paper casts the search for an optimal, arbitrage‑free rebalancing as a convex optimization problem when the trading function is log‑concave (which includes the ubiquitous constant‑product market maker). The objective maximizes the sum of log‑liquidity across all pools, subject to (i) non‑negativity of final pool sizes, (ii) liquidity non‑reduction constraints (F_i(x_i′) ≥ F_i(x_i)), and (iii) linear conservation constraints linking the transfer variables δ to the new pool balances. Because the objective is concave in the log‑domain and the constraints are linear, the problem is a standard convex program with a unique global optimum that can be solved efficiently by existing solvers.

The framework is further extended to “mixed rebalancing”, where some CFMMs refuse direct pool‑to‑pool transfers or act as price oracles (e.g., centralized exchanges). In such cases, those pools adjust their prices through ordinary trades while the remaining pools perform direct transfers. The same convex formulation applies, allowing the system to harvest arbitrage profits from non‑participating pools or oracle markets while still guaranteeing an arbitrage‑free, Pareto‑efficient final state.

Finally, the authors discuss practical implications: a “rebalancing‑as‑a‑service” could periodically monitor a network of CFMMs, detect arbitrage‑prone states, and execute the optimal rebalancing automatically. This not only shields LPs from value leakage but also raises overall liquidity, reduces slippage for traders, and maximizes capital efficiency across the ecosystem. The work thus provides a rigorous, computationally tractable foundation for future AMM protocols that proactively defend liquidity providers while improving market quality.