Automated Arbitrage Market Makers (A2MMs)

• A2MMs are algorithm-driven platforms in decentralized finance that close price discrepancies using adaptive pricing and zero-expected-profit principles.
• They utilize methods such as Kalman filtering, adaptive bonding curves, and convex optimization to mitigate loss-versus-rebalancing and distribute trading profits.
• Empirical studies show that A2MMs improve capital efficiency and reduce MEV risks by ensuring near-zero expected losses and optimal liquidity provider returns.

Automated Arbitrage Market Makers (A2MMs) are a class of algorithmic market platforms in decentralized finance (DeFi) designed to algorithmically close and capture price discrepancies between automated market maker (AMM) pools and external reference prices. By algorithmically identifying and neutralizing arbitrage opportunities, A2MMs fundamentally alter the distribution of trading profits and risks among liquidity providers (LPs), arbitrageurs, and protocol operators. This article surveys the motivations, mathematical foundations, operating principles, and empirical performance of A2MMs, as well as implementation and governance considerations.

1. Motivation and Problem Statement

In the canonical CFMM—such as those underpinning Uniswap v2/v3—prices are derived from fixed bonding curves (e.g., $x y = k$). When the external or “true” market price $p_{\text{ext}}$ changes, informed arbitrageurs trade against pools until the AMM’s marginal price tracks $p_{\text{ext}}$, inducing realized losses to the pool (loss-versus-rebalancing, LVR):

$$\text{LVR per trade} = (p_{\rm ext} - p_{\rm AMM}) \times \Delta x$$

This dynamic leads to adverse selection and recurrent “toxic order flow” problems (Nadkarni et al., 2024, McMenamin et al., 2022), often overwhelming fee income and reducing LP returns. A2MM protocols systematically address this by either (a) algorithmically updating marginal prices to achieve zero-expected-profit for adversarial arbitrageurs, or (b) auctioning or redistributing anticipated arbitrage profits on-chain, minimizing external leakage.

2. Algorithmic Foundations

A2MMs are rooted in both DeFi-specific market microstructure and classical market making theory:

• Zero-Expected-Profit Principle: The A2MM sets its swap pricing so that, conditioned on all available information (historical order flow, reference prices, etc.), the expected difference between the instantaneous pool quote and the true external price is zero. Formally, for trade “signals” $p^{t}_{\rm trad}$ and trade history $\mathcal H_{t-1}$:

$$p^t = \mathbb{E}[p_{\rm ext}^t | \mathcal H_{t-1}, p^t_{\rm trad}]$$

This is a direct adaptation of the Glosten–Milgrom microstructure model (Nadkarni et al., 2024).

• Adaptive Bonding Curves: The optimal bonding curve $g_t(p)$ satisfies an integral equation whose solution tracks the conditional external price. This yields a differential equation (for additive/GBM price models):

$$(\beta_t(p) - p)\,g_t'(p) + \beta_t'(p)\,[g_t(p) - x_0^t] = 0$$

where $\beta_t(p)$ is the filtered estimate of the external price conditioned on trade history.

• Kalman Filtering and Oracle-Free Estimation: External price is inferred via on-chain/off-chain Kalman filtering, eliminating reliance on external oracles and achieving fully endogenous parameter adaptation (Nadkarni et al., 2024).
• Defensive and Mixed Rebalancing: Defensive rebalancing uses convex optimization to identify direct asset transfers (“rebalancings”) across pools (including passive/active AMMs and oracles), ensuring Pareto-efficient, arbitrage-free configurations. The resulting convex program, unique under log-concave invariants, distributes arbitrage profit efficiently while maintaining non-negative liquidity for all participating pools (Devorsetz et al., 26 Jan 2026).
• Closed-form Arbitrage Optimization: For geometric-mean (Balancer-type) N-token pools, closed-form solutions for optimal arbitrage trades can be obtained, supporting efficient and parallelized on-chain execution (Willetts et al., 2024).

3. Design Patterns and Protocol Architectures

A2MMs can be organized along several architectural axes:

A2MM Design	Price Discovery Logic	Arbitrage Profit Allocation	Key Implementation Features
Endogenous Adaptive	Zero-profit price quoting	LPs retain (near-)all expected value	Kalman filtering, adaptive bonding curves
Centralized Auction (Diamond)	Sealed-bid or periodic conversion auctions	Arbitrage profit split: fraction $\beta$ rebated to pool	Collateralized block producer auctions of LVR (McMenamin et al., 2022)
Atomic Cross-pool	Joint multi-pool routing + atomic arbitrage	Swap initiator/user	Atomic batch swaps, trade routing (Zhou et al., 2021)
Defensive Rebalancing	Global liquidity-maximizing convex opt	All participating LPs via direct transfers	Convex program, no-LP-loss guarantee (Devorsetz et al., 26 Jan 2026)
Closed-form Multi-Asset	Global maximization over N-token cycles	Arbitrageur or protocol	GPU-parallel closed formulae (Willetts et al., 2024)

The above organizational table strictly reflects factual distinctions from the cited works.

Key Mechanics in Detail

• Adaptive Curves: The pool’s curve parameters are recalibrated block-by-block (or at every trade) off-chain, using recent trade data and an adaptive filter (see Kalman filter algorithm in (Nadkarni et al., 2024)). The updated parameters and the corresponding proof of correct filtering (often via SNARK/zk-proof) are then submitted on-chain, updating swap logic.
• Direct Arbitrage Capture and Distribution: Methods such as Diamond (McMenamin et al., 2022) require arbitrageurs (often the block proposer) to post collateral and execute the “arbitrage move” on behalf of the protocol. A deterministic fraction of realized arbitrage gain is rebated to the pool, with further reconciliation via periodic vault conversions.
• Atomic Multi-pool Routing: On-chain algorithms atomically split input swaps and route trades across multiple pools to maximize user output and neutralize arbitrage-induced divergence. Any residual price gaps are closed by a two-point arbitrage executed in the same transaction (Zhou et al., 2021).
• Defensive/Mixed Rebalancing: A global optimization routes direct asset transfers and trades across an active set of CFMMs and price oracles, guaranteeing that no LP is made worse off, whence all arbitrage surplus accrues to the involved set rather than external arbitrageurs (Devorsetz et al., 26 Jan 2026).

4. Mathematical Analysis and Closed-Form Results

• Optimal Adaptive Curve for Arbitrage Minimization: For additive (Gaussian) or geometric Brownian motion price models, the optimal adaptive curve is constructed from the one-step-ahead Kalman estimate. The Kalman gain $K_t$ is

$$K_t = \frac{\operatorname{Var}_{\text{prediction}}}{\operatorname{Var}_{\text{prediction}} + \eta^2}$$

leading to tractable closed-form bonding curves (Nadkarni et al., 2024).

• Empirical Upper Bound on Arbitrage-Induced Loss: In protocols such as Diamond, formal proofs show that LVR is capped at a $(1-\beta)$-fraction of that incurred by standard CFMMs, with no expected-value loss in vault token conversion through either auction or futures settlement under martingale asset price assumptions (McMenamin et al., 2022).
• Pareto Efficiency and Convexity in Defensive Rebalancing: Any configuration in which further direct rebalancing would strictly reduce some pool’s liquidity is necessarily arbitrage-free and vice versa. The global optimum is unique due to strict log-concavity, and is solvable with polynomial complexity for small numbers of pools (Devorsetz et al., 26 Jan 2026).
• No-Arbitrage Bands and Price Dynamics: With trading fee $\gamma$, AMM price process is constrained within $[\gamma P_t, P_t/\gamma]$, creating a Skorokhod-reflected process (Najnudel et al., 2024). The tradeoff between tightness of the peg and the frequency of arbitrage trades can be analytically optimized given expressions for expected boundary-hit rates.
• Closed-form Arbitrage for N-token Pools: For geometric mean (Balancer-type) AMMs, all nontrivial trade “signatures” are exhaustively evaluated, and the global profit-optimal arbitrage trade is available in analytic form, supporting on-chain and GPU parallelization (Willetts et al., 2024).

5. Implementation and Algorithmic Considerations

A2MM deployment typically involves hybrid on-chain/off-chain architectures:

• On-chain Components: Smart contracts enforce pool state transitions, collateral management, swap routing, and handle multi-pool atomic execution. Zero-knowledge proofs may be used for off-chain filter updates or solution verifications (Nadkarni et al., 2024).
• Off-chain Components: Filtering and state estimation (notably, adaptive Kalman filters) are generally offloaded to trusted off-chain or layer-2 coprocessors, which process trade history, update parameter estimates, and submit state proofs on-chain.
• Algorithmic Complexity: Convex program solvers for defensive rebalancing are polynomial in $n$, the number of pools. Closed-form multi-asset arbitrage is exponential in $n$ due to signature enumeration but highly parallelizable, yielding practical implementations for moderate pool sizes (Willetts et al., 2024).
• Atomicity and MEV Mitigation: All routes, swaps, and rebalancing moves are executed atomically, eliminating outsized MEV extraction by external searchers and reducing consensus instability and network-layer overhead by up to 32.8% in simulation (Zhou et al., 2021).
• Pricing Reference and Oracle Use: Many A2MM architectures use only endogenous (trade-based) estimation of the reference price, avoiding external price feeds. When oracles are included (e.g., in mixed rebalancing), timely and accurate feeds are essential for minimizing stale-price arbitrage leaks (Nadkarni et al., 2024, Devorsetz et al., 26 Jan 2026).

6. Empirical Performance and Comparative Evaluation

Simulation and backtesting studies reveal:

• Near-zero Expected Losses: Adaptive Kalman filter A2MMs exhibit near-zero expected loss across a wide parameter regime, outperforming static CFMMs especially under volatility and adversarial perturbations (Nadkarni et al., 2024).
• MEV and Fee Reduction: Atomic, on-chain A2MMs recapture the majority of value from multi-pool arbitrage, reducing block-space and P2P overhead by over 30%. Net user fee reductions of approximately 90% are documented (Zhou et al., 2021).
• LVR Mitigation: Protocols such as Diamond consistently outperform traditional CFMMs and HODL strategies by ~1% in simulated daily returns over annual horizons, with advantage scaling in volatility and LVR-rebate parameter $\beta$ (McMenamin et al., 2022).
• Multi-asset Efficiency: Closed-form solvers for N-token AMMs produce slightly greater arbitrage profits and much lower computational latency than convex solvers, enabling exploitation of granular, fleeting opportunities and providing edge in “dueling arbitrageur” scenarios (Willetts et al., 2024).
• Defensive Rebalancing Efficacy: Convex optimization-based rebalancing restores pools to arbitrage-free configurations without impairing any participant’s liquidity, and supports mixed strategies linking both DEX and CEX liquidity reservoirs (Devorsetz et al., 26 Jan 2026).

7. Limitations, Extensions, and Open Directions

Prominent limitations include:

• Collateral and Liveness: Block-proposer collateralization in auction-based A2MMs can constrain participation; dynamic adjustment to $\beta$ or auxiliary “CoW-bundling” can reduce overhead (McMenamin et al., 2022).
• Front-running and Adversarial Behavior: Some A2MMs, including Diamond, do not alter intra-block ordering or sandwich attack surface. Defensive mechanisms remain an area for research integration (McMenamin et al., 2022).
• Parameter Tuning and Adaptivity: Empirical results show that truncated-history filters maintain optimality under non-stationarity, but fast regime shifts remain challenging (Nadkarni et al., 2024).
• Complexity of Multi-asset and Multi-pool Coordination: While GPU-parallelization renders N-token closed-form arbitrage practical for moderate $n$, scaling to dozens of assets remains an active engineering concern (Willetts et al., 2024).
• Oracle Integration: For mixed-rebalancing, the security and liveness of oracles, especially under rapid market stress, is a critical design constraint (Devorsetz et al., 26 Jan 2026).

Ongoing directions include concentrated liquidity extensions, volatility-adaptive parameterization, and integration with advanced fair-ordering consensus or minor MEV-protection layers (McMenamin et al., 2022, Nadkarni et al., 2024).

---

In summary, Automated Arbitrage Market Makers synthesize microstructure theory, algorithmic filtering, atomic multi-pool routing, and convex optimization to deliver markedly improved value retention, capital efficiency, and systemic stability in AMM-driven markets. The technical literature demonstrates robust performance gains for LPs, with rigorously characterized guarantees under a range of price models, implementation regimes, and adversarial behaviors (Nadkarni et al., 2024, Zhou et al., 2021, Devorsetz et al., 26 Jan 2026, McMenamin et al., 2022, Willetts et al., 2024).