Automated Market Makers (AMMs) Overview

• Automated Market Makers are decentralized, algorithm-driven systems that use mathematical invariants to continuously manage asset pools and set prices.
• They employ core mechanisms such as constant-product and constant-sum models to ensure liquidity without traditional order books, while addressing risks like impermanent loss.
• Advanced architectures integrate concepts like concentrated liquidity, external oracle pricing, and scalable shard designs to optimize fee income, risk management, and execution fairness.

Automated Market Makers (AMMs) are algorithmic agents, typically implemented as smart contracts on blockchains, that continuously manage pools of assets and autonomously set swap prices via a mathematical rule, enabling decentralized trading without order books or intermediaries. The AMM concept is foundational to the infrastructure of decentralized finance (DeFi), providing liquidity and facilitating on-chain, permissionless trading. Their operation, risk structure, and mathematical underpinnings have been rigorously formalized in the academic literature, and their design continues to evolve to address economic, executional, and scalability challenges.

1. Formal Definition and Core Mechanism

An Automated Market Maker holds reserves $(x, y)$ (for a two-asset AMM), defines an invariant or potential function $u(x, y)$ (or $\varphi(x, y)$), and supports swaps such that the post-trade state remains on the same indifference surface: $u(x+\Delta x, y-\Delta y) = u(x, y)$ (Bichuch et al., 2022, Loesch, 2022, Bartoletti et al., 2021). The pricing rule is entirely internal—the price of one asset in terms of the other is calculated as the ratio of partial derivatives: $P(x, y) = \frac{u_x(x, y)}{u_y(x, y)}.$ This mechanism sidesteps the need for order books and enables continuous liquidity.

The most prevalent class of AMMs are Constant Function Market Makers (CFMMs); the canonical examples include:

• Constant-product: $x \cdot y = k$ (Uniswap v2), yielding a marginal price $y/x$ and slippage scaling with trade size.
• Constant-sum: $x + y = k$ (mStable), offering zero slippage but susceptible to reserve depletion.
• Weighted geometric mean: $\prod_{i=1}^n x_i^{w_i} = k$ (Balancer), enabling multi-asset/portfolio pools.

These invariants underpin price discovery, slippage behavior, and liquidity sensitivity (Kirste et al., 2023, Loesch, 2022).

2. Axiomatic and Geometric Frameworks

Axiomatic characterizations abstract away from specific invariants to generalize desirable AMM properties. The fundamental axioms on the utility function $u(x, y)$ include:

• Unboundedness from below/above (no zero-reserve free lunch, infinite liquidity),
• Strict monotonicity (more reserves $\implies$ higher utility),
• Continuity and quasiconcavity (well-behaved indifference surfaces),
• Scale-invariance (homogeneity under scaling),
• Inada boundary/marginal behavior (marginal price diverges as reserves vanish),
• Single-crossing curvature (ensures unique, monotonic pricing) (Bichuch et al., 2022).

The swap-size function

$$Y(x; a, b) := \sup\{ y \mid u(a+x, b-y) \geq u(a, b) \}$$

is strictly increasing and concave in $x$; marginal pricing is determined by $P(a, b) = Y'(0; a, b) = u_x(a, b) / u_y(a, b)$. The geometric point of view regards the AMM as a strictly convex manifold $S_A = \{R: u(R) = k\}$, where arbitrage leads to stable points minimizing portfolio value along $S_A$ with respect to external prices (Engel et al., 2021, Tiruviluamala et al., 2022).

3. Impermanent Loss and Liquidity Provider Risks

Impermanent loss (IL), or divergence loss, measures the shortfall experienced by a liquidity provider (LP) versus a static portfolio held off-chain when the external market price moves. For a two-asset constant-product AMM, the closed-form for IL, given an initial price ratio $\mu$ and final ratio $\mu'$, is: $$\mathrm{IL}(\mu') = 2\sqrt{\mu'/\mu} / (1 + \mu'/\mu) - 1.$$ In $n$-asset pools, or more general AMMs, IL is a function of the geometry of invariant surfaces and can be further reduced for General Geometric Mean Market Makers (G3Ms), where IL depends solely on exchange-rate quotients (exchange-rate level independence—ERLI) (Tiruviluamala et al., 2022). More complex invariants (e.g., Curve's StableSwap) introduce additional IL degrees of freedom, complicating the LP risk profile.

Fee income, arbitrage loss, and opportunity cost jointly determine LP incentive to participate. Empirical results show that higher asset volatility or low correlation increases IL and reduces LP deposit flows (Capponi et al., 2021). Fee design, including marginal rather than per-swap fees, can mitigate trade splitting arbitrage and align user-LP incentives (Bichuch et al., 2022).

4. Advanced AMM Architectures

4.1 Concentrated Liquidity and Range Order Models

Uniswap v3 introduced concentrated liquidity (CLMMs), allowing LPs to specify price intervals $[a, b]$ in which their liquidity is active. This boosts capital efficiency but increases exposure to "range-outs" (e.g., being left solely with one asset if price exits $[a, b]$). The current and withdrawable token balances for an LP with liquidity $L$ and active price $p$ are determined via (Monga, 23 Jul 2024, Tang et al., 15 Nov 2024): $$\text{if}\ p \in [a, b]: \quad x^r = L \left(\frac{1}{\sqrt{p}} - \frac{1}{\sqrt{b}}\right),\quad y^r = L(\sqrt{p} - \sqrt{a})$$ Game-theoretic analyses reveal that, under pro-rata fee sharing and budget constraints, Nash equilibria are unique and characterized by "water-filling": low-budget LPs exhaust budgets across all price buckets, richer LPs absorb residual profitable intervals (Tang et al., 15 Nov 2024). However, real-world LPs often deviate from equilibrium, preferring wider ranges and infrequent reallocation.

Stochastic control approaches formalize the LP's interval-choice tradeoff: narrow intervals maximize fee income but expose LPs to frequent reallocations and high divergence loss, while wide intervals reduce capital efficiency. Simulation and sample-average approximation (SAA)+MIP techniques, as in (Zeller et al., 23 Apr 2025), support optimal interval selection under realistic fee and reallocation structures.

4.2 Oracle-Integrated and Multi-Asset AMMs

Mechanisms like UAMM (Im et al., 2023) and batch-clearing AMMs (Chan et al., 14 Feb 2024) integrate external price oracles to set swap prices, eliminating arbitrage when oracle prices are efficient. UAMM introduces a target-balance anchor and only applies constant-product liquidity when the internal pool assignment would depart from oracle-aligned proportions, thus minimizing impermanent loss. Batch-clearing AMMs settle all block trades at uniform batch prices and provide arbitrage-resilience and sequencing-fairness, important for MEV mitigation.

Multi-asset AMMs can be globally optimal in profit and risk under suitable convexity and transport duality structures (Curry et al., 14 Feb 2024). For assets with stationary spreads (e.g., pegged assets, liquid staking tokens), specialized AMMs utilizing Ornstein-Uhlenbeck process filtering and quadratic control achieve tight risk–fee frontiers (Bergault et al., 12 Nov 2024).

4.3 Scalable and Sharded AMM Architectures

Scalability frameworks such as SAMM (Chen et al., 8 Jun 2024) distribute trading and liquidity provision across multiple independent AMM "shards", each a local CPMM. SAMM employs a bounded-ratio polynomial fee function that discourages trade splitting and enforces self-balance across shards. Game-theoretic analysis (Subgame-Perfect Nash Equilibrium) ensures that traders and LPs distribute load optimally and the system achieves near-linear throughput scaling up to the parallelism limits of the underlying blockchain.

5. Execution Costs, Statistical Arbitrage, and Market Microstructure

AMMs directly encode execution costs as a function of pool depth and invariant convexity. For constant-product mechanisms, slippage per trade of size $y$ is proportional to $Z^{3/2}/\kappa\,|y|$ (where $Z$ is marginal price, $\kappa$ is pool depth). Execution models estimate trading costs and optimal strategies for liquidity takers under stochastic convexity and statistical arbitrage across CEX/DEX venues (Cartea et al., 2023, Monga, 23 Jul 2024).

Empirical evidence confirms that AMM execution costs and market impact curves closely match theoretical convexity-based predictions (Monga, 23 Jul 2024). "Lead-lag" dynamics and cross-venue price spillovers highlight the role of off-chain order books as effective price oracles for on-chain AMMs.

6. Mechanism Design, Order Flow, and Miner Extractable Value (MEV)

AMMs are vulnerable to order-sequencing attacks and MEV extraction. Miners (or sequencers) can profitably front-run/back-run trades (the "sandwich" attack), extracting all price-improvement rent at the LP’s and trader’s expense (Bartoletti et al., 2021). Formal models demonstrate that, in the absence of exogenous sequencing guarantees, optimal adversarial strategies layer "Dagwood sandwich" attacks—front-running each honest user’s action and then resetting minted-token prices for maximum gain.

Batch auction mechanisms (Chan et al., 14 Feb 2024) and fair ordering protocols eliminate intra-block MEV by executing all trades at uniform batch prices and guarantee incentive compatibility for honest trading under sequencing fairness. Empirical and formal proofs confirm arbitrage-resilience—that no risk-free profit can be extracted by combining block trades.

7. Design Taxonomy, Best Practices, and Practical Trade-offs

AMM design is multidimensional. The comprehensive taxonomy in (Kirste et al., 2023) enumerates governance, pricing, liquidity, and trading dimensions:

Dimension	Influence	Examples
Governance	Parameter tuning, upgrades	DAO vs. team
Pricing	Price discovery, slippage	CPMM, Oracle, StableSwap, Sum
Liquidity	Structure, flexibility	Open vs. permissioned
Trading	Execution guarantees	Slippage tolerance, limit orders

Three archetypes capture most deployed AMMs:

• Price-discovering LP-based: e.g. CPMM/CFMM; internal invariant, subject to arbitrage, prone to impermanent loss.
• Price-adopting LP-based: e.g. oracle AMMs; prices set by external feed, minimizing arbitrage and impermanent loss.
• Supply-sovereign (bonding curve): e.g. token issuance, with controlled supply-price evolution.

Parameters—fee levels, slippage curves, volatility-adaptive bands—and invariant selection must be tuned to asset correlation, desired liquidity profile, and LP risk tolerance. Concentrated liquidity, dynamic fees, and stable-swap curves are preferred under high-correlation pairs; broad-range, internal price discovery under volatile pairs.

Practical deployments must track gas costs, feature concentration incentives, and integrate risk-mitigating features (e.g., external insurance, imbalance surcharges). Empirically, LP strategies that closely match Nash equilibrium water-filling in CLMMs outperform naive, wide-range, or infrequent updates (Tang et al., 15 Nov 2024).

Conclusion

Automated Market Makers constitute a rigorously formalized, compositional, and decentralized class of on-chain market microstructures. Their core geometric, axiomatic, and economic properties are now well-understood, informing both practical designs and ongoing research—including risk-minimizing mechanisms, oracle integration, multi-asset generalizations, and scalable architectures. The explicit encoding of liquidity, risk, and execution costs makes AMMs both transparent and adaptable, but their continued evolution is driven by a balance between capital efficiency, LP risk, execution fairness, and system scalability.