A Deep Dive into Automated Market Maker Design

Automated Market Makers (AMMs) have become the backbone of modern decentralized finance. They enable token swaps, liquidity provision, and price discovery without relying on traditional order books. This article examines the design principles that underpin AMMs, focusing on the broader class of Generalized Market Makers (GMMs). For a broader perspective, see /revealing-how-amms-and-gmms-shape-modern-trading.

From Order Books to Liquidity Pools

Traditional exchanges match buyers and sellers through an order book. Each trade consumes an existing order, and prices move gradually as supply and demand shift. In a decentralized environment, publishing order books requires significant coordination and incurs latency. AMMs resolve this by maintaining a continuous, self‑balancing pool of tokens. Users trade against the pool, and the pool’s invariant guarantees that the product of token reserves remains constant (or follows another defined function). This design eliminates the need for counterparties and allows instantaneous trades.

The key insight behind AMMs is that liquidity can be represented mathematically as an equation linking reserves. When a user swaps one token for another, the equation constrains how reserves must adjust. By solving the equation for the new reserve levels, the protocol determines the exchange rate that keeps the invariant satisfied.

The Constant Product Formula

The most widely known AMM model is the constant product formula, used by Uniswap v2 and many other platforms. The invariant is:

x * y = k

where x and y are the reserves of two assets and k is a constant. A swap modifies the reserves such that the product stays unchanged. The exchange rate is derived from the derivative of the invariant, yielding the familiar slippage curve:

price = (x + Δx) / (y - Δy)

The constant product model is mathematically simple, ensures liquidity across a wide price range, and is resistant to large price swings. However, it also suffers from high slippage for large trades and can lead to significant impermanent loss for liquidity providers (LPs) when the token pair’s relative value diverges.

Constant Sum and Other Simple Invariants

The constant sum invariant (x + y = k) is suitable for stable‑coin pairs where the ratio of the assets should remain close to 1:1. The price curve is linear, providing low slippage for small trades, but the pool quickly runs out of one asset if a large imbalance occurs. As a result, constant sum pools are rarely used for volatile asset pairs.

Other simple invariants include the constant mean and constant elasticity of substitution (CES) formulas, each offering a different balance between price stability and liquidity depth. Designers choose an invariant that aligns with the intended use case, whether it be a volatile exchange or a stable‑coin swap.

Generalized Market Makers (GMMs)

Generalized Market Makers (GMMs) expand the AMM concept by allowing the invariant to be any differentiable function F(x, y) = k. This generalization enables more sophisticated pricing curves that can adapt to different market conditions, fee structures, or token characteristics. The design space for F includes:

• Piecewise functions that switch between different invariants at certain price thresholds.
• Logarithmic or exponential functions that adjust slippage dynamically.
• Custom functions that incorporate external data, such as oracle prices or liquidity depth.

A GMM is defined by three components:

1. Invariant function F(x, y).
2. Fee schedule that determines how much of the trade volume is routed to LPs versus protocol treasury.
3. Liquidity weighting that may prioritize certain token amounts or liquidity providers.

Because GMMs encompass a wide array of possible invariants, they can emulate existing AMMs (e.g., Uniswap, Curve) while offering novel features such as hybrid price curves or dynamic fee models.

Designing an Invariant: Key Considerations

1. Market Volatility

Highly volatile pairs benefit from invariants that provide tighter slippage curves near the current price. For example, a power curve can be steeper in the mid‑range, reducing slippage for moderate trades while still maintaining depth for large orders.

2. Liquidity Provision Incentives

The shape of the invariant influences the risk of impermanent loss. A shallow curve near the equilibrium price reduces loss for LPs but may attract fewer traders due to higher slippage for large swaps. Balancing LP incentives with trader appeal is a core design trade‑off.

3. Computational Efficiency

Since every trade must solve for the new reserve levels, the invariant’s analytical solution must be fast to compute on‑chain. Complex functions that require iterative numerical methods increase gas costs and can become a barrier to adoption.

4. Predictable Gas Costs

When gas consumption varies widely with trade size or price, users face uncertainty. Simple invariants (product or sum) yield predictable gas usage, whereas custom functions may introduce variability.

Liquidity Provision and Impermanent Loss

Liquidity providers deposit equal‑value amounts of each asset into the pool and receive liquidity tokens representing their share. Their earnings come from trading fees and any appreciation of the pool’s total value.

Impermanent loss arises when the relative price of the two assets diverges from the initial deposit ratio. LPs may end up holding more of the depreciated asset and less of the appreciated one, causing their share of the pool to be worth less than a simple holding of the underlying tokens. The loss is called “impermanent” because it can be mitigated if the price reverts to the original ratio.

Strategies to mitigate impermanent loss in GMMs include:

• Dynamic fee adjustment: Higher fees during volatile periods can compensate LPs.
• Liquidity weighting: Assigning higher weights to more stable tokens reduces exposure.
• Hybrid invariants: Combining constant product with stable‑coin segments (as in Curve) limits loss for pairs with high price correlation.

Slippage and Price Impact

Slippage is the difference between the expected price (based on pool reserves) and the execution price. For a constant product pool, slippage grows quadratically with trade size. GMMs can be engineered to control slippage by shaping the invariant’s curvature. For example, a log‑arithmic invariant yields a flatter slippage curve for large trades, attracting high‑volume traders.

A practical way to present slippage to users is via a “price impact” slider that dynamically updates as the trade size changes. This transparency helps traders decide whether to split a large order across multiple pools.

Advanced Features in Modern AMMs

1. Concentrated Liquidity

Protocol designs such as Uniswap v3 allow LPs to supply capital only within specific price ranges. This increases capital efficiency but requires careful modeling of liquidity distribution across the invariant’s domain. Concentrated liquidity can be represented as a piecewise GMM where the invariant applies only to the active range.

2. Meta‑Swaps and Layered Pools

Meta‑swaps aggregate liquidity from multiple underlying pools to provide deeper markets. Each layer may use a different invariant, creating a multi‑tiered GMM. Traders benefit from lower slippage, while LPs can choose to provide liquidity in the layer that best matches their risk profile.

3. Cross‑Chain and Interoperable AMMs

Cross‑chain bridges enable AMMs to swap assets across different blockchains. The invariant must account for cross‑chain fees and slippage, potentially requiring a composite function that integrates on‑chain liquidity pools with off‑chain price feeds.

Risk Management and Governance

Decentralized AMMs are governed by on‑chain rules. However, the invariant’s choice influences systemic risk. For instance, a highly convex invariant may amplify price swings, exposing the pool to flash‑loan attacks. Governance proposals can adjust the invariant, fee schedule, or LP reward structures.

Auditability is critical: the mathematical proof that the invariant is correctly enforced by the smart contract must be straightforward to verify. Using standard, well‑tested invariants reduces complexity and mitigates bugs.

Real‑World Examples of GMMs

• Uniswap v2 – Constant product invariant, simple fees.
• Curve Finance – Piecewise invariant that uses a “stable” curve for highly correlated assets, reducing impermanent loss.
• Balancer – Supports multiple token pools with configurable weighting, essentially a GMM with arbitrary exponents.
• SushiSwap v3 – Concentrated liquidity, effectively a multi‑layer GMM.

Each protocol illustrates a different way of leveraging the GMM framework to meet specific market needs.

Future Directions

1. Adaptive Invariants
   Future AMMs may dynamically adjust the invariant based on market volatility or LP participation. Machine learning models could predict optimal curvature settings, feeding them back into the protocol.

2. Protocol‑agnostic Layered AMMs
   Building a modular layer that can wrap any underlying AMM allows the creation of composite invariants. This could enable cross‑protocol arbitrage opportunities while keeping each layer’s logic transparent.

3. Integration with Synthetic Asset Platforms
   Synthetic tokens require unique pricing curves due to their underlying real‑world assets. GMMs can incorporate oracle data directly into the invariant, enabling precise price discovery.

4. Improved Impermanent Loss Mitigation
   New mathematical techniques, such as dynamic rebalance algorithms, can help LPs automatically adjust their positions within the pool to minimize loss.

5. User‑Defined Invariants
   Giving power users the ability to design custom invariants (within safety constraints) could foster innovation. A sandbox environment would let developers test new formulas before deploying them to production.

Conclusion

Automated Market Makers have transformed how decentralized exchanges operate, and the Generalized Market Maker framework provides a powerful lens through which to understand, design, and innovate on top of this technology. By carefully selecting the invariant function, fee structure, and liquidity weighting, protocol designers can craft markets that balance liquidity depth, price stability, and LP incentives. As the DeFi ecosystem matures, we can expect to see increasingly sophisticated GMMs that adapt to market conditions, reduce impermanent loss, and provide richer trading experiences.