Automated Market Maker

An automated market maker (AMM) is a type of decentralized exchange that uses a mathematical formula – rather than an order book – to price trades. Traders swap against a liquidity pool instead of matching with a counterparty.

Traditional exchanges work by matching buyers and sellers: someone posts a bid, someone posts an ask, and the exchange matches them. This requires active market makers who continuously quote prices and enough participants on both sides to maintain a liquid book. On a blockchain, where every operation costs gas and blocks are seconds apart, maintaining an on-chain order book is expensive and slow. AMMs side-step the problem entirely. A smart contract holds reserves of two tokens, and a pricing function determines the exchange rate based on the ratio of those reserves. Anyone can trade at any time, and anyone can supply liquidity.

Pricing models#

The pricing formula is what defines an AMM’s behavior – how it sets prices, how much slippage traders face, and how capital-efficient it is.

Constant product (x * y = k)#

The simplest and most widely deployed model. Two token reserves, x and y, are constrained so their product never changes. When a trader buys token Y by depositing token X, x increases and y decreases, but x * y stays at k. This means the price rises as supply shrinks – a natural supply-demand curve. See constant product formula for a detailed walkthrough.

Uniswap V2 and SushiSwap use this model. It is simple, robust, and works for any token pair, but it spreads liquidity across the entire price range from zero to infinity, which is capital-inefficient.

Constant sum (x + y = k)#

The reserves sum to a constant, giving zero slippage at any trade size. Zero slippage also means the quoted price never moves away from an arbitrageur, so the moment the external price leaves the peg the whole reserve of the cheaper asset can be bought out. In practice this only works for pairs that are tightly pegged, and then only alongside another curve.

Constant mean (weighted pools)#

Balancer generalizes the constant product to multiple tokens with arbitrary weightings. Instead of a 50/50 two-token pool, a pool can be 80/20 ETH/USDC, or hold five tokens at once. The invariant becomes the weighted geometric mean of all reserve balances.

Concentrated liquidity#

Uniswap V3 introduced concentrated liquidity, where LPs choose a price range in which their capital is active. Within that range the pool behaves like a much deeper constant-product pool. Outside it, the LP’s position is inactive. Capital efficiency improves by roughly the ratio of the full curve to the chosen range — a factor in the thousands for a tight stablecoin band — and the cost is a position that has to be re-ranged as the price moves. Uniswap V4 extends this with hooks that allow custom logic on every swap.

StableSwap (Curve)#

Curve’s StableSwap invariant blends a constant-sum and constant-product curve, producing very low slippage near the peg and increasing slippage as the price diverges. It is optimized for assets that should trade near 1:1 (stablecoins, liquid staking tokens).

How a swap works#

1. The pool holds reserves of Token A and Token B. The invariant is a * b = k.
2. A trader sends da of Token A to the pool. The new Token A reserve becomes a + da.
3. The contract computes the new Token B reserve: b' = k / (a + da).
4. The trader receives b - b' of Token B (minus a fee, typically 0.3%).
5. The fee is added to the reserves, slightly increasing k and rewarding LPs.

Because the curve is convex, larger trades push the price further – this is slippage. Price impact scales with the trade as a fraction of the reserves, so a $100 swap against a $10M pool moves the price by a few thousandths of a percent and the same swap against a $100K pool moves it a hundred times as much.

Roles in an AMM#

Traders swap one token for another, paying a fee on each trade.

Liquidity providers (LPs) deposit tokens into the pool and receive LP tokens representing their share of the reserves. They earn a pro-rata share of trading fees but bear the risk of impermanent loss – the divergence in value between holding tokens in the pool versus holding them outright.

Arbitrageurs keep AMM prices aligned with the broader market. When the AMM price diverges from the price on other venues, arbitrageurs trade against the pool until the prices converge, pocketing the difference.

Risks and limitations#

• Impermanent loss is the dominant risk for LPs. In volatile pairs, IL can exceed fee income, making the position a net loss.
• Slippage on large trades is inherent to the bonding curve. Concentrated liquidity and deeper pools reduce it but don’t eliminate it.
• MEV (maximal extractable value) – sandwich attacks and front-running target AMM traders. A searcher sees a pending swap, trades ahead of it (pushing the price), and trades behind it (profiting from the movement). Private mempools and intent-based routing (UniswapX) are the main countermeasures.
• Smart contract risk – the pool is only as safe as its code. Uniswap’s v2 core has been live and unmodified since 2020, holding billions across every market condition since; a fork with edits, or a new curve, has none of that record behind it.

Notable AMM protocols#

External links#

• Wikipedia – Automated market maker
• Uniswap V2 – How it works
• Uniswap V3 – Concentrated liquidity
• Curve – StableSwap whitepaper