How decentralized exchanges work: pools, prices and the x × y = k rule

What is a decentralized exchange?#

ethereum.org defines a decentralized exchange (DEX) as an Ethereum app “that lets you swap tokens with peers on the network”, noting that DEXs are not subject to the geographical restrictions of centralized exchanges ^[2]. The app is a collection of smart contracts: programs that live on the blockchain and run exactly as written. ethereum.org stresses that when you trade on a DEX you never give up control of your assets ^[3].

Federal Reserve researchers list three practical differences from a centralized exchange (CEX). A CEX can take and pay out ordinary money; a DEX cannot, which is why stablecoins serve as the dollar-like asset on DEXs. A CEX uses market makers and a limit order book; a DEX uses an automated market maker. And a CEX often checks your identity, while a DEX runs on permissionless smart contracts ^[4]. For a side-by-side view, see CEX vs DEX.

Centralized exchange vs decentralized exchange

Why don’t most DEXs use an order book?#

On a stock exchange, buyers and sellers post orders at different prices in an order book, and professional market makers keep updating them. Uniswap’s documentation argues that order books need intermediary infrastructure to host and match orders, and active market makers with sophisticated tools — features that fit poorly on a blockchain where anyone can create a token and most tokens have low liquidity ^[5]. Read more about the traditional model in order books and liquidity.

An automated market maker (AMM) replaces the order book with a liquidity pool: a smart contract holding reserves of two tokens. You trade against the pool itself rather than against another person’s order, and the price updates automatically as the balance of the two tokens changes ^[1].

How does the constant-product formula set the price?#

When you take one token out of the pool, you must put enough of the other in to keep the product constant ^[7]. In Uniswap v2 a 0.30% fee is taken from the amount you pay in before the formula is checked, and the fee stays in the pool for the liquidity providers ^[6]. The result: small trades get a price close to the pool’s current price, while large trades push the price further against themselves ^[1].

Your trade moved the pool’s price up by almost 7%. That gap between the price you see and the price you get is what traders call slippage; the part caused by your own trade size is called price impact.

Computed from x × y = k with the fee applied to the input amount. In a pool ten times deeper (1,000 ETH and 3,000,000 USDC), the same 10,000 USDC trade would be only about 0.63% worse than 3,000.

Do all DEXs price trades with x × y = k?#

No. Michael Egorov’s 2019 StableSwap paper says the constant-product rule suits assets like ETH but works poorly for coins that are meant to stay at the same price: for them, price slippage is “enormous” unless the pool holds very large funds ^[8]. The opposite extreme is a constant-sum rule, x + y = constant, which always prices one coin at exactly one of the other ^[8]. The paper notes that this does not work in a fluctuating market unless the price is adjusted all the time ^[8].

StableSwap is a middle ground built for stablecoin pools. Near balance it behaves almost like constant-sum, so the price changes slowly and slippage stays small; as the pool becomes lopsided it shifts toward constant-product ^[8]. That shift means it still offers some liquidity at any price, unlike a pure constant-sum pool ^[8]. A setting called the amplification coefficient (A) controls the blend: the lower A is, the closer the pool behaves to constant-product ^[8]. The paper used A = 100 when calculating slippage and compared that with using Uniswap with 100× leverage ^[8].

The StableSwap paper gives the full equation for any number of coins; this table shows only the idea behind it.

Who puts the tokens in the pool, and what do they earn?#

Anyone can become a liquidity provider (LP) by depositing both tokens of a pair ^[1]. New deposits must match the pool’s current ratio. In return, the pool mints liquidity tokens that represent the provider’s share; every trade pays a 0.3% fee that is shared pro rata among LPs, and burning the liquidity tokens returns the provider’s share of both reserves plus accumulated fees ^[5].

Providing liquidity is not risk-free. When prices move, the pool automatically rebalances, which can leave a provider worse off than if they had simply held the two tokens. The Uniswap v2 whitepaper calls this impermanent loss ^[6]. We work through the numbers in liquidity pools and impermanent loss.

How does a DEX price stay in line with other markets?#

The pool itself has no idea what ETH costs elsewhere. Arbitrage traders close the gap: if the pool’s price differs from the wider market by more than the fee, they trade against it until it lines up again. The Uniswap v2 whitepaper notes that, for this reason, the price offered by Uniswap tends to track the relative market price of the assets ^[6]. ethereum.org describes DEX arbitrage as the simplest and most competitive form of maximal extractable value (MEV) ^[9].

What happens when you press “swap”?#

1. Pick the tokens and amount

   The app shows how much you will receive; the AMM gives instant feedback on rates and slippage ^[7].

2. Approve the token (first time only)

   Many contracts must first be approved on the token contract before they can move your tokens ^[7]. Approvals are powerful — see token approvals.

3. Sign the transaction

   Your wallet asks you to confirm and pay a network fee. See Ethereum gas fees.

4. The contract checks the rule

   A router contract computes the trade and the pool contract checks the x × y = k rule; if the pool cannot be paid in full, the whole transaction is reverted ^[6].

5. Tokens arrive in your wallet

   The swap is final. Interactions with smart contracts are irreversible ^[10].

How can you limit slippage when you swap?#

Uniswap’s guide calls its router contract the easiest way to swap tokens safely, and each router swap takes a safety limit ^[11]. When you sell an exact amount, the limit is the minimum amount you will receive — called amountOutMin in the code; when you buy an exact amount, it is the maximum you are willing to pay ^[11]. Uniswap’s guide says the limit should be worked out from a price source outside the pool, and warns that without one, trades can be front-run “for considerable loss” ^[11].

Why outside the pool? The guide explains that a contract which treats the current Uniswap price as “fair” is vulnerable to manipulation: a bad actor could place trades just before and after yours — a sandwich attack — so that you trade at a much worse price ^[11]. For people trading on their own behalf, it calls the trader’s own off-chain observation of the current price the best check to pass into the trade ^[11].

The quote already includes the fee and your own price impact; the minimum adds a margin for the price moving before your transaction is confirmed. Uniswap’s guide adds that sandwich attacks are mitigated by trading in extremely liquid pools and/or at low values ^[11].

What are the risks of using a DEX?#

• Price impact and slippage in small pools, as the table above shows. A minimum amount out caps how much worse than the quote you accept ^[11].
• Sandwich attacks. Bots watch for large pending DEX trades, buy just before and sell just after. ethereum.org says users who are sandwiched face increased slippage and worse execution ^[9].
• Fake or worthless tokens. Uniswap’s design is permissionless: anyone can swap, provide liquidity or create new markets ^[1]. Nothing stops a copycat token with a familiar name.
• Smart contract and app risk. EU supervisors warn that crypto service providers have suffered cyber-attacks and severe operational problems ^[12].
• No recourse. There is no customer-support desk that can reverse a swap sent with the wrong token or amount ^[10].

What mistakes do beginners make here?#

• Swapping a large amount in a small pool

  Price impact grows quickly as your trade becomes a bigger share of the pool. Check the quoted amount out before you sign.

• Trusting a token because it appears in the app

  Pools are permissionless. Verify the contract address from an official source.

• Approving unlimited spending without thinking

  An approval lets a contract move your tokens later. Review and revoke approvals you no longer need.

• Forgetting the network fee

  Each swap is a blockchain transaction. On a small trade, the network fee can be larger than the price difference you were chasing.

• Assuming a DEX is “safer” than a CEX in every way

  You avoid trusting an exchange with custody, but you take on smart contract risk, front-running and the full responsibility for mistakes.

• Swapping without a minimum amount out

  Uniswap’s guide warns that a swap without a price check from outside the pool can be front-run for considerable loss ^[11]. Check the minimum you will receive before you sign.

Frequently asked questions#

The bottom line#

A decentralized exchange replaces the company and the order book with a pool and a formula. That makes trading open to any wallet at any hour, but it also means the price you get depends on how big your trade is compared with the pool, and that nobody can undo your mistakes.

Next, see what liquidity providers actually earn and risk in liquidity pools and impermanent loss, or look up DEX and slippage in the glossary.

Sources#