Automated market maker

Reading depth

Each view includes the earlier layers; the complete engineer or auditor page is shown by default.

BeginnerCore purpose and mental modelPractitionerAdds mechanisms and economicsEngineer or auditorAdds state and verification

Dependency check

Advanced detail assumes prerequisite knowledge

Before using practitioner mechanisms or engineer and auditor checks, review any unfamiliar dependency. These paths come from canonical depends_on relationships; Web3 Academy does not track completed lessons.

Review 6 dependencies

• AssetRequired through Asset versus claim

  1. Automated market maker
  2. Decentralized exchange
  3. Asset versus claim
  4. Asset

• ClaimRequired through Asset versus claim

  1. Automated market maker
  2. Decentralized exchange
  3. Asset versus claim
  4. Claim

• TokenRequired through Asset versus claim

  1. Automated market maker
  2. Decentralized exchange
  3. Asset versus claim
  4. Token

• Asset versus claimRequired through Decentralized exchange

  1. Automated market maker
  2. Decentralized exchange
  3. Asset versus claim

• Decentralized exchangeDirect prerequisite

  1. Automated market maker
  2. Decentralized exchange

• Liquidity poolDirect prerequisite

  1. Automated market maker
  2. Liquidity pool

An automated market maker is a trading mechanism that accepts or rejects exchanges according to a programmed rule over market state and liquidity.

Category: TradingOpposed arrows category cue

System record

Start with the economic purpose, participants, resources, and entitlements before studying implementation detail.

Why it exists

Automated market makers provide deterministic on-chain quotes and settlement without requiring every trade to match a newly posted opposing order.

Traditional-finance analogy

Automated dealer or market-making rule is the closest comparison recorded for this concept.

Where the analogy stops

• An AMM exposes pooled inventory through a public state-transition rule instead of letting a dealer choose each quote using private inventory and risk limits.
• Different AMMs use different curves, ranges, oracles, auctions, hooks, and fee rules; constant product is one model rather than the definition of every AMM.

Main actors

• ActorTrader
• ActorLiquidity provider
• ActorPool and pricing contracts
• ActorArbitrageur
• ActorGovernance, hook, or fee administrator

Assets and claims

Assets — controlled or transformed resources

Assets are resources the mechanism moves, holds, values, or transforms.

• AssetInput and output assets
• AssetPool reserves

Claims — entitlements and corresponding dependencies

Claims are rights to value, repayment, redemption, control, or another party's performance; each depends on an obligation or system that must honor it.

• ClaimLiquidity position or pool-share claim

An automated market maker, or AMM, accepts or rejects proposed trades using a programmed rule over liquidity and market state.

Why it exists

An AMM provides on-chain quotes and settlement without waiting for a newly posted opposing order for every trade. The rule makes execution predictable from contract state, but it does not know a universal fair price. External trading and arbitrage connect its state to other markets.

Traditional-finance analogy

An automated dealer is a useful analogy. A dealer chooses quotes from inventory, information, and risk limits; an AMM exposes a public state-transition rule. Some AMMs use external information or auctions, so even this distinction is not absolute. Constant product is one AMM, not the definition of all AMMs.

Constant-product example

For reserves x units of token X and y units of token Y, the simplified zero-fee rule is:

x × y = k
spot price of X in Y = y / x

If a trader adds X and removes Y, x rises, y falls, and the marginal price moves along the curve. With a fee retained in the pool, the post-trade product can increase. Exact formulas, rounding, protocol fees, and range behavior depend on the implementation.

Table: Constant-product example

AMM family	Main pricing state	Important difference
Constant product	Reserve product	Full-range two-asset baseline
Stable-swap curve	Reserves plus amplification-like parameters	Lower impact near a target region
Weighted pool	Reserves and weights	Multi-asset and unequal-weight exposure
Concentrated liquidity	Price, ticks, and active range liquidity	Depth changes across price intervals
Oracle or auction assisted	Pool state plus external information or bids	Adds information and timing dependencies

Actors, assets, and claims

Traders exchange reserve assets. Liquidity providers own pooled shares or range-specific pooled claims. Arbitrageurs trade discrepancies with external markets. Pool, hook, router, and position-manager contracts execute the rules. Governance or administrators may control fee, hook, pause, or upgrade parameters.

Step-by-step mechanism, state changes, and flows

1. Read reserves, active liquidity, fees, and relevant parameters.
2. Calculate or propose input and output amounts.
3. Transfer or account for the input.
4. Transfer the output.
5. Enforce the invariant, fee, price-limit, and callback conditions.
6. Record new reserves, price state, liquidity state, and fees.

Table: Step-by-step mechanism, state changes, and flows

State transition	Reserve X	Reserve Y	LP claim and fee state
Add liquidity	Rises by accepted X	Rises by accepted Y	Share or range-specific claim is created
Swap X for Y	Rises by settled X input	Falls by settled Y output	Retained fee follows the pool’s accounting
Remove liquidity	Falls by the provider’s X amount	Falls by the provider’s Y amount	Burned or collected claim is reduced
Failed trade	Reverts to its pre-trade state	Reverts to its pre-trade state	No successful trade or fee claim is recognized

Capital flow follows trader input/output and provider reserves. Claim flow follows LP positions. Return flow is trader fees and any separate incentives. Risk flow includes stale pricing, adverse selection, manipulation, thin or inactive liquidity, faulty math, callbacks, hooks, and privileged control.

Return source and loss allocation

Traders pay fees allocated under the pool rules. LP inventory gains or loses value as prices and reserve composition change. Arbitrage profit is funded by the executable difference between the AMM and another venue, net of costs; part of that transfer can be an adverse-selection cost to LPs.

Engineer or auditor lens

Specify units, domains, fee placement, and rounding for every formula. Test zero and near-zero liquidity, maximum inputs, exact input/output, reserve limits, overflow, callback payment, reentrancy, token behavior, tick crossings, inactive ranges, hooks, donations, and price limits. A manipulable instantaneous pool price is not automatically safe as an oracle for another protocol.

Uniswap’s cross-version overview and v2 pair implementation provide a constant-product lineage. Curve’s StableSwap paper shows a different curve objective. These are concrete mechanisms, not evidence that one curve dominates every market or that every implementation is safe. The living Uniswap references were reviewed 2026-08-10; the versioned contract and paper remain implementation artifacts rather than current parameter claims.

Common misunderstandings

• “The AMM reports the fair price.” It reports a price implied by its current state and rule.
• “x × y = k defines every AMM.” Curves, ranges, weights, oracles, auctions, hooks, and fees vary.
• “The invariant alone proves safety.” Authorization, accounting, token behavior, callbacks, control, and integration still matter.

Next separate price impact from total execution slippage.

Machine-readable model

Key equations

Canonical expressions come from the structured concept record. KaTeX renders the notation, while the plain-text expression and variable table keep its meaning and units inspectable without JavaScript. Read the narrative above for the model's domain, assumptions, and rounding rules.

1. Equation 1

   $$\mathrm{x} \times \mathrm{y} = \mathrm{k}$$

   Plain-text equation: x * y = k

   Variables and units for equation 1

   Symbol	Meaning	Unit
   x	Recorded token X reserve amount	token X reserve units
   y	Recorded token Y reserve amount	token Y reserve units
   k	Constant-product reserve invariant	token X units multiplied by token Y units

2. Equation 2

   $$\mathrm{spotPriceXInY} = \frac{\mathrm{y}}{\mathrm{x}}$$

   Plain-text equation: spotPriceXInY = y / x

   Variables and units for equation 2

   Symbol	Meaning	Unit
   spotPriceXInY	Spot price of token X quoted in token Y	token Y units per token X unit
   y	Recorded token Y reserve amount	token Y reserve units
   x	Recorded token X reserve amount	token X reserve units

Interactive module

Test the mechanism

Change reserves, trade size, fees, and the external market price; inspect execution, pool state, and arbitrage direction.

Open Constant-product AMM lab on its full lab page

Model inputs

Controls

Token X reserves

Unit: X · Must be greater than 0

X

Current amount of Token X held by the pool.

Token Y reserves

Unit: Y · Must be greater than 0

Y

Current amount of Token Y held by the pool.

Trade direction

Choose the asset sent to the pool

X to YSwap Token X for Token YY to XSwap Token Y for Token X

Trade size

Unit: X · Must be greater than or equal to 0

X

Gross amount of the input token sent by the trader.

Fee rate

Unit: % · 0% through 5%

%

Percentage charged on the gross input amount.

External market price

Unit: Y/X · Must be greater than 0

Y/X

Reference price in Token Y per Token X.

Reset allShare current state

Reserve geometry

Constant-product price curve

• Before trade
• After trade

The pool moves from 10,000.000000 X and 20,000.000000 Y to 11,500.000000 X and 17,398.112305 Y along the constant-product reserve curve.

Inspect all curve data points

Constant-product curve samples; each row holds the pre-trade reserve product fixed

Point	X reserve	Y reserve
1	4,500.000000 X	44,444.444444 Y
2	4,833.125000 X	41,381.094013 Y
3	5,166.250000 X	38,712.799419 Y
4	5,499.375000 X	36,367.769065 Y
5	5,832.500000 X	34,290.612945 Y
6	6,165.625000 X	32,437.911809 Y
7	6,498.750000 X	30,775.149067 Y
8	6,831.875000 X	29,274.540298 Y
9	7,165.000000 X	27,913.468248 Y
10	7,498.125000 X	26,673.335000 Y
11	7,831.250000 X	25,538.707103 Y
12	8,164.375000 X	24,496.669984 Y
13	8,497.500000 X	23,536.334216 Y
14	8,830.625000 X	22,648.453535 Y
15	9,163.750000 X	21,825.126177 Y
16	9,496.875000 X	21,059.559065 Y
17	9,830.000000 X	20,345.879959 Y
18	10,163.125000 X	19,678.986532 Y
19	10,496.250000 X	19,054.424199 Y
20	10,829.375000 X	18,468.286489 Y
21	11,162.500000 X	17,917.133259 Y
22	11,495.625000 X	17,397.923123 Y
23	11,828.750000 X	16,907.957307 Y
24	12,161.875000 X	16,444.832725 Y
25	12,495.000000 X	16,006.402561 Y
26	12,828.125000 X	15,590.742996 Y
27	13,161.250000 X	15,196.124988 Y
28	13,494.375000 X	14,820.990227 Y
29	13,827.500000 X	14,463.930573 Y
30	14,160.625000 X	14,123.670389 Y
31	14,493.750000 X	13,799.051315 Y
32	14,826.875000 X	13,489.019095 Y
33	15,160.000000 X	13,192.612137 Y
34	15,493.125000 X	12,908.951551 Y
35	15,826.250000 X	12,637.232446 Y
36	16,159.375000 X	12,376.716302 Y
37	16,492.500000 X	12,126.724269 Y
38	16,825.625000 X	11,886.631254 Y
39	17,158.750000 X	11,655.860712 Y
40	17,491.875000 X	11,433.880016 Y
41	17,825.000000 X	11,220.196353 Y

Scenario explanation

What is happening

You send 1,500.000000 X to the pool and receive 2,601.887695 Y. That adds X and removes Y, moving the pool price from 2.000000 to 1.512879 Y per X. The average execution price is 1.734592 Y per X, with 13.2704% price impact. The 4.500000 X fee remains in pool inventory for LP claim holders.

• Capital flow
• Claim flow
• Return flow
• Risk flow

Inventory movement

Reserve bars

Each outlined bar is the reserve before the trade; each solid bar is the reserve after. Labels and values carry the same meaning without color or motion.

• Token X10,000.000000 → 11,500.000000 X
• Token Y20,000.000000 → 17,398.112305 Y

Deterministic output

Results

Constant-product AMM results. Display values are rounded; calculations retain Decimal precision.

Metric	Value	Unit	Meaning
Spot price before	2.000000	Y per X	Y reserve ÷ X reserve
Quoted output	2,601.887695	Y	Amount sent from the pool
Average execution price	1.734592	Y per X	Gross-input execution price
Price impact	13.2704%	percent	Deterioration from spot
Fee paid	4.500000	X	Retained in the pool
X reserve after	11,500.000000	X	Post-trade pool inventory
Y reserve after	17,398.112305	Y	Post-trade pool inventory
Invariant before	200,000,000.000000	X·Y	x × y before
Invariant after	200,078,291.505372	X·Y	Includes retained fee
Pool price after	1.512879	Y per X	Post-trade Y ÷ X
Arbitrage direction	Sell Y, receive X	trade direction	Moves pool toward market
LP inventory change	+1,500.000000 X; -2,601.887695 Y	token amounts	Capital flow, not LP profit

Balance-sheet view

Pool state transition

Pool assets and LP claims before, during, and after the swap. Fee attribution is included in the gross trader flow, not added twice.

Pool account	Before trade	Trader flow	Fee attribution	After trade
Token X asset	10,000.000000 X	+1,500.000000 X	4.500000 X retained	11,500.000000 X
Token Y asset	20,000.000000 Y	-2,601.887695 Y	—	17,398.112305 Y
LP share claims	Pro-rata claim on pool assets	No shares minted or burned	Fee accrues inside pool assets	Supply unchanged; asset mix changed
Invariant (x × y)	200,000,000.000000 X·Y	Pricing curve applied	Retained input fee can increase k	200,078,291.505372 X·Y
Pool price	2.000000 Y per X	Moves against the trade	Fee included in execution	1.512879 Y per X

Guided comparison

Guided scenarios

Load a scenario, then compare its execution price, reserve movement, and invariant.

1. 01Small trade, low impactSee why a trade that is small relative to pool reserves stays close to the starting spot price.
2. 02Medium trade, visible slippageCompare execution price with spot price as the same pool absorbs a larger order.
3. 03Large trade, high impactObserve the nonlinear reserve and price movement created by a trade equal to half of the starting X reserve.

Assurance contract

Security properties

These structured statements define desired behavior. Their stable IDs can bind tests, invariants, specifications, audit findings, or proof results without turning descriptive review advice into an assurance claim.

Desired · not evaluated: No test, audit, or proof result is implied until scoped evidence is linked to this property.

1. SP-01Desired · not evaluated

   Every accepted trade satisfies the selected invariant and fee rule under explicit rounding

2. SP-02Desired · not evaluated

   No accepted trade can produce output without accountable input or make reserves negative

3. SP-03Desired · not evaluated

   Reserve, liquidity, and fee state remain reconciled after swaps, deposits, and withdrawals

4. SP-04Desired · not evaluated

   Spot-state manipulation is not treated as an external fair-price oracle without an explicit defense

Prerequisites

Direct dependencies recorded by this concept's canonicaldepends_on relationships.

• Decentralized exchangePath 20

  A decentralized exchange is a protocol that coordinates token trades through blockchain transactions rather than a single custodial exchange ledger.

• Liquidity poolPath 21

  A liquidity pool holds assets under shared accounting and trading rules so participants can exchange against available reserves.

Learn next

Concepts that directly depend on this one, ranked by the recommended learning path. Check each page for additional prerequisites.

• Price impactPath 26

  Price impact is the change in a venue's quoted or marginal price caused by the trade itself relative to the venue's available depth.

• Impermanent lossPath 28

  Impermanent loss is the value difference between an AMM liquidity position and a stated hold benchmark after relative asset prices change.

Knowledge check

Quiz

Answer in your own words, then open the model answer.

What problem does Automated market maker exist to address?

Model answer

Automated market makers provide deterministic on-chain quotes and settlement without requiring every trade to match a newly posted opposing order.

Mark as understood