AMM Invariant

The most common AMM invariant, used by protocols like Uniswap V2. It states that the product of the quantities of two tokens in a pool (x and y) must remain constant (k). This creates a hyperbolic bonding curve where price changes are continuous and liquidity is theoretically infinite.

• Price is the slope of the curve at any point.
• Impermanent Loss occurs because the formula rebalances reserves as trades execute.

A hybrid invariant designed for trading pegged assets (e.g., stablecoins). It combines the constant product formula with a constant sum formula (x + y = k) to create a flatter curve within a defined price range.

• Low Slippage: Enables large trades with minimal price impact for correlated assets.
• Concentrated Liquidity: The invariant allows liquidity providers to focus capital where it's most effective.

An evolution where liquidity providers can set custom price ranges for their capital, making the invariant active only within those bounds. This increases capital efficiency.

• Virtual Reserves: The invariant x * y = k is calculated using virtual reserves, not the real token amounts in the pool.
• Tick System: Prices are discretized into ticks, each with its own liquidity depth and local invariant state.

Transaction fees are integrated into the invariant's operation. For a constant product pool, fees are typically taken in the incoming asset, which slightly increases the constant k after a trade.

• Protocol Fee: A portion may be directed to governance or treasury.
• Dynamic k: The invariant's constant grows over time as fees accumulate, rewarding liquidity providers.

AMM invariants provide a built-in price oracle. The instantaneous price is derived directly from the reserve ratio. For manipulation resistance, protocols like Uniswap V2 use the time-weighted average price (TWAP) calculated from cumulative price data stored by the invariant's state.

• On-Chain Data: The invariant's state is the single source of truth for these price feeds.

Extends the concept to pools with N tokens and customizable weights. The invariant is ∏ (balance_i ^ weight_i) = k (the product of each reserve raised to its weight).

• Weighted Pools: Allows for pools with non-50/50 distributions (e.g., 80/20).
• Multi-Asset Exposure: A single pool can hold many tokens, with the invariant governing all pairwise relationships.

The most common invariant, used by Uniswap v2 and others. It states that the product of the quantities of two tokens in a pool (x * y) must remain constant (k) before and after a trade. This creates a hyperbolic bonding curve where price changes are continuous and liquidity is always available, but slippage increases with trade size. The spot price of Asset A in terms of Asset B is given by Price_A = y / x.

A hybrid invariant designed for low-slippage swaps between pegged assets like stablecoins. It combines the constant product formula with a constant sum formula (x + y = k), creating a flatter curve within a defined price range. This allows for extremely efficient trades near the peg, while the constant product component ensures liquidity and acts as a fallback if the price deviates significantly.

An evolution where liquidity providers (LPs) can allocate capital to a specific price range, rather than the full (0, ∞) curve. The invariant becomes x * y = L², where L represents liquidity. This increases capital efficiency for LPs and reduces slippage for traders within the active range. The pool's overall curve is the aggregation of all individual, segmented liquidity positions.

The invariant is the direct cause of impermanent loss (divergence loss). When the market price of the pooled assets changes, the AMM's rebalancing mechanism—enforced by the invariant—causes the pool's value to diverge from simply holding the assets. For a constant product pool, the loss is greater the larger the price movement. This is the fundamental risk-reward trade-off for liquidity providers.

Invariants can be generalized for pools with more than two assets. For example, a Balancer pool with n tokens and weights w_i uses the invariant: ∏ (balance_i)^{w_i} = k. This allows for customizable, multi-token pools where the invariant ensures the weighted geometric mean of the balances remains constant, enabling complex portfolio-like liquidity provision.

AMM reserves, governed by their invariant, provide a decentralized source of price data. The time-weighted average price (TWAP) oracle, pioneered by Uniswap, reads cumulative price data from the pool state over an interval. This mitigates manipulation, as attacking the price requires moving the pool's reserves against the invariant, which becomes prohibitively expensive over longer time windows.

The most common AMM invariant, used by Uniswap V2 and others. It states that the product of the quantities of two tokens in a pool (x and y) must remain constant (k).

• Price is determined by the ratio of the reserves: Price of X = y / x.
• Slippage increases with trade size because each trade moves the price along a hyperbolic curve.
• This model provides infinite liquidity but bounded reserves, ensuring a price for any trade size.

The opportunity cost liquidity providers (LPs) face when the price of deposited assets changes compared to simply holding them. It is a direct consequence of the invariant's rebalancing mechanism.

• Mechanism: When the external market price diverges, arbitrageurs trade against the pool to re-align it, changing the reserve ratios. LPs end up with more of the depreciating asset and less of the appreciating one.
• Maximized during high volatility; it is 'impermanent' only if prices return to their original ratio.

A hybrid invariant designed for stablecoin pairs (e.g., USDC/DAI) that approximates a constant sum formula (x + y = k) near parity but reverts to a constant product curve at extreme prices.

• Benefit: Drastically reduces slippage and impermanent loss for pegged assets within a defined 'amplification coefficient' range.
• Trade-off: Higher complexity and vulnerability if the peg breaks, as the pool can be drained along the flatter curve.

AMM prices are often used as on-chain oracles. The invariant's continuous pricing can be exploited if an attacker can move the pool price significantly within a single transaction.

• Flash Loan Attack: A borrower uses a flash loan to perform a large, imbalanced swap, skewing the price, triggering downstream contracts (like lending protocols) at the manipulated rate, and then arbitraging the price back.
• Mitigations: Use time-weighted average prices (TWAPs), as implemented by Uniswap V3 oracles, which are costly to manipulate over longer time intervals.

An evolution where LPs can allocate capital within custom price ranges, making the invariant active only within those bounds.

• Mechanism: The global x*y=k curve is replaced by many virtual curves over segmented price intervals. Liquidity becomes a 'liquidity density' function.
• Impact: Greatly increases capital efficiency for LPs who accurately predict price ranges, but introduces more active management and the risk of liquidity being 'out of range' and earning no fees.

The invariant is a core smart contract security property; its immutability ensures the pool cannot be drained through standard swaps.

• Verification: Protocol audits rigorously check that all state-changing functions (swaps, adds, removes) preserve the invariant.
• Rounding Direction: A critical implementation detail. Swaps must always round in favor of the pool (increase k slightly) to prevent gradual reserve depletion from many small trades.
• A broken invariant represents a critical failure, potentially allowing infinite minting or draining of assets.

The most common AMM invariant, expressed as x * y = k, where x and y are the reserve quantities of two assets and k is a constant. This formula ensures that the product of the reserves remains unchanged by trades, creating a hyperbolic bonding curve where price changes are a function of the trade size relative to the pool's liquidity.

• Key Property: Price impact increases non-linearly as the pool's reserves become imbalanced.
• Example: Used by Uniswap V2 and many foundational AMMs.

A smart contract that holds reserves of two or more tokens, enabling permissionless trading via an AMM invariant. Liquidity providers deposit an equal value of both assets to fund the pool, earning fees from trades.

• Components: Token reserves, the invariant formula, and a fee mechanism.
• Role: Serves as the automated counterparty for all swaps, replacing traditional order books.

The graphical representation of the price relationship between two assets in a pool, derived from its invariant. The curve's shape dictates pricing and slippage.

• Constant Product: Forms a hyperbolic curve, ensuring infinite liquidity but variable price.
• StableSwap: Creates a flatter curve within a price range for stablecoin pairs, reducing slippage.
• Function: Visually shows how the price of asset Y in terms of X changes as reserves shift.

The difference between the expected price of a trade and the executed price, caused by moving along the bonding curve. It is a direct consequence of the AMM invariant and the trade's size relative to pool liquidity.

• Calculation: Larger trades consume more liquidity, moving the price further along the curve.
• Mitigation: Traders set slippage tolerance limits; pools with deeper liquidity (higher TVL) exhibit lower slippage for standard trade sizes.

A risk for liquidity providers where the value of their deposited assets diverges from simply holding them, due to price changes governed by the invariant. It occurs when the external market price of the pooled assets changes after deposit.

• Mechanism: The AMM invariant forces the pool to rebalance reserves along the curve, selling the appreciating asset and buying the depreciating one.
• Offset: Providers aim to earn enough trading fees to compensate for this potential loss.

An AMM design that allows liquidity providers to allocate capital within a specific price range, deviating from the full-range model of a basic constant product pool. This makes the invariant active only within the chosen bounds.

• Benefit: Provides greater capital efficiency and deeper liquidity where it's most needed.
• Invariant Impact: The constant product formula x * y = k still applies, but only for the active price interval. Used by Uniswap V3.