Lesson 1 — AMMs explained: Uniswap V2, V3, V4, Curve, Balancer

**Uniswap V2** is the canonical constant-product AMM. The pool holds two reserves (call them x of token A and y of token B). The product x·y is held constant across every swap: if a trader removes Δx of token A, they must add enough B so that (x − Δx)(y + Δy) = x·y. Solving for the price impact gives the swap output and the slippage. The smaller your trade relative to the reserves, the smaller the slippage. The larger your trade, the more nonlinearly slippage grows. The constant-product formula is symmetric — it provides liquidity at every price from zero to infinity — which means capital is spread thin everywhere and concentrated nowhere.

**Uniswap V3** introduced concentrated liquidity. LPs deposit liquidity within a chosen price range — say between $1,800 and $2,200 for ETH/USDC — and capital is active only within that range. Inside the range, the effective depth is much greater than V2 for the same capital, so traders see less slippage. Outside the range, the position holds 100 percent of one asset and earns nothing. The trade-off: LPs must choose ranges and monitor them, which is a more involved operation than V2's set-and-forget model. The protocol uses a 'tick' system where the price space is discretized into log-spaced intervals; positions are bounded by ticks.

**Uniswap V4** (released 2024–2025) keeps V3's concentrated-liquidity model and adds 'hooks' — programmable callbacks that execute before or after each pool operation. Hooks let pool creators add features like custom oracles, dynamic fees, MEV protection, or limit orders without forking the protocol. V4 also moves all pool accounting into a single 'PoolManager' singleton contract, which dramatically reduces gas costs for multi-hop swaps. The risk surface expands accordingly — every hook is custom code that can have its own bugs.

**Curve Finance** uses a different formula entirely: the StableSwap invariant, which behaves like a constant-sum formula (x + y = k, near-zero slippage) when prices are near 1:1 and like a constant-product formula (x·y = k, normal AMM slippage) when they diverge. This is the right curve for assets that should trade at a fixed ratio — stablecoins, ETH and stETH, BTC and WBTC — and it's why Curve dominates stablecoin-pair trading. The cost is higher computational complexity and a more complex oracle surface that has been exploited multiple times (most notably the July 2023 Vyper-compiler reentrancy exploit affecting Curve, JPEG'd, Alchemix, and Metronome for ~$70M).

**Balancer** generalizes the AMM model further: any number of tokens in a pool, with arbitrary weights summing to 100 percent. A 'BAL-WETH-USDC 33/33/33' pool prices each pair against the others using a weighted geometric mean. Balancer's design enables index-fund-style baskets and was a foundation for protocols like Element and Aura. Like Curve, the additional complexity expands the audit surface.

**Where slippage actually comes from.** Three sources, often confused. (1) Price impact — the deterministic movement caused by your trade against the curve. Large trades against shallow pools cause large price impact. This is what the AMM math computes. (2) Sandwich-attack slippage — value extracted by MEV bots that detect your pending transaction in the mempool, front-run with a buy that pushes price against you, then back-run with a sell to close out. This is recoverable in principle with private mempools (Flashbots Protect, MEV-Share). (3) Adverse selection / informed flow — sophisticated traders pick off LP positions when external markets have moved but the AMM price hasn't yet, effectively buying from LPs at stale prices. This is a structural cost of being an LP and is the main driver of LVR (loss-versus-rebalancing) covered in academic literature.

**The right question for any AMM deposit.** Not 'what's the APY?' but 'on which curve, against what flow, with what fee tier, in what price range?' The APY headline number assumes specific volume, fees captured, and absence of adverse selection. The breakdown matters because the same APY can come from sustainable fee yield or from being adversely-selected by every informed trader.