Weighted Automated Market Maker (Weighted AMM)

Weighted AMMs generalize the constant product formula which is characteristic of standard 50/50 AMMs like Uniswap V2. For a liquidity pool containing $n$ different tokens, where each token $i$ has a current balance (liquidity) $B_i$ and a pre-defined, fixed weight $W_i$ (such that the sum of all weights $\sum_{i=1}^{n} W_i = 1$), the core invariant or value function $V$ that the AMM strives to keep constant (ignoring fees) during trades is often defined as: $$V = \prod_{i=1}^{n} B_i^{W_i}$$ This formula implies that for any swap to occur between two tokens in the pool (say, Token X and Token Y, which are part of the $n$ tokens), the balances of all tokens in the pool must adjust such that this weighted geometric mean of the balances remains constant. The price of one token relative to another is determined by the ratio of their balances and their weights. For example, in a two-token pool with Token A (balance $B_A$, weight $W_A$) and Token B (balance $B_B$, weight $W_B$), the spot price of A in terms of B can be derived as $rac{B_B/W_B}{B_A/W_A}$. This design allows for several interesting use cases: * **Index-like Pools**: Pools can be created to mimic a target asset allocation, like a crypto index fund. * **Reduced Impermanent Loss (IL) Strategies**: LPs can give higher weight to assets they are more bullish on or perceive as less volatile, potentially altering their IL profile compared to a 50/50 pool. * **Liquidity Bootstrapping Pools (LBPs)**: A variation where weights can change over time, often used for initial token distribution.