Weighted Geometric Mean Market Maker

A Weighted Geometric Mean Market Maker is an automated market maker (AMM) whose pricing curve is defined by the equation ∏ x_i^{w_i} = k, where x_i represents the reserve amount of asset i, w_i is its non-negative weight summing to 1, and k is a constant. This generalizes the classic Constant Product Market Maker (x * y = k), which is a special case where all weights are equal (0.5, 0.5). By adjusting the weights w_i, liquidity providers can control the pool's price sensitivity and capital allocation, creating pools that are more stable for certain asset pairs or tailored to specific trading expectations.

The core innovation is adjustable curvature. A higher weight for an asset makes the pool's reserves of that asset deplete more slowly as it is traded, making its price more stable within the pool. For example, a stablecoin pair like USDC/DAI might use a 50/50 weight for efficiency, while a volatile pair like ETH/DAI might use an 80/20 weight to reduce impermanent loss for the ETH side and provide deeper liquidity around the current price. This flexibility allows for more capital-efficient liquidity pools compared to a one-size-fits-all constant product curve.

Prominent implementations include Balancer, where pools can have 2 to 8 assets with customizable weights. This enables the creation of index-like pools or portfolios that automatically rebalance through arbitrage. The weighted geometric mean formula ensures the pool remains homogeneous, meaning liquidity can be added in any proportion of the underlying assets, though often in a specific ratio to minimize slippage for the depositor. The marginal price of an asset is determined by the ratio of its reserve to its weight, and all trades must preserve the invariant k.

From a technical perspective, the spot price of asset i in terms of asset j is given by (x_j / w_j) / (x_i / w_i). This shows how weights directly influence pricing. The slippage profile is less steep near the equilibrium point for assets with higher weights, benefiting high-volume, low-spread trading pairs. This design makes Weighted G3Ms a foundational primitive for DeFi applications requiring customized AMM logic, serving as the engine for decentralized exchanges, liquidity vaults, and protocol-owned liquidity strategies.

A Weighted Geometric Mean Market Maker operates on the invariant (V = \prod_{i=1}^{n} R_i^{w_i}), where (R_i) is the reserve of asset (i) and (w_i) is its non-negative weight summing to 1. This generalizes the classic Constant Product Market Maker (like Uniswap V2's (x * y = k)) by allowing for asymmetric liquidity provision and customizable price curves. The weights control each asset's influence on the pool's price sensitivity; a higher weight makes the pool's value more sensitive to changes in that asset's reserve, allowing for tailored fee structures and reduced impermanent loss for specific trading pairs.

The core mechanism adjusts spot prices algorithmically based on the ratio of reserves and their weights. The instantaneous spot price of asset (i) in terms of asset (j) is derived as (P_{i/j} = (R_j / w_j) / (R_i / w_i)). This allows pools to concentrate liquidity around a target price more efficiently than a standard constant product curve. Prominent implementations like Balancer V1 utilize this model to create multi-asset pools (e.g., with 3 assets weighted 80%/10%/10%), enabling complex portfolio management and gas-efficient swaps between any two assets in the pool without direct trading pairs.

Key advantages of the WGMMM model include capital efficiency for correlated assets and customizable pool governance. By setting unequal weights, liquidity providers can express a market view, such as holding a larger reserve of a stablecoin. However, this introduces complexity in managing divergence loss (impermanent loss), which is minimized when assets move according to their weight proportions. This design is foundational for index pools, managed portfolios, and decentralized fund management on-chain, providing a flexible primitive beyond simple token swaps.

A Weighted Geometric Mean Market Maker (G3M) is defined by the constant product invariant, expressed as ∏ x_i^{w_i} = k. Here, x_i represents the reserve amount of asset i, w_i is its predefined, constant weight (where ∑ w_i = 1), and k is the invariant constant. This formula is a generalization of the Constant Product Market Maker (CPMM) used by Uniswap V2, which is the special case where all weights are equal (e.g., 0.5 for a two-asset pool). The invariant ensures that for any trade, the weighted geometric mean of the reserves remains constant, automatically setting the exchange rate based on the ratio of assets in the pool.

The weights (w_i) are the critical tuning parameter. A pool with weights (0.8, 0.2) will hold 80% of its value in the first asset and 20% in the second, making it much less sensitive to price changes in the dominant asset. This design allows for capital efficiency for correlated assets (like stablecoin pairs) and enables the creation of Balancer-style pools that can contain multiple tokens with custom weightings. The price impact of a trade is derived by differentiating the invariant, resulting in a slippage curve that depends directly on the chosen weights and the size of the trade relative to the reserves.

Compared to a constant sum invariant (which enables zero slippage but can be drained) or a constant mean invariant, the geometric mean provides a robust, convex curve that guarantees liquidity is never fully depleted. This mathematical property ensures the AMM can provide continuous liquidity across all prices, functioning as a decentralized price oracle. The invariant is recalculated after every trade, mint, and burn, with k increasing when liquidity is added (minting LP tokens) and decreasing when it is removed (burning LP tokens).

In practice, this model powers advanced DeFi primitives. For example, a liquidity bootstrapping pool might start with a weight configuration like (0.98, 0.02) for a new token and a stablecoin, gradually shifting weights to (0.50, 0.50) to smooth price discovery. The flexibility of the weighted geometric mean invariant is foundational to concentrated liquidity models as well, where the invariant is applied over a specific price range, requiring even more complex integrals (like in Uniswap V3) to track the constant k within that interval.