Various Use Cases Based on the PMM Algorithm

Use Case 1: Active Market Making Following External Market Prices

This use case is applicable to mainstream coins like ETH and BTC, which have substantial external markets. PMM can adjust order book prices proactively to reduce impermanent loss (IL) and achieve higher capital utilization compared to AMM. Set as follows:

• Set the price 'i' to the external market price.

• Set 'k' to a value less than 1.

• Subsequent deposits (or withdrawals) must operate on both assets simultaneously according to the current asset ratio.

This use case is referred to as the "Standard Pool".

Use Case 2: Automatic Market-Making Algorithm with Low Capital Requirements

This use case applies to long-tail coin markets where there is no subjective pricing capability, and pricing is entirely left to the taker. Typically, long-tail coins are new assets with insufficient sell-side liquidity under the AMM framework, resulting in long-term liquidity shortages. PMM algorithm can effectively address this issue. Set as follows:

• Set the price 'i' to the initial issuance price.

• Set 'k' arbitrarily.

• The initial deposit can be in any asset ratio without changing the market price.

• Subsequent deposits (or withdrawals) must operate on both assets simultaneously according to the current asset ratio.

This use case is referred to as the "Single-Token Pool".

Use Case 3: Fully Customizable Market-Making Algorithm

This use case is for ambitious market makers, whether they are project teams or individuals, who want complete control over the liquidity pool. All parameters can be adjusted, and market makers can flexibly price assets based on their judgment. Market makers can deposit or withdraw a single asset without affecting the market price. For example:

• Market making near ETH=1600USDT with a small 'k' to gain competitive liquidity and earn significant fees. When anticipating an ETH price increase, withdraw ETH inventory to reduce unilateral risk without compromising buy-side liquidity.

• Projects issuing new tokens provide tokens without funds. Start selling tokens at $1. To allow more people to buy tokens at a reasonable price, a small 'k' is set to moderate token price increases. Later, the project can withdraw assets from the pool directly without harming sell-side liquidity, ensuring users can still buy plenty of inexpensive tokens.

Set as follows:

• Only market makers can deposit and withdraw, allowing single-asset operations.

This use case is referred to as the "Private Pool".

Use Case 4: Stablecoin Trading

Set as follows:

• Set 'i' to 1.

• Set 'k' to 0.01 (the smaller this value, the closer it is to rigid stablecoin exchange).

The algorithm's performance can match that of Curve and is more flexible, allowing it to be closer to rigid exchange or closer to AMM.

This use case is referred to as the "Anchored Pool".

Use Case 5: Degradation to AMM

PMM algorithm performs the same as AMM under the following conditions:

• Set 'k' to 1.

• Deposits and withdrawals must involve both assets simultaneously according to the current price ratio.

This implies that AMM is a special case of PMM. The derivation is as follows:

The price formula for AMM is xy = k, where k is a constant, and x and y represent the quantities of two tokens. To avoid ambiguity in letters, we define the constant product in the AMM formula as "Const" and rewrite it as y = Const/x.

$xy =Const$ -> $y=\frac{Const}{x}$

Where y represents the quote token inventory, x represents the base token inventory, and the price can be written as the derivative of y (note the absolute value because prices are positive).

$P=|\frac{Δy}{Δx}|=\frac{Const}{x^2}$

Returning to the PMM formula for k=1, we can write it as:

$P = i(1-k + k(\frac{B_0}{B})^2)$

$P = \frac{iB_0^2}{B^2}$

We observe that the PMM price formula has a constant in the numerator and the base token's inventory in the denominator. Simply set:

$iB_0^2=Const$

Now, the PMM pricing formula aligns with the AMM pricing formula.