Constant Product Market Maker

Liquidity pools on GnoSwap utilize the Constant Product Formula to achieve deterministic pricing of any token in a pool, ensuring that trades can occur at any price within the range of (0, ∞). Liquidity pools maintain a constant k value, which is the product of x and y, representing the quantity of token x and token y in the pool, respectively. This mechanism results in the following formula:

$$x *y=k$$

When a trade occurs in a pool, the values of x and y change. Let's assume that ∆ represents the change in the quantity of a token. If a trader sells ∆x amount of token x, the pool receives ∆x amount of token x and pays out ∆y amount of token y (minus the swap fee) as the output in a way that satisfies the equation below:

$$(x+∆x)(y-∆y)=k$$

Based on the above formula, we can plot a graph that illustrates changes in token reserves in a pool after a trade.

To illustrate this concept, let's consider a scenario where Pool A holds 10 $GNOT and 10,000 $USDC. If Alice wishes to exchange 1 $GNOT for $USDC in Pool A, an equation reflecting the current reserves and ∆x (change in $GNOT) is set up as follows:

$$k = 10*10,000 = 100,000 = (10+1)(10,000-∆y)$$

Solving for ∆y, we get:

$$∆y = 10,000-(100,000/11) = 909.090909...$$

Therefore, we can expect Alice would receive 909.090909... $USDC if she were to sell 1 $GNOT in Pool A.

When adding liquidity to an existing pool, liquidity providers must deposit an amount of token x and token y that is proportional to the current x : y ratio in the pool. Once liquidity is added, the value of k adjusts to the new values of x and y.

As k is proportional to the liquidity of the pool, this structure can be seen as a virtual order book where liquidity is evenly distributed across the entire price range.