How does this bonding curve work?

The formula y = L / (1 + e^(-k(x-x0))) defines a sigmoid bonding curve where the token price follows an S-shaped progression as more tokens are bought. Here, x represents the current token supply, L is the maximum price, k controls how steep the curve is, and x₀ is the midpoint where the price increases most rapidly.

When x is small, the exponent is negative and large in magnitude, so the price stays low. As x approaches x₀, the curve enters its steepest phase, causing the price to rise quickly. After passing this point, the growth slows down, and the price gradually approaches L without exceeding it.

This creates a balanced incentive structure: early buyers get low prices, mid-phase buyers experience rapid price increases, and late buyers face a capped price instead of extreme spikes.

Advantages and limitations

One of the main advantages of this model is its smooth and controlled price behavior. Unlike curves that approach infinity, the sigmoid curve has a natural upper limit, making it more accessible and user-friendly.

It also provides a more realistic progression, with slow growth at the beginning, rapid expansion in the middle, and stabilization at the end.

However, the model is more complex to implement, especially when calculating trade costs. Additionally, since the price is still determined by a formula, it does not reflect real market demand and may not align with actual perceived value.