Dynamics Weights

The price of the $i$th token in the pool in terms of our internal numeraire, scale, is simply

$$P_i = \frac{s_i}{\alpha_i},$$

where $s_i$ is the token's scale and $\alpha_i$ is the token's reserve so changes in the price of a token can be expressed as

$$P_i + \Delta P_i = \frac{s_i + \Delta s_i}{\alpha_i + \Delta \alpha_i}.$$

With Multiswap, there are two extreme ways to change the price of a token:

1. Trading: Holding the scale constant ($\Delta s_i = 0$) and changing the reserves; or
2. Reallocating: Holding the reserves constant ($\Delta \alpha_i = 0$) and changing the scale.

Recall that the weight of a token is given by

$$w_i = \frac{s_i}{s_0},$$

where $s_0$ is the scale of the LP token given by

$$s_0 = \sum_{j=1}^n s_j$$

so that changing the scale of a token also changes the weights.

In this simple tutorial, we focus on how changing the scale of a token affects the prices and weights.