| Variable | Description |
|---|---|
| Omnipool reserves of asset |
|
| LRNA in subpool for asset |
|
| Shares in asset |
|
| Weight cap for asset |
For a given state variable
- For all assets
$i$ in Omnipool, the invariant$R_i Q_i$ should not decrease due to a swap. This means that after a swap for all assets$i$ in Omnipool:
-
$R_iQ_i$ should be invariant, but one is calculated from the other. If e.g.$R_i^+$ is calculated it may have error up to$1$ , in which case the product$R_i^+Q_i^+$ may have error up to$Q_i^+$ . If$Q_i^+$ is calculated, then the product has error up to$R_i^+$ . Thus we should always be able to bound the error by$max(R_i^+,Q_i^+)$ , giving us
- Add liquidity should respect price
$\frac{Q_i}{R_i}$ . This means$\frac{Q_i}{R_i} = \frac{Q_i^+}{R_i^+}$ , or$Q_i R_i^+ = Q_i^+ R_i$ . What is most important here is not which direction we round but the accuracy. So we must test that
- Adding liquidity in asset
$i$ should keep the ratio of assets per shares constant. We round so as to not decrease the assets per share of asset$i$ ,$\frac{R_i}{S_i}$ ; that is, we favor the other LPs over the LP currently adding liquidity, to avoid any potential exploit. This means,$\frac{R_i^+}{S_i^+}\geq \frac{R_i}{S_i}$ , so
- Adding liquidity needs to respect weight caps. That is,
- Withdraw liquidity should respect price
$\frac{Q_i}{R_i}$ . This means$\frac{Q_i}{R_i} = \frac{Q_i^+}{R_i^+}$ , or$Q_i R_i^+ = Q_i^+ R_i$ . Allowing for rounding error, we must check
- Withdraw liquidity in asset
$i$ should keep the ratio of assets per shares constant. We round so as to not decrease the assets per share of asset$i$ ,$\frac{R_i}{S_i}$ ; that is, we favor the other LPs over the LP currently withdrawing liquidity, to avoid any potential exploit. This means$\frac{R_i^+}{S_i^+}\geq \frac{R_i}{S_i}$ , so