The permissionless lending needs to achieve the following.

• Pay interest to lenders because they lock up their deposit $\gets$ need to determine the interest rate $r$.
• Borrowers can get their collateral back if they pay back their debt

Current stage

LPs are also lenders.

Each borrower has two state variables: Collateral Ratio ($\text{CR}$) and Utilization Ratio ($\text{UR}$). Borrower borrows $y_0$ amount of $Y$ by collateralizing $x_0$ amount of $X$ at price ${\large p}{\scriptsize\text{borrow}}$. If the current price of $X$ in $Y$ is ${\large p}{\scriptsize\text{current}}$, then

$$ \text{CR}=\frac{{\large p}{\scriptsize\text{current}}x_0}{y_0}=\frac{{\large p}{\scriptsize\text{current}}}{{\large p}_{\scriptsize\text{borrow}}}. $$

Aggregate Liquidation LP should support interest collection. This can happen only if ${\large p}{\scriptsize\text{current}}>{\large p}{\scriptsize\text{borrow}}$. Then there must be enough liquidity to the left of ${\large p}{\scriptsize\text{current}}$ and to the right of ${\large p}{\scriptsize\text{borrow}}$. This is NOT Jain’s idea, where the liquidity to cover the debt is at the same place in the price space.

For each borrow position, we need to keep track of the liquidity that is available to liquidate it. We introduce

$$ \text{UR}\coloneqq\frac{\text{principal}+\text{interest}}{\text{unlocked range order tokens to the right}}=\frac{\text{P}+\text{I}}{\text{UT}}, $$

where

$$ \text{UT}=\sum_{\text{tick }:p_a>p_{\text{borrow}},\ p_b<p_{\text{current}}}\text{unlocked tokens in tick}. $$

Regarding $\text{UR}$

• We want $\text{UR}<1$ as it guarantees the solvency and independence of the protocol.
• It helps determine the interest rate $r$. We can decrease $r$ as $\text{UR}$ goes up to a threshold. After this, lender can initiate a process to force the borrower to pay back debt. This is different from liquidation because at this time there may not be a drop in $p_c$ and $\text{CR}$ is high. This might happen if someone comes to borrow a lot at some tick and $\text{UT}$ decreases a lot. One benefit of this is we don’t need to use LP’s liquidity to do this.
• The other approach is if borrowing at some tick causes other positions to be closable, then we do not allow that borrow. This is better than the above approach for two reasons
  • User experience: Old borrower is forced to pay back debt because new borrower comes to borrow.
  • It is counterintuitive that higher $\text{UR}$ leads to a smaller $r$.
• if above a certain threshold, we can
  • make certain positions close-able
  • do not allow some borrowing

If UR is still shooting up, we can close someone’s position using some rule such as the person with the lowest CR