# Mining without collateral
**U(Borrowing FIL to onboard)> U(mining without collateral)> U(not onboarding).**

## Standard mining with Collateral

Define

* $C$ is the amount of FIL required as collateral to seal a minimal sector.
* $R$ is the total expected reward over the lifetime of this sector.
* $M$ is the total operational costs of this sector (this includes costs of ongoing sector proofs, as well as opportunity costs for having capital commited for sector lifetime).

We will then focus on the problem of a given SP which has an amount of storage power available for onboarding, but not enough FIL available to onboard all sectors. We denote:
* $N_F*C$ is the amount of liquid FIL the SP has available for onboarding.
* $N_D$ is the amount of storage power the SP has available for onboarding (as a multiple of minimal sector size).

We will then consider the constraint that this SP has more storage power available, than FIL to seal it with, or,
$$ N_D>N_F.$$

We will also assume the utilities described above are given simply by the expected ROI of each strategy times the number of sectors sealed.
$$U= N_{sectors sealed}* ROI_{ sector}$$

First let us evaluate the expected ROI for a single minimal sector, which is given by
$$ROI=\frac{C+R}{C+M}$$
Note that this is the **total ROI** and not the **FIL on FIL ROI**, as external non-FIL costs have been accounted by including the operational costs, $M$.

**Assumption:** We assume onboarding a minimal sector is profitable, such that a positive incentive to onboard exists, that is we assume
$$\frac{C+R}{C+M}>1,$$
or equivalently
$$ R>M.$$
(In reality this is not always true).

If the SP will only seal the sectors they have FIL to seal, the total ROI is
$$ROI=N_F\left(\frac{C+R}{C+M_{N_F}}\right),$$
where we introduced $M_{N_F}$ as the operational costs per sector, when $N_F$ are onboarded, where $M_{N_F}\le M$ to account for advantages of scale.

This is what we call in the "Not onboarding" strategy, as there is an additional supply of storage power, $N_D-N_F$ that does not get onboarded. We then have for the right hand side of our inequality:
$$U({\rm not\,\,onboarding})\,\,=N_F\left(\frac{C+R}{C+M_{N_F}}\right).$$

We now examine the utility for an SP that doesn't have enough FIL to seal all sectors, but borrows the additional needed FIL at some market rate to seal the remaining sectors.

We start by assuming the inequality:
$$U({\rm borrowing\,\,FIL\,\,to\,\,onboard})\,\,>U({\rm not\,\,onboarding})$$
holds. That is, the market rates are such that it would be rational to seal your sectors using your own funds if available, than to borrow funds.

Let's assume one can borrow an amount $C$ of FIL for the minimal sector duration, after which it can be repaid, plus an additional total interest $I$ (note this is total interest, not an interest rate).

One could borrow all the FIL required to seal a sector, then obtain all the reward from the sector's lifetime, then return the borrowed FIL plus interest, which gives an ROI
$$ROI_{borrow}=\frac{R}{I+M}$$
The inequality above implies that
$$\frac{R}{I+M}<\frac{C+R}{C+M},$$
or the total interest satisfies,
$$I> \frac{R(C+M)}{C+R}-M.$$
This is a lower bound on total interests for the lending market, under which SP's would be incentivized to never use their own funds to seal sectors.

An upper bound can be obtained by the requirement that it is still profitable to borrow FIL to seal sectors, or
$$U({\rm borrowing\,\,FIL\,\,to\,\,onboard})\,\,>1,$$
which gives the constraint,
$$I< R-M.$$

Given the above constraints, The utility for an SP who is able to borrow FIL to seal the remaining $N_D-N_F$ they don't have capital for, is
$$U({\rm borrowing\,\,FIL\,\,to\,\,onboard})=N_D\left[\frac{(N_D-N_F)R+N_F(C+R)}{(N_D-N_F)(I+M)+N_F(C+M)}\right]$$
$$=N_D\left[\frac{N_DR+N_FC}{(N_D-N_F)I+N_DM+N_FC}\right]$$

And the constraint that this is better than not onboarding additional sectors is satisfied as,
$$N_D\left[\frac{N_DR+N_FC}{(N_D-N_F)I+N_DM+N_FC}\right]>N_F\left(\frac{C+R}{C+M_{N_F}}\right),$$
as long as the previously defined constraints are satisfied.

## Mining without collateral

Suppose the SP is allowed to mine without placing any collateral, and in return, they will only received a fraction $Y$ of the total rewards.

We first want to satisfy the fact that the SP will pay collateral for the first $N_F$ sectors they do have collateral for, so the utility of sealing those sectors without collateral should be smaller than sealing with collateral,

$$N_F\frac{YR}{M}<N_F\left(\frac{C+R}{C+M}\right)$$
or,
$$Y< \frac{M}{R}\left(\frac{C+R}{C+M}\right)$$

We can then assume that the first $N_F$ sectors are sealed with collateral, then the utility of sealing the rest of the sectors without collateral is then
$$U=N_D\left[\frac{(N_D-N_F)YR+N_F(C+R)}{N_DM+N_F C}\right],$$

Requiring that onboarding without collateral is better than not onboarding means
$$N_D\left[\frac{(N_D-N_F)YR+N_F(C+R)}{N_DM+N_F C}\right]>N_F\left(\frac{C+R}{C+M}\right),$$
or
$$ Y>\frac{1}{R(N_D-N_F)}\left[(N_DM+N_FC)\frac{N_F}{N_D}\left(\frac{C+R}{C+M}\right)-N_F(C+R)\right]$$

Requiring that onboarding without collateral is worse than borrowing to onboard gives us
$$Y<\frac{1}{R(N_D-N_F)}\left\{(N_DM+N_FC)\left[\frac{N_DR+N_FC}{(N_D-N_F)I+N_DM+N_FC}\right]-N_F(C+R)\right\}$$

This can be simplified in the case where the SP has no funds available at all, $N_F=0$, where we get
$$0<Y<\frac{M}{I+M}$$

## Too many unknowns

Trying to fix a value of $Y$ this way requires knowledge of operational costs, interest rates, and funds available to each SP, which is unrealistic to have all this information at hand, to design a mechanism based on this.

### Self regulating approaches
An alternate approach is to have a dynamic mechanism to find the appropriate market value of $Y$.

One simple alternative is to have the fraction of rewards that are not earned by SPs, $(1-Y)R$, be given back to SPs that are paying full collateral as additional reward.
In this mechanism the higher the percentage of the network that is sealing without collateral, the higher the incentive to gets to seal with collateral.

Suppose the network has total QAP $P$, and a fraction $f$ of that power has been sealed without collateral.
The total reward per sector for a fully collateralized sector is
$$R\frac{(1-f)+(1-Y)f}{1-f}=R\frac{1-Yf}{1-f}$$

We then have the constraint that the utility of sealing a sector without collateral should be smaller than sealing with collateral
$$Y< \frac{M}{R}\left(\frac{C+R\frac{1-Yf}{1-f}}{C+M}\right),$$
note that $Y$ appears on the right hand side as well here. The right hand side grows monotonically and goes to infinity as $f\to 1$, therefore for some value of $Y$, there is a maximum proportion $f$ of the network power that can be sealed without collateral.

### Fixing $f$ with an EIP-1559 like discovery mechanism for $Y$
For a given $Y$, there will always be a fraction of network power without collateral, $f$, such that at that point it becomes more profitable to seal with collateral.

For a given value of $Y$, we might not be happy with the value $f$ that the network found, and if we would like to lower $f$, we can adjust $Y$.

We could choose to have an automated $Y$-discovery mechanism, based on how close the network is to the desired target $f_{\rm target}$, inspired by EIP-1559, we could design something like
$$Y_{t+1}=Y_t\left[1+\alpha(f_t-f_{\rm target})\right]$$
*Have to modify this law, the sign is wrong, below target should increase $Y$, and should modify to stay in range $0<Y<1$*