# Incentives to extend sectors --- issue 44

:::info
:hammer: Draft - work in Progress
:::

### Context
[Issue 44](https://github.com/protocol/CryptoEconLab-private/issues/44) is about incentivizing longer sector lifetimes. The decision to extend a sector is becoming important as the network begins to see waves of sector expiration. And furthermore, sector lifetime is relevant to [Issue 50](https://github.com/protocol/CryptoEconLab-private/issues/50) on [FIL+ Indefinite Term Limits](https://www.notion.so/pl-strflt/FIL-indefinite-term-limits-08079d7ec25c4cae839a3bf95a82df26), so having a handle on this is doubly valuable.

### Sector expiry outcomes
When a sector expires there’s two possible outcomes: it can be resealed, or permanently terminated. We assume mining is always economically favorable, so termination at expiry is not rational, reducing our problem space in this document to only considering reseal vs extension.

### Reseal decision factors
The decision to reseal depends on the reseal opportunity value. This depends on how much collateral can be recaptured if the initial pledge drops, and also the fixed and variable costs associated with reseal, namely labor, gas and hardware. If the reseal opportunity value is negative, a rational SP will extend, but if it’s positive they’ll reseal.

**Network perspective**:
The network favors extension over resealing
* Because resealing is a waste of resources, it should be minimized.
* Because resealing involves closing a sector, which has a non-zero risk of no return which should be minimized
* Longer sector lifetime results in more FIL locked as collateral
* Longer commitment signals confidence

As to why the network might prefer resealing over extension, there’s no obvious argument.

**SP perspective:**
Factors that support resealing are
* It gives an opportunity to recapture tied-up capital
* Extension has an associated message gas cost

But SP may favor extension
* Because labor costs are less than resealing
* Because less hardware costs than resealing
* Smaller gas costs than resealing

### Questions
* The generic question is what’s the optimal extension policy given the objective of maximizing network health.
* A concrete and addressable question is, given the tokenomic [specification](https://spec.filecoin.io/algorithms/cryptoecon/) of MinimumSectorLifetime, InitialPledgeCap and InitialPledgeFunction, what values minimize sector expirations and miner collateral requirements?
* We can an answer using two approaches:
* Heuristic economic analysis
* Empirical-data based model inferences

### Heuristic analysis
* Increasing minimum sector lifetime parameter
1. **Benefit:** Since $\text{MinimumSectorLifetime} \sim 1/\text{ExpiryFrequency}$, expiry must become more infrequent which benefits the network.
2. **Side effect:** miners have less flexibility (increased operational risk) and may be discouraged from engaging with the network. But this risk is hard to quantify.
3. **Conclusion:** increase MinimumSectorLifetime, but uncertainty as to how much. More empirical analysis needed.
* Increase collateral pledge
1. **Effect:** decision boundary becomes sharper.
2. **Benefit:** system behavior more deterministic, less affected by uncertainty induced by gas/labor/pledge variability. More security for clients.
3. **Downside:** Miners pay more, doesn’t directly help extension.
4. **Conclusion:** avoid.
* Decrease collateral pledge
1. **Benefit:** lower barrier to entry for SP.
2. **Downside:** insecurity for clients data hurt demand for the token current FIL-on-FIL return is still attractive.
3. **Conclusion:** decrease in InitialPledge may be worthwhile. Uncertainty substantial due to difficulty valuing the change in security. More empirical analysis needed.
* Make subsequent collateral pledges be fixed to the value of the first initial pledge value
1. **Benefit:**
* More collateral is locked, probably.
* Longer commitment from SPs and stronger guarantee for clients although SPs may choose to drop the deals. However, it can be argued that it is not a given that SPs who only take short-term deals can get FIL+ deals from clients in the long run (reputation feedback).
* Simplicity, no change from the status quo but we accept that SPs sometimes may find it more reasonable to reseal vs extend.
2. **Downside:**
* since pricing is fixed at the start, mispricing leads to misaligned incentives (excessive collateral locked from SP for level of storage provided).
* And initial price has to be cached.
3. **Conclusion:** plausible, and simple, could be a winner. But relies on the assumption of initial pledge always going down. Check this empirically. If true, support.
* Subsequent reseals are pledge free
1. **Benefit:** miners regain capital.
2. **Downside:** lower security degrades network health.
3. **Conclusion:** avoid.
* Recalculate pledge collateral at extension
1. **Benefit & Downside**:
* SPs will always favor extension over resealing (but is this really a benefit?)
* Lower pledge collateral locked on the network.
* But more storage will stay.
* Probable that all sectors on the network will choose the minimum lifetime, % locked in circulating supply decreases, ceteris paribus.
2. Potentially, we can consider increasing the minimum sector lifetime but also allowing recalculation at reseal.
* In this case, we are accepting that all sectors will be of the minimum sector lifetime (say 1 year) since there is no incentive to do longer sectors (which might be okay)
* We need more analysis on the macro impact on the circulating supply

**Conclusions:** the most viable strategy seems to be to increase sector lifetime and potentially decrease collateral pledge with recalculate pledge on renewal. Generating new values for these parameters needs data-driven analysis.

### Reseal model for empirical analysis

**Goal**
Given a model and data we can examine rational SP extend/reseal choice in terms of
1. Different estimates of the hardware and labor costs SPs face
2. Forecasting to look at reseal/extend balance in future
3. Testing collateral requirements and reseal costs counterfactuals, e.g.:
* Decreasing the collateral requirements
* Making subsequent reseals have different collateral requirements to the initial collateral pledge
* Introducing a mechanism coupling reseal collateral lease and reseal costs to incentive extension

**Data**
We have [data from sentinel](https://lilium.sh/lily/models.html) for initial pledge by epoch $t$ and miner_id $m$. We can get feel for this by looking at the distribution since the start of 2021:
Or we can summarise the time-series for hourly mean initial pledge:

We also have access to data for gas, index by epoch and actor. But for labor and hardware assumptions will be needed.

**Model1**
We can frame the analysis in terms of the decision to reseal a sector after the minimum term. We define the 180 day reseal opportunity for sector $s$ at time $t$ in terms of two factors, the change in collateral requirements (via initial pledge) and reseal costs:

\begin{align*}
\text{ResealOpportunity}_{s,t} := \Delta\text{collateral}_{s,t} + \text{ResealCosts}_{s,t}
\end{align*}

Here the potential collateral realease is

\begin{align*}
\Delta\text{collateral}_{s,t}= \text{InitialPledge}_{m,t}/N^\text{seal}_{m,t} - \text{InitialPledge}_{m,t{-180}}/N^{\text{seal}}_{m,t-180}
\end{align*}

Since InitialPledge is only known by miner_id, and not sector_id, we divide through by $N^\text{seal}_{m,t}$, the number of sectors that miner $m$ seals at time $t$.

The reseal costs are given by

\begin{align*}
\text{ResealCosts}_{s,t} = \text{Gas}_{s,t} + \text{Labor}_{s,t} + \text{Hardware}_{s,t}
\end{align*}

where we assume the gas associated with a sector is $\text{Gas}_{s,t}=\text{Gas}_{m,t}/N^\text{seal}_{m,t}$.

The rational decision boundary is given by
\begin{align*}
D_{s,t}=\begin{cases}
\text{Reseal} & \text{ResealOpportunity}_{s,t}>0\\
\text{Extend} & \text{ResealOpportunity}_{s,t}<0
\end{cases}
\end{align*}
which we can examine for different assumptions and scenarios.

**Model2**

Beyond the expected rational decision, we can try to explain the expiration data that's actually observed, and learn the importance of the different contributions from collateral release and reseal costs. One approach is to aggreate up from individual sector levels. For example, we can work with total daily ($d$) expirations, $N^\text{expire}_{d}$, since we know the total number of sectors, terminations and reseals:

\begin{align*}
N^\text{expire}_{d} = N_{d} - N^\text{terminate}_{d} - N^\text{reseal}_{d}\,.
\end{align*}

We can model these counts with a negative binomial likelihood

\begin{align*}
N^\text{expire}_{d} \sim \text{NegativeBinomial}(\mu_d,\phi)
\end{align*}

where $\mu_d=\text{exp}(\text{LatentResealOpportunity}_{d})$, and latent reseal opportunity is

\begin{align*}
\text{LatentResealOpportunity}_{d} = \beta_1 \Delta\text{collateral}_{d} + \beta_2\text{Gas}_{d} + \beta_3\text{HardwareAndLabour}_{d}
\end{align*}

where $\beta_i$ are the coefficients that explain the relative importance of average daily sector collateral release ($\Delta\text{collateral}_{d}$) vs gas vs other costs. This seems fairly do-able.

**Model3**

Another approach may be to model individual sector status $E_{s,d}$, as Bernoulli random variables

\begin{align*}
E_{s,d} \sim \text{Bernoulli}(\text{Logistic}(\text{LatentResealOpportunity}_{d})).
\end{align*}
This may be somewhat difficult, more thought needed.

***Status:*** I'll try getting the data together for the simplest model first.