# Pledge Token

The idea for an alternate token, to be used to pay Consensus pledge, was introduced [here](https://www.notion.so/Filecoin-pledge-token-8855f4a974ec4b8da306e8667f4682aa). Here we present a brief analysis on the motivation of this token, and its ability to provide stability to data onboarding in Filecoin.

## Examine lack of stability of status quo

A potential problem that is pointed out [here](https://www.notion.so/Filecoin-pledge-token-8855f4a974ec4b8da306e8667f4682aa) is that the current formulation for the Consensus Pledge may lead to a destabiizing dynamic.

The consensus pledge that a sector labeled by $i$, with QAP $p_i$ has to lock is given by
$$C_t^i=0.3 S_t\frac{p_i}{P_t},$$
where $C_t^i$ is the consensus pledge paid by sector $i$ at time of onboarding $t$, $S_t$ is the FIL circulating supply at time $t$, and $P_t$ is the total network QAP at time $t$.

The consensus pledge is then directly proportional to the circulating supply, which my present a stability issue. The risk is that if net network onboarding rateis reduced, there will be higer amount of circulating supply available (because it was not locked up in new sectors), which increases the cost of the collateral. *This in turn reduces the FIL-on-FIL ROI of that sector*, where more collateral has to be paid for the same reward. This means that onboarding new sectors is further disincentivized, creating a negative feedback look: if onboarding decreases, this creates even more incentive to decrease onboarding.

This analysis is strictly true in terms of the consensus pledge increasing when onboarding is reduced. It is however questionable whether it is always true that this results in a decrease of sector ROI. This is questionable because a decrease in onboarding also means that the sector has a larger share of the total network QAP, so it would be receiving a larger amount of reward.

## Satus quo change in ROI

Let us first define **sector lifetime FIL-on-FIL ROI* as

$$ROI=\frac{\rm Initial\,Pledge+Reward\,earned\,over\,sector\,lifetime}{\rm Initial\,Pledge}-1$$
$$= \frac{\rm Reward\,earned\,over\,sector\,lifetime}{\rm Initial\,Pledge}$$

**simplification:** for now we will assume the initial pledge=consensus pledge, ignoring the storage pledge portion, since this is generally significantly smaller.

The expected reward for sector, $i$, which was onboarded at time $t_0$, and stayed online for a period $T$, is given by
$${\rm Reward\,earned\,over\,sector\,lifetime}=p_i\int_{t_0}^{t_0+T}\frac{R_t}{P_t}dt,$$
where $R_t$ is the total rate of new token minted as block reward at time $t$.

The ROI for a sector onboarded at time $t_0$ is then
$$ROI_{t_0}=\frac{P_{t_0}\int_{t_0}^{t_0+T}\frac{R_t}{P_t}dt}{0.3S_{t_0}}$$

Here is easier to see the argument for why ROI does not necessarily decrease for the individual sector, if there is a downturn in onboarding.

We can assume $S_{t0}$ is generally a function (or functional) of the entire history of network power, $S_{t_0}[P_t]$, where we understand if power decreases, circulating supply will increase, but not necessarily in a linear manner.

**Let us define the potential problem:** If there are two alternate possible trajectories for onboarding after $t_0$, where we could have either $P_t^1$ or $P_t^2$ for $t>t_0$, with $P_t^1>P_t^2$ for $t>t_0$.

Suppose that at time $t_1>t_0$ a new sector $i$ is onboarded, will the ROI for that sector be larger for trajectory $P_t^1$ than for $P_t^2$? If this is case then it is true the assertion that there is a destabilizing mechanism in how the consensus pledge is formulated.

The question is then, is it true that
$$\frac{P_{t_1}^1\int_{t_1}^{t_1+T}\frac{R_t}{P_t^1}dt}{0.3S_{t_1}[P_t^1]}>\frac{P_{t_1}^2\int_{t_1}^{t_1+T}\frac{R_t}{P_t^2}dt}{0.3S_{t_1}[P^2_t]}\,?$$

*note* that we are assuming $R_t$ is independent of the network power (which is only true when network power is above baseline).

The key insight here is that **it is not that clear which side is larger**, as it depends on the precise trajectory of onboarding $P_t^{1,2}$, as well as the precise way that circulating supply responds to onboarding rates.

In particular if $P_t^2$ is *very quickly* decreasing, throughout the lifetime of the sector, this may indeed make the right hand side of the inequality larger. This is because the circulating supply is only computed at the initial time, $t_1$, depending on the total network QAP at that time. However if the network QAP keeps decreasing after that, the total rewards received by the sector will become larger.

**The larger $P_t^1-P_t^2$ grows for times $t>t_1$, the less likely for the inequality to hold true, and at some point the right-hand-side can become larger**

So the conclusion is that whether it is true that the consensus pledge definition leads to a destabilizing dynamic, depends on the exact trajectory of network QAP, but it is not universally true or false.

## Pledge Token ROI

In the proposed pledge token mechanism, consensus pledge would need to be paid in a new token we call PFIL. One is always able to buy PFIL from the network using FIL with a conversion rate 1FIL=1PFIL. However the network will not buy back PFIL, so the price of PFIL is only capped to be a maximum of 1FIL, but in the free market the price could drop below that.

More generally the market exchange rate at a given time $t$ would be given by
$$1{\rm FIL}=n_t{\rm PFIL},$$
with $n_t\ge 1$.

The amount of PFIL to be paid as consensus pledge depends on the circulating supply of FIL, discounting the circulating PFIL.

The FIL-on-FIL ROI of sector $i$ being onboarded at time $t_0$ is
$$ROI=\frac{n_{t_0}P_{t_0}\int_{t_0}^{t_0+T}\frac{R_t}{P_t}dt}{0.3S_{t_0}}+(\frac{n_{t}}{n_{t+T}}-1).$$

The important new element here is the introduction of the variable exchange rate, $n_t$. If the network is onboarded at a time where there is excess PFIL in the market, and $n_{t_0}>1$, then this is generally good for the sector's ROI. This is because the reward is given in FIL, while the collateral is paid in discounted PFIL.

We point out that this could also bring a new risk of sectors being *undercollateralized*, of ROI being so large, based on initial $n_{t_0}$, that the collateral is not large enough to incentice stability (or in the extreme case, not enough to cover a termination fee).

We point out that there is also a potential, that with the PFIL, the ROI will be even decrease further than it would, without the PFIL, the opposite of solving the problem it was trying to solve.

This would happen if $n_{t+T}>n_t$, this can start deteriorating the ROI. That is, at the end of the sector's lifetime, the initial pledge is given back to the SP, but this was in PFIL, which may have already lost even more of its market value by the time $t_0+T$, if onboarding was reduced more at that time.

**There is even a new potential for total ROI to be negative**, if PFIL devalue enough, a risk that didn't exist with the status quo initial pledge formulation.

We can then question, **under what scenario is the PFIL better for ROI than the status quo?** We can answer that by examining the inequality
$$\frac{n_{t_0}P_{t_0}\int_{t_0}^{t_0+T}\frac{R_t}{P_t}dt}{0.3S_{t_0}}+(\frac{n_{t}}{n_{t+T}}-1)>\frac{P_{t_0}\int_{t_0}^{t_0+T}\frac{R_t}{P_t}dt}{0.3S_{t_0}},$$
from which we can get the condition
$$n_{t_0+T}< \frac{n_{t_0}}{1-(n_{t_0}-1)\frac{P_{t_0}\int_{t_0}^{t_0+T}\frac{R_t}{P_t}dt}{0.3S_{t_0}}}.$$
This inequality tells us how much PFIL is allowed to drop in value after sector onboarding, for PFIL to remain a good deal for the sector ROI.

We can look at a simple example to get some better intuition. We can assume at onboarding time, there was no excess of PFIL in the market, so its market value was $n_{t_0}=1$. In this case then there is the condition reduces to
$$n_{t_0+T}<n_{t_0}.$$
That is, if at onboarding time 1PFIL=1FIL, then if PFIL drops in value by any amount,the pledge token mechanism would be a bad deal for the SP.

## Conclusion

The Pledge token is an interesting proposal worth investigating more, however it is not always a net positive. Without the pledge token, the destabilizing dynamic only exists under certain conditions, and not universally. The pledge token only improves this dynamic in some cases, and not universally. So it doesn't seem the pledge token is a universally good solution, but simply a different one. If it does offer some advantage some of the time, it is unclear if this advantage is large enough to justify the added complexity and risk.