# Layer 2 Staking Mechanism

Here we study the economic details of a possible mechanism where token holders may stake their tokens for a given period of time, in exchange for a yield on their stake at the end of a given period.

The assumption is that this service is built on an existing blochain with a native mining mechanism, such as providing storage in filecoin.

The basic idea in a general Proof of Stake mechanism is that a token holder would stake an amount $S$, for a period of time $T$, and at the end of that period would obtain a return,
$$R=(1+y)S,$$
for a positive number $y$.

A proposed advantage of PoS is the linearity of the return on investment, $R\sim S$, whereas in other mechanisms that involve using and operating hardware to mine new tokens, typically economies of scale would favor larger miners with more initial capital.

We imagine a mechanism is built such that a token holder locks $S$, these funds can be used as a loan to a miner, who can use it to grow capacity and run their equipment for a period of time, $T$, after which the return on investment will be
$$R=(1+y)f(S),$$
where $f(S)$ is a function that describes the economies of scale, such that it is generally superlinear.

In the Filecoin case, further restriction comes from the fact that there is a minimum unit of storage, one sector.

We can define the minimum stake, $S_{\rm min}$, where $M=f(S_{\rm min})$ is the amount of funds it takes to buy and operate one sector until it breaks even and generates $yf(S_{\rm min})$ of profit.

We can define $t$ as the time it takes for one sector to break even and generate $yf(S_{\rm min})$ profit.

The staked amount can be increased discretely, by funding $n$ sectors. Stakes can be in discrete amounts,
$$S_n=f^{-1}(n\cdot M).$$
$S_n$ increases *sublinearly* with $n$.

We point out that $T$ and $y$ are related, and only one variable can be chosen independently. We can therefore denote the yield as a function $y(T)$. This function can be computed from the miner reward, and it is more generally a function also of the absolute time $y(t,T)$, since miner reward will decay over time.

#### Adjusting $t$ to keep $R$ linear.

One of the advantages of PoS is that it doesnt disproportionately benefit larger miners, as is the case with comitting storage capacity.

The total return could be kept constant instead by letting smaller token holders lock their funds for a longer time. We define a linearized return,
$$R_L=(1+y_L(t))S,$$
and adjust we can adjust parameters like $T$, such that $R=R_L$, or
$$(1+y(t,T))f(S)=(1+y_L(t))S,$$
from which we find
$$T=y^{-1}\left(\frac{S}{f(S)}(y_L(t)+1)-1,t\right),$$
where we defined the inverse function, $y^{-1}$, as
$$y(t,y^{-1}(A,t))=A$$

To summarize, it is possible to vary the locking period $T$ such that return on investment remains linear.