# Reward refactoring, economic details

We explore the economic details of the refactoring mechanism proposed in https://docs.google.com/document/d/1-I7E-KHD9KPmUtdnrSSp7y4o9yboUfxZO2zv6mEo27k/edit#heading=h.5msz3otbwuue.

Currently, consensus power and miner reward are determined by the QAP of the miner, relative to the total network QAP. In the current implementation, the amount of reward per sector for a given time period can be expressed as
$$Reward\,\, per\,\, sector =\frac{R}{P_R}\frac{q}{Q},$$
where $R(P_R)$ is the total amount of new rewards FIL minted, which depends on the total network raw power, $P_R$, via the baseline minting mechanism. $P_Q=QP_R$ is the total network QAP, $q$ is the quality multiplyer for that given sector, and $Q$ is the average network quality multiplyer. If a sector has a higher than average quality multiplyer, $q>Q$, then their fraction of the rewards will be increased.

It is proposed that rewards be divided into a consensus reward that depends only on raw network power, and a Deal subsidy that involves the quality multiplyers. This has a computational power that one does not need to track the quality multiplyer for the consensus reward, which improves the scalability of the network.

It is also proposed that consensus power should correspond only to raw power, since this directly corresponds to the amount of resources used.

The new proposed reward per sector for a period of time can then be expressed as
$$ Proposed\,\,Reward\,\,per\,\,sector=a\frac{R}{P_R}+(1-a)\frac{R}{P_R}\frac{q}{Q},$$
where the first term corresponds to the consensus reward, and second to the deal subsidy. We have defined $a\in(0,1)$, which determines the distribution between the two types of rewards.

## Keeping total reward structure unchanged.

Suppose we would like to split the consensus reward and deal subsidy in a way that the total reward retains the current structure, where it is proportional to the sector qap. One way this could be acheived is by replacing the total reward in the deal subsidy term, by some amount, $M$, such that
$$ Proposed\,\,Reward\,\,per\,\,sector=a\frac{R}{P_R}+(1-a)\frac{M}{P_R}\frac{q}{Q}.$$

We can fix $M$ by requiring
$$\frac{R}{P_R}\frac{q}{Q}=a\frac{R}{P_R}+(1-a)\frac{M}{P_R}\frac{q}{Q}.$$
The solution is then
$$M=\frac{QR}{q}\frac{\left(\frac{q}{Q}-a\right)}{(1-a)}$$

## Fixing $a$ by linear proportion of verified deals (PROBABLY NOT THE BEST APPROACH, SEE NEXT SECTION)

Furthermore, we can fix $a$ by the following consideration. If $Q=1$, it means that there are no verified deals in any sector, so the deal subsidy term should vanish. As $Q$ increases, the proportion of the deal subsidy to the total reward increases (here we assume this happens **linearly**). If $Q=10$, then all of the sectors contain only verified deals, and only 10% of the total reward should be coming from consensus reward. We can then parametrize this as
$$a=\frac{11-Q}{10}.$$
Note that then the total fraction of reward that is given as consensus reward is then informed by the average network QAP. So even though individual sector qap's do not need to be tracked, the average QAP is still needed.

Substituting for $a$ we then have,
$$M=\frac{QR}{q}\frac{\left(\frac{q}{Q}-\frac{11-Q}{10}\right)}{\left(1-\frac{11-Q}{10}\right)}.$$

To summarize then, the rewards are of the form,
$$Consensus\,\,reward=\frac{(11-Q)}{10}\frac{R}{P_R},$$
$$Deal\,\,subsidy=\frac{(Q-1)}{10}\frac{QR}{q}\frac{\left(\frac{q}{Q}-\frac{11-Q}{10}\right)}{\left(1-\frac{11-Q}{10}\right)}\frac{q}{P_RQ},$$
$$Total\,\,reward=\frac{R}{P_R}\frac{q}{Q}.$$

## Fixing $a$ by maximum consensus reward, without any reimbursment.

If we were to choose $a$ to be too high, then the subsidy adjustment means some people would actually need to *return* money, for not having , rather than receiving a subsidy.

Here we fix $a$ by choosing the maximum value possible, while ensuring that no one has to return any any payment, and everyone receives either zero or positive subsidy.

We return to our expression,
$$M=\frac{QR}{q}\frac{\left(\frac{q}{Q}-a\right)}{(1-a)}.$$
We see we satisfy our requirment by choosing,
$$a=\frac{1}{Q},\,\,\,\,\,,\,1-a=\frac{Q-1}{Q}.$$
The miners with the fewest possible amount of verified deals will have $q=1$. So we see from the previous expression, if $a>1/Q$, $M$ will be negative for some of the miners, which means they will be required to return payment.

We then have,
$$M=\frac{R}{q}\frac{(q-1)}{\left(1-\frac{1}{q}\right)}=R,$$
and,
$$Consensus\,reward=\frac{1}{Q}\frac{R}{P_R},$$
$$Deal\,subsidy=\frac{Q-1}{Q^2}\frac{Rq}{P_R}.$$