Math behind the rewards contract

LR - Block number when the last rebalancing took place. Certain variables would be rebalanced when the state change is necessary. Those events would include:

• adding or modifying staked tokens balance by the user
• changing parameters used to calculate boost, such as rewards per block

ULR - Time of rebalancing that occurrs when the user modifies his position (ipToken, pwToken delegation). It is saved under individual staking position.

User power-up

User power-up is calculated for each user whose position changes. That includes staking or unstaking ipTokens or modifying the delegated balance of pwIPOR.

The user's power-up is governed by the below function:

$$UPU = \begin{cases} 10\cdot \left(\frac{pwIpor}{ipToken} \right) +0.2 & \frac{pwIpor}{ipToken} < 0.01 \\ 4\cdot \left(\frac{pwIpor}{ipToken} \right) +0.26 & \frac{pwIpor}{ipToken} < 0.02 \\ 3\cdot \left(\frac{pwIpor}{ipToken} \right) +0.28 & \frac{pwIpor}{ipToken} < 0.03 \\ 2\cdot \left(\frac{pwIpor}{ipToken} \right) +0.31 & \frac{pwIpor}{ipToken} < 0.04 \\ 1\cdot \left(\frac{pwIpor}{ipToken} \right) +0.35 & \frac{pwIpor}{ipToken} < 0.05 \\ VS + \log_2\left(HS +\frac{pwIpor}{ipToken}\right) & \frac{pwIpor}{ipToken} \ge 0.05 \\ \end{cases}$$

VS - Vertical Shift
HS - Horizontal Shift
pwIpor - User's pwIPOR balance delegated to a particular asset
ipToken - User's staked ipToken balance

The curve can be swapped by the DAO for a different one at a later stage in time, affecting all future rebalancings. The user's power-up value is fixed and can only be changed if the rebalancing is triggered.

Aggregate Power-up

Aggregate Power-up is a synthetic summary of all power-ups across all users. It's used to calculate individual rewards in relation to the rest of the pool.

$$APU = \sum_{\# \text{all users whitch stake }ipTokens \geq 1} (ipTokens \cdot UPU)$$

APU - Aggregate Power-up
UPU - User Power-up

Accrued rewards

Accrued rewards are tracked over time to allow for the calculation of users' rewards between events of rebalancing.

$$\operatorname{accruedRewards} = \operatorname{accruedRewards}_{LR} + \operatorname{rewardsPerBlock} ~\cdot~ \left(\text{blockNumber} - \operatorname{blockNumber}_{LR}\right)$$

Composite Multiplier

$$CM = \frac{blockRewards}{APU}$$

The composite multiplier (CM) accounts for the rewards per "boosted unit of liquidity staked" per block.

$$CMC = \sum_{\# blocks}CM_{Block}$$

Composite Multiplier Cummulative (CMC) is a sum of Composite multipliers for every block until now.

Account Mining Rewards

User rewards at block n can be calculated using the below formula. Block ULR is the block with the state of compositeMultiplierCumulative (CMC) at the time of the last rebalancing of user rewards.

$$userRewards_{n}​=ipToken * UPU~*~\left(\operatorname{CMC}_{\operatorname{n-1}}​−CMC_{ULR-1}​\right)$$

The $CMC_{ULR-1}$ is saved when the user modifies his position. It is calculated to reflect the state without accounting for the current block.

​

$$CMC_{ULR-1} = CMC_{LR-1} + \left(URL-LR \right) \cdot CM_{LR}$$

When the user modifies their position, the most recent CMC is saved per the user's record.

Basic assumptions

VerticalShift ⇒ VS ⇒$10^{-4}$ $\leq$ VS $\leq$ 3

HorizontalShift ⇒ HS ⇒ 1 $\leq$ HS $\leq$ $10^{3}$

ipToken ⇒ 1 $\leq$ ipToken

pwIpor ⇒ 0 $\leq$ pwIpor $\leq$ 25 $\cdot$ $10^{6}$

$blockRewards $\leq$ 100

$totalRewards = 25 $\cdot$ $10^{6}$