# 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:

```math
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.

```math
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.

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

### Composite Multiplier

```math
CM = \frac{blockRewards}{APU}
```

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

```math
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.

```math
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.

​

```math
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}`