> For the complete documentation index, see [llms.txt](https://docs.ipor.io/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://docs.ipor.io/dao-governance/tokenomics/usdipor-token-legacy/math-behind-the-rewards-contract.md). # 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 \ \&#xNAN;***HS*** - Horizontal Shift\ \&#xNAN;***pwIpor*** - User's pwIPOR balance delegated to a particular asset\ \&#xNAN;***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\ \&#xNAN;***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 $$\cdot$$ $$10^{6}$$ --- # Agent Instructions This documentation is published with GitBook. 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