System Level Reward Logic & Equations

## 2. System Level Reward Logic & Equations

### 2.1 Determining Overall Reward QTY for Liquidity Providers

The amount of assets to be attracted depend on the available reserve, as the aim is to use it at its highest potential and thus efficiently provide liquidity. The smaller amount in reserve (ETH or assets) will be used as basis to calculate the overall qty of rewards as to not attract liquidity that could not be deployed.
\begin{align*} &\text {IF ETH_{inPOA} \leq SumAssets_{inPOA}} \\ \\ &\ \ \ \text {THEN ETHRewards = \frac{ETH_{inReserve} \times POAMultiplier \times APR_{ETH}}{TOKEPrice}} \\ \\ &\ \ \ \text {AND AssetRewards = \frac{ETH_{inReserve} \times POAMultiplier \times APR_{Assets}} {TOKEPrice}} \\ \end{align*}
\begin{align*} &\text {IF SumAssets_{inPOA} \leq ETH_{inPOA}} \\ \\ &\ \ \ \text {THEN ETHRewards = \frac{SumAssets_{inPOA} \times POAMultiplier \times APR_{ETH}}{TOKEPrice}} \\ \\ &\ \ \ \text {AND AssetRewards = \frac{SumAssets_{inPOA} \times RPOAMultiplier \times APR_{Assets}} {TOKEPrice}} \end{align*}
Please note that in it's current state the total qty of rewards does not follow the above logic but is predetermined and is currently set to:
• 20,000 TOKE per week for LPs in Token Reactors (project tokens)
• 37,240 TOKE per week for LPs in Pair Reactors (stables and ETH)
• 88,340 TOKE per week for LDs

## 3. Reactor Level Reward Logic and Equations

In order to incentivize behavior that is beneficial to the system and by extension to all participants, the above calculated rewards are only paid out in full if the system is in balance. This way both the interests of the system and the participants are aligned.
The reward equations that govern Tokemak only pay the maximum global rewards if all participants (Liquidity Directors [LDs] and Liquidity Providers [LPs]) have properly allocated in the pursuit of maximal APR. In this case, Tokemak runs at peak efficiency and there is maximal liquidity from LPs and maximal crowd-sourced liquidity direction and collateralization from LDs.
In order to determine the rewards to be paid out both the assets provided to a reactor by LPs as a percentage of all LP provided assets (except for the base asset such as ETH) and the TOKE staked to the reactor as a percentage of all TOKE staked across reactors has to be known.

### 3.1 Token Reactor LP Equations

Definition of variables:
• i is the i-th reactor
• lp_token(i) is the % of the overall LP Token Reactor assets in the i-th Reactor (normalized to $terms) • ld_token(i) is the % of the LD-staked TOKE to Token Reactors in the i-th Reactor • r_lp_token(i) is the % of the cycle’s LP Token Reactor rewards that are allocated to the LPs in Token Reactor i • R_lp_token is the maximum TOKE rewards budgeted to LPs across all Token Reactors each Cycle • R_lp_token(i) is the quantity of TOKE rewards that LPs in Token Reactor i will earn this Cycle The % of the overall rewards that go to LPs in Token Reactor i is governed by the following equation: $r_{lp\_token}(i)=lp\_token(i)^ \frac{2}{3} \times ld\_token(i)^ \frac{1}{3}$ The following equation can be used to determine the actual rewards for LPs in Token Reactor i: $R_{lp\_token}(i)=R_{lp\_token} \times r_{lp\_token}(i)$ The implications of the above equations is that Token Reactor LPs only receive maximal rewards (globally) if the % of Token Reactor LP assets in Token Reactor i matches the % of Token Reactor LD TOKE staked to Token Reactor i. In layman’s terms, either side controls the other. The equations are coupled. Stake more TOKE to juice the rewards for LPs and attract more tokens to the reactor. Deposit more tokens to juice the rewards for LDs staking TOKE to that reactor. Everything will balance given greedy participants that seek APR. ### 3.2 Pair Reactor LP Equations Definition of variables: • i is the i-th Reactor • lp_pair(i) is the % of the overall LP Pair Reactor assets in the i-th Reactor (normalized to$ terms)
• ld_pair(i) is the % of the LD-staked TOKE to Pair Reactors in the i-th Reactor
• r_lp_pair(i) is the % of the cycle’s LP Pair Reactor rewards that are allocated to the LPs in Pair Reactor i
• R_lp_pair is the maximum TOKE rewards budgeted to LPs across all Pair Reactors each Cycle
• R_lp_pair(i) is the quantity of TOKE rewards that LPs in Pair Reactor i will earn this Cycle
The % of the overall rewards that go to LPs in the Pair Reactor i is governed by the following equation:
$r_{lp\_pair}(i)=r_{lp\_pair}(i)^{\frac {2}{3}} \times ld\_pair(i)^{\frac {1}{3}}$
The below equation shows how to compute the actual rewards for LPs in each Pair Reactor i:
$R_{lp\_pair}(i)=R_{lp\_pair} \times r_{lp\_pair}(i)$

### 3.3 LD Equations

Definition of variables:
• i is the i-th reactor
• lp(i) is the % of the overall LP assets in the i-th Reactor (normalized to \$ terms)
• ld(i) is the % of the LD-staked TOKE in the i-th Reactor
• r_ld(i) is the % of the cycle’s LD rewards that are allocated to the LDs in Reactor i
• R_ld is the maximum TOKE rewards budgeted to LDs across all Reactors each Cycle
• R_ld(i) is the quantity of TOKE rewards that LDs in Reactor i will earn this Cycle
Note that these equations are global: covering assets and TOKE staked across both Pair Reactors and Token Reactors. This is very important to maintain balance.
The % of the overall rewards that go to LDs in Token Reactor i is governed by the following equation:
$r_{ld}(i)=ld(i)^{\frac{2}{3}} \times lp(i)^{\frac{1}{3}}$
The below equation shows how to compute the actual rewards for LDs in each Reactor i:
$R_{ld}(i)=R_{ld} \times r_{ld}(i)$
The implications of the above equations is that LDs only receive maximal rewards (globally) if the % of global LP assets in Reactor i matches the % of LD TOKE staked to Reactor i.
Note on Future Updates
These equations will continue to evolve and expand, and the next big update will balance TVL in Pair Reactors with TVL in Stable Reactors. We will also soon introduce the concept of Reserves and the expanded equations that balance Reserves, Liquidity Providers, and Liquidity Directors in the very near future.
Total Reward Quantities
The reward quantities as of Sept 30, 2022 are as follows:
• 10,000 TOKE / Cycle for Token Reactor LPs
• 37,240 TOKE / Cycle for Pair Reactor LPs
• 29,800 TOKE / Cycle for LDs
• 39,000 TOKE/ Cycle for TOKE/ETH Sushi LPs
*Please note these rewards quantities are subject to change. Follow along in the #dev-feed channel in Discord for the latest updates.