The Maths

Subject to change. Initial ideas and concepts of the formulas for reward staking.

Formula 1: Daily Reward Calculation (Staking)

$$r = \frac{R \times 10^{18}}{10^{18} + k \times P} \times \frac{10^{18}}{10^{18} + m \times S}$$

The interest rate for staking rewards is designed to be dynamic, adjusting based on the number of players and the total amount staked. This mechanism ensures a fair distribution of rewards and incentivizes early participation while balancing the system as more players join and more tokens are staked.

Description of Components:

• R = is the initial annual percentage rate (APY), set to 77% initially.
• k = is a constant that adjusts the impact of the number of players on the interest rate.
• P = is the total number of players participating in the staking.
• m = is a constant that adjusts the impact of the total staked amount on the interest rate.
• S = is the total amount of tokens staked
• 10^{18} = is used to maintain precision and avoid floating-point errors.

Formula 2: Decreasing Reward Over Time

$$R(t) = r * e^{-kt}$$

Reward Decay Over Time for $SLOTH Formula:

Description of Components:

• R(t) = Reward at time .
• r = Initial reward rate at time ‘t=0’ .
• e = Base of the natural logarithm.
• k = Decay constant determining rate of reward decrease.
• t = Time (days) since the staking began.

Formula 3: PRIDE Coin Earnings from Treasury

$$E=T/N-B$$

PRIDE Coin Earnings Calculation:

Description of Components:

• E = Earnings from the treasury per PRIDE coin when burned.
• T = Total funds available in the treasury designated for distribution.
• N = Initial total number of PRIDE coins (7,777 to start).
• B = Total number of PRIDE coins burned to date.

Formula 4: Purity Earned Calculation

$$\text{Purity Earned} = \frac{\text{amountStaked} \times \text{adjustedIssuanceRate} \times \text{duration}}{\text{totalAmountStaked} \times \text{numberOfPlayers} \times 1e6}$$

Purity Earned Calculation

Description of Components:

• {amountStaked} is the amount of tokens staked by the user.
• {adjustedIssuanceRate} is the rate at which purity is issued, adjusted for the remaining duration.
• {duration} is the staking duration in seconds.
• {totalAmountStaked} is the total amount of tokens staked by all players.	•	\text{numberOfPlayers} is the total number of players staking tokens.
• 1e6 is a constant to maintain fixed-point arithmetic precision.

Formula 5: Purity Redistribution

$$\text{playerShare} = \left( \frac{\text{playerStakeTime} \times \text{purityAmount}}{\text{totalTimeStaked}} \right)$$

Purity Redistribution Calculation

When a player unstakes, the accumulated purity is redistributed among the remaining players. The formula for redistributing purity:

• {playerStakeTime} is the product of the player’s staked amount and the staking duration.
• {purityAmount} is the total purity to be redistributed.
• {totalTimeStaked} is the sum of the staking times for all remaining players.

These formulas provide a framework for implementing a dynamic reward system within the 7DS Coins ecosystem, allowing for both incentivization of long-term engagement and active management, depending on the characteristics desired for each Sin Coin.