Appendix B

The reward mechanism in Core has a lot of moving parts. For a better understanding of the reader, this section presents two simple examples to simplify the inner workings of the reward mechanism in Core.

Part 1

Let’s assume there are two validators, both of which have been elected:

Validator A, which has:

• Two units of delegated hash power.

• One unit of CORE stake.

• One unit of non-custodial BTC stake.

Validator B, which has:

• One unit of hash power.

• Three units of CORE stake.

• Two units of non-custodial BTC stake.

Assumptions

Let’s assume that there are 10 total units of Bitcoin hash power on the Core network. Validator A has 20% of the hash power (i.e., $\frac {2} {10}$), and Validator B has 10% of the hash power (i.e., $\frac {1} {10}$).

Let’s also assume there are 20 total units of CORE staked on the network. Validator A has 5% of the CORE staked (i.e., $\frac {1} {20}$), and Validator B has 15% of the CORE staked (i.e., $\frac {3} {20}$).

Let’s further assume there are 10 total units of non-custodial BTC staked on the network. Validator A has 10% of the BTC staked (i.e., $\frac {1} {10}$), and Validator B has 20% of the BTC staked (i.e., $\frac {2} {10}$).

For this example, let’s assume the following variables for simplicity in calculation:

• m = $\frac {1} {6}$ represents the weight assigned to hash power in the hybrid score calculation.
• k = $\frac {1}{2}$ represents the weight assigned to CORE staking in the hybrid score calculation.
• l = $\frac {1}{3}$ represents the weight assigned to BTC staking in the hybrid score calculation.

Hybrid Score Calculation

The hybrid score for a validator combines contributions from hash power, CORE staking, and BTC staking using the following formula:

$$S = (\frac{rHp}{tHp} × m) + (\frac{rSp}{tSp} × k) + (\frac{rBp}{tBp} × l)$$

Where, m + k + l = 1

• rHp: Hash power delegated to the validator.

• tHp: Total hash power on the network.

• rSp: CORE tokens staked to the validator.

• tSp: Total CORE tokens staked on the network.

• rBp: BTC staked to the validator.

• tBp: Total BTC staked on the network.

• S: Hybrid score of the validator.

For Validator A:

$$S = (\frac{2}{10}×\frac{1}{6} ) + (\frac{1}{20}×\frac{1}{2} ) + (\frac{1}{10}×\frac{1}{3} ) = \frac{11}{120}$$

For Validator B:

$$S = (\frac{1}{10}×\frac{1}{6} ) + (\frac{3}{20}×\frac{1}{2} ) + (\frac{2}{10}×\frac{1}{3} ) = \frac{19}{120}$$

Reward Distribution

The validators share their rewards with the delegators delegating Hash, CORE, or BTC. The delegator's rewards are distributed across three components—hash power delegation (rH), CORE staking delegation (rS), and BTC staking delegation (rB). These are calculated as follows:

$$rH = \frac{(\frac{rHp}{tHp} × m)}{S} × R$$

$$rS = \frac{(\frac{rSp}{tSp} × k)}{S} × R$$

$$rB = \frac{(\frac{rBp}{tBp} × l)}{S} × R$$

Where:

• rH: Rewards allocated based on hash power.

• rS: Rewards allocated based on CORE staking.

• rB: Rewards allocated based on BTC staking.

• R: Total rewards allocated for all delegators of a validator after deduction of validator commission.

For Validator A:

For this example, assume R=1000.

• $rH =$$rH = \frac{(\frac{2}{10}×\frac{1}{6})}{S}× R = \frac{\frac{1}{40}}{\frac{11}{120}}× 1000 = \frac{4000}{11}$
• $rS = \frac{(\frac{1}{20}×\frac{1}{2})}{S}× R = \frac{\frac{1}{40}}{\frac{11}{120}}× 1000 = \frac{3000}{11}$
• $rB = \frac{(\frac{1}{10}×\frac{1}{3})}{S}× R = \frac{\frac{1}{30}}{\frac{11}{120}}× 1000 = \frac{4000}{11}$

For Validator B:

• $rH =$$rH = \frac{(\frac{1}{10}×\frac{1}{6})}{S}× R = \frac{\frac{1}{60}}{\frac{19}{120}}× 1000 = \frac{2000}{19}$
• $rS = \frac{(\frac{3}{20}×\frac{1}{2})}{S}× R = \frac{\frac{3}{40}}{\frac{19}{120}}× 1000 = \frac{9000}{19}$
• $rB = \frac{(\frac{2}{10}×\frac{1}{3})}{S}× R = \frac{\frac{2}{30}}{\frac{19}{120}} × 1000 = \frac{8000}{19}$

Rewards Per Unit

The rewards per unit of hash power and CORE staking are calculated by dividing the validator rewards by the respective delegation amounts:

$$rHu = \frac{rH}{rHp}$$

$$rSu = \frac{rS}{rSp}$$

As for BTC staking, the rewards per unit are further subdivided based on the delegators’ dual staking yield tiers. Assuming for this example that there are 4 boosted yield levels (PBASE, P₁, P₂, …, Pₘₐₓ) with dual staking yield multipliers (e, f, g, and h). The reward per unit for BTC staking will be calculated as follows

• rBu of PBASE = $\frac{rB}{rBp} × e$
• rBu of P₁ = $\frac{rB}{rBp} × f$
• rBu of P₂ = $\frac{rB}{rBp} × g$
• rBu of Pₘₐₓ = $\frac{rB}{rBp} × h$

Where:

• rHu: Hash power rewards per unit.

• rSu: CORE staking rewards per unit.

• rBu of PBASE is the BTC staking rewards per unit for PBASE delegator

• rBu of PLevel1 is the BTC staking rewards per unit for PLevel1 delegator

• rBu of PLevel2 is the BTC staking rewards per unit for PLevel2 delegator

• rBu of PMAX is the BTC staking rewards per unit for PMAX delegator;

• Yield Multipliers: Each reward tier has a specific multiplier (e,f,g,h, ..., etc) that is multiplied to rewards earned per unit of BTC staked.

Assuming that the Yield Multipliers for each tier are as follow:

• $e = 20 \%$
• $f = 35 \%$
• $g = 80 \%$
• $h = 1000 \%$

For Validator A:

• rHu = $\frac{rH}{rHp} = \frac{4000}{11} ÷ 2 = \frac{ 2000}{11}$
• rSu = $\frac{rS}{rSp} = \frac{3000}{11} ÷ 1 = \frac{ 3000}{11}$
• rBu of PBASE = $\frac{rB}{rBp} × e$ = $(\frac{4000}{11} \div 1 ) × 20\%$ = $\frac{700}{11}$
• rBu of P₁ =$\frac{rB}{rBp} × f$ = $(\frac{4000}{11} \div 1 ) × 35\%$ = $\frac{1400}{11}$
• rBu of P₂ =$\frac{rB}{rBp} × g$ = $(\frac{4000}{11} \div 1 ) × 80\%$ = $\frac{3200}{11}$
• rBu of Pₘₐₓ = $\frac{rB}{rBp} × h$ = $(\frac{4000}{11} \div 1 ) × 1000\%$ = $\frac{40000}{11}$

For Validator B:

• rHu = $\frac{rH}{rHp} = \frac{2000}{19} ÷ 1 = \frac{ 2000}{19}$
• rSu = $\frac{rS}{rSp} = \frac{9000}{19} ÷ 3 = \frac{ 3000}{19}$
• rBu of PBASE = $\frac{rB}{rBp} × e$ = $(\frac{4000}{19} \div 2 ) × 20\%$ = $\frac{40}{19}$
• rBu of P₁ =$\frac{rB}{rBp} × f$ = $(\frac{4000}{19} \div 2 ) × 35\%$ = $\frac{70}{19}$
• rBu of P₂ =$\frac{rB}{rBp} × g$ = $(\frac{4000}{19} \div 2 ) × 80\%$ = $\frac{1600}{19}$
• rBu of Pₘₐₓ = $\frac{rB}{rBp} × h$ = $(\frac{4000}{19} \div 2 ) × 1000\%$ = $\frac{20000}{19}$

Part 2

Here, we’ll work through an identical example, except we’ll make a few different assumptions about the relationships between different quantities.

Validator A:

• 60 units of delegated hash power.

• 5,000,000 units of CORE stake.

• 400 units of non-custodial BTC stake.

Validator B:

• 30 units of hash power.

• 15,000,000 units of CORE stake.

• 200 units of non-custodial BTC stake.

Assumptions

Let’s assume the following:

• Total hash power (tHp): 300 units. Then, Validator A: rHp = 60, Validator B: rHp = 30

• Total CORE staked (tSp): 100M units. Then, Validator A: rSp = 5M, Validator B: rSp = 15M

• Total BTC staked (tBp): 4,000 units. Then, Validator A: rBp = 400, Validator B: rBp = 200

• Weights: $m = \frac{1}{6}$, $k = \frac{1}{2}$, $l = \frac{1}{3}$
• Yield Multipliers: $e= 20\%$ $f= 35\%$ $g= 80\%$ and $h = 1000\%$

Hybrid Score Calculation

$$S = (\frac{rHp}{tHp} × m) + (\frac{rSp}{tSp} × k) + (\frac{rBp}{tBp} × l)$$

For Validator A:

$$S = (\frac{60}{300}×\frac{1}{6} ) + (\frac{5M}{100M}×\frac{1}{2} ) + (\frac{400}{4000}×\frac{1}{3} ) = \frac{11}{120}$$

For Validator B:

$$S = (\frac{30}{300}×\frac{1}{6} ) + (\frac{15M}{100M}×\frac{1}{2} ) + (\frac{200}{4000}×\frac{1}{3} ) = \frac{13}{120}$$

Reward Distribution

For this example, assume R=1000, then

For Validator A:

• $rH =$$rH = (\frac{60}{300}×\frac{1}{6}) \div \frac{11}{120} × 1000 = \frac{4000}{11}$
• $rS = (\frac{5M}{100M}×\frac{1}{2}) \div \frac{11}{120} × 1000 = \frac{3000}{11}$
• $rB = (\frac{400}{4000}×\frac{1}{3}) \div \frac{11}{120} × 1000 = \frac{4000}{11}$

For Validator B:

• $rH =$$rH = (\frac{30}{300}×\frac{1}{6}) \div \frac{13}{120} × 1000 = \frac{2000}{13}$
• $rS = (\frac{15M}{100M}×\frac{1}{2}) \div \frac{13}{120} × 1000 = \frac{9000}{13}$
• $rB = (\frac{200}{4000}×\frac{1}{3}) \div \frac{13}{120} × 1000 = \frac{2000}{13}$

Rewards Per Unit

For Validator A:

• rHu = $\frac{rH}{rHp} = \frac{4000}{11} ÷ 60 = \frac{ 200}{33}$
• rSu = $\frac{rS}{rSp} = \frac{3000}{11} ÷ {5M} = \frac{ 3000}{55M}$
• rBu of PBASE = $\frac{rB}{rBp} × e$ = $(\frac{4000}{11} \div 400) × 20\%$ = $\frac{2}{11}$
• rBu of P₁ =$\frac{rB}{rBp} × f$ = $(\frac{4000}{11} \div 400 ) × 35\%$ = $\frac{7}{22}$
• rBu of P₂ =$\frac{rB}{rBp} × g$ = $(\frac{4000}{11} \div 400 ) × 80\%$ = $\frac{8}{11}$
• rBu of Pₘₐₓ = $\frac{rB}{rBp} × h$ = $(\frac{4000}{11} \div 400 ) × 1000\%$ = $\frac{100}{11}$

For Validator B:

• rHu = $\frac{rH}{rHp} = \frac{2000}{13} ÷ 30 = \frac{ 200}{39}$
• rSu = $\frac{rS}{rSp} = \frac{9000}{13} ÷ 15M = \frac{ 9}{195,000}$
• rBu of PBASE = $\frac{rB}{rBp} × e$ = $(\frac{2000}{13} \div 200 ) × 20\%$ = $\frac{40}{19}$
• rBu of P₁ =$\frac{rB}{rBp} × f$ = $(\frac{2000}{13} \div 2 ) × 35\%$ = $\frac{7}{26}$
• rBu of P₂ =$\frac{rB}{rBp} × g$ = $(\frac{2000}{13} \div 2 ) × 80\%$ = $\frac{8}{13}$
• rBu of Pₘₐₓ = $\frac{rB}{rBp} × h$ = $(\frac{2000}{13} \div 2 ) × 1000\%$ = $\frac{100}{13}$