Block Production & Rewards

Proof-of-Liquidity governs block rewards and token emissions on Berachain using the $BGT token. This page explains the mathematical principles behind validator selection, block rewards, and emissions calculations.

Validator Selection

The network maintains an active set of 69 validators who are eligible for block production. Selection criteria include:

• Only top 69 validators by $BERA stake are included in active set
• Block proposal probability is proportional to staked $BERA and does not affect reward amounts
• Stake limitations per validator:
  • Minimum: 250,000 $BERA
  • Maximum: 10,000,000 $BERA

A given Validator's probability of selection for producing a block is the proportion of its stake's weight to the total stakes of the active set.

$BGT Emissions Structure

When a validator produces a block, $BGT tokens are emitted through two emission components:

1. Base Emission

   • Fixed amount equal to a base rate parameter (B)
   • Paid directly to block-producing validator

2. Reward Vault Emission

   • Variable amount dependent on validator's boost (x)
     • i.e. percentage of total $BGT delegated to the validator
   • Distributed to Reward Vaults selected by validator
     • Proportional to weights configured in the validator's Reward Allocation
     • Validators receive Incentives from projects based on amounts directed to their Reward Vaults

Validator Boosts

Boost is a crucial metric that determines a validator's reward emissions:

• Calculated as the percentage of $BGT delegation a validator has compared to the total $BGT delegated in the network
• Expressed as a decimal between 0 and 1
• Example: If a validator has 1000 $BGT delegated and the network has 10000 total $BGT delegated, their boost would be 0.1 (10%) Higher boost leads to higher reward emissions, subject to the emission formula

$BGT Emissions Per Block

The total $BGT emitted per block is calculated using the following formula:

$$emission=[B+max(m,(a+1)(1-\frac{1}{1+a{x}^b})R)]$$

Parameters

Parameter	Description	Impact
x (boost)	Fraction of total $BGT delegated to validator (range: [0,1])	Determines $BGT emissions to Reward Vaults
B (base rate)	Fixed amount of $BGT for block production	Determines baseline validator rewards
R (reward rate)	Base $BGT amount for reward vaults	Sets foundation for reward emissions
a (boost multiplier)	Boost impact coefficient	Higher values increase boost importance
b (convexity parameter)	Boost impact curve steepness	Higher values penalize low boost more severely
m (minimum boosted reward rate)	Floor for reward vault emissions	Higher values benefit low-boost validators

Sample Emissions Chart

Using the following sample parameters, we can visualize how emissions scale with $BGT delegation:

$$B=0.4,R=1.1,a=3.5,b=0.4,m=0$$

Max Block Inflation

$BGT emissions grow with the amount of boost a validator has, up to a cap. The maximum theoretical block emission occurs at 100% boost:

$$max\mathbb{E}[\text{emission}]=[B+max(m,aR)]$$

$BGT Distribution

$BGT is emitted to reward vaults on a per-block basis via the Distributor by invoking the distributeFor function. This invocation creates $BGT that is then claimable by Reward Vault stakers.

TIP

Rewards are created on a per-block basis; however, the distribution of rewards is done over a three-day period.

Rewards are streamed linearly over this period to depositors proportionally to their deposit amounts. The reward window is reset each time new rewards are added.

Distribution Example

On Berachain, $BGT is distributed per block, meaning that the three-day distribution period is consistently being pushed to "start" on the current block. Thus, this period should be viewed as a sliding window based on the emissions at any given time during the previous three days.

A more real-world example with simplified numbers can be used to understand distribution currently (taking place over 9 days):

• 3 $BGT distributed daily, for a total of 27 over 9 days
• 1 depositor, owning all the deposits

The distribution graph would be as follows:

Legend

• Emitted: Total number of $BGT distributed and available
• Claimable: Total number of $BGT able to be claimed by depositors
• Daily Reward: Daily number of $BGT marked as claimable based on emitted tokens unlocks

This results in the depositor receiving an increasing amount of $BGT daily until rewards reach a saturation point after three days where all rewards are actively being distributed. Given that rewards are distributed on a frequent basis, the reward rate on a new reward vault should normalize after the initial three-day period.

Reward duration periods incentivize ecosystem alignment with depositors via this distribution mechanism rather than allowing rewards to be instantly claimed.

Calculating Boost APR

Boost APR is shown throughout the Berachain Hub.

Boost APR % is calculated using ranges of blocks, defined by a starting block and an ending block. At the time the percentages are calculated, the APR calculator samples the prices of all tokens (in $BERA).

Initial variables:

• $R_c$ : cycle rate
• $B_s$ : starting block
• $t_{B_s}$ : timestamp of starting block
• $B_e$ : ending block
• $t_{B_e}$ : timestamp of ending block
• $IT$ : incentive token
• $P_{T,t_{B_e}}$ : price of token $T$ at the end time of the block range

These variables are collected for the particular validator:

• $I_{T,b}$ : incentive amount received by validator at given block $b$ per given token $T$
• $Boos{t}_{t_{B_e}}$ : total BGT boost of the validator at ending time of the block range

Then, the $APR$ for that validator is the sum over the range of all received incentives priced at the end time of the block range, divided by the total boost at the end time, priced with BERA price at final time, annualized.

$$R\ast c=\frac{\sum \ast T\in IT(\sum \ast {b=B_s}^{B_e}I\ast T,b)\times P\ast T,t\ast B\ast e}{Boost\ast t\ast B_e\times P\ast BERA,t\mathrm{\_}{B}_e}$$$$APR=R\ast c\times \frac{T\ast year}{t\ast B_e-t\ast B_s}$$

Note that by “incentive received” we only consider incentives distributed by the RewardVault to the BGTIncentiveDistributor, tracked by the event BGTBoosterIncentivesProcessed (see contract code).

Amount in this calculation is net of commission, so per token it’s given by:

$$I\mathrm{\_}T,b=I{R}_T\times DistributedBGT\times (1-C)$$

with:

• $I{R}_T$ : incentive rate of token $T$
• $C$ : validator commission on incentives