Performance Based Inflation

Berachain's emissions system includes a variable component (R) that controls total BGT issuance per block, but this parameter is currently static. Inflation does not respond automatically to incentive demand, meaning excess BGT may be emitted during low-demand periods while opportunities are missed during high demand.

Performance-Based Inflation makes R responsive to market conditions, thus emitting only as much BGT as the market can efficiently absorb.

Original BGT Emissions Formula

The current per-block BGT emission function can be expressed as:

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

Component Breakdown

Separating the formula into its functional components:

$$\begin{aligned} \text{Floor BGT emissions with NO boost}(B) &= B \\[6pt] \text{Floor BGT emissions with boost}(F) &= \max\!\left(m,\right) \\[0pt] \text{Boost Concentration Modifier}(C) &= (a+1)\left(1-\frac{1}{1+a x^{b}}\right) \\[6pt] \text{BGT Supply Modifier}(R) &= R\ \\[6pt] \end{aligned}$$

Simplified:

• B = Absolute minimum BGT emitted per block, even when no boosts are present

• F = Minimum emission level when boosts exist, preventing emissions from collapsing

• C = Allocates emissions based on boost amount; convex curve penalizes excessive concentration

• R = Controls total BGT issuance per block

  • R=1: baseline emissions

  • R>1: increased inflation

  • R<1: reduced inflation

Making R Responsive

The inflation multiplier R becomes responsive to incentive market efficiency, converging toward a target clearing rate while dynamically adjusting emissions.

Definitions:

• R_t: Emission control variable at time t

• R_t-1: Previous period value

• b_t: Observed incentive rate (or clearing efficiency)

• b^*: Target incentive rate

• d_t = b_t - b^*: Deviation from target

Interpretation:

• d_t > 0: market clearing above target

• d_t < 0: market clearing below target

Adjustment Models

1. Fixed Step Binary Adjustment

$$R_t = R_{t-1} + \begin{cases} +\delta & \text{if } d_t > 0 \\ -\delta & \text{if } d_t < 0 \end{cases}$$

Adjusts emissions by a fixed amount based solely on direction. Magnitude of deviation does not matter.

Characteristics:

• Prioritizes stability

• Slow to react

• Low precision

Parameters:

• Step size (upward): 0.0005

• Step size (downward): 0.0008

Asymmetric steps allow faster contraction during low demand while being cautious on expansion.

2. Gradual Linear Response

$$R_t = R_{t-1} + k \cdot d_t$$

Adjustments scale proportionally with deviation from target.

Characteristics:

• Small deviations → small adjustments

• Large deviations → larger adjustments

• Smooth convergence without abrupt jumps

Parameters:

• Sensitivity k: 0.002

At a 0.2 deviation from target, this produces a daily adjustment of 0.0004.

3. Gradual Convex Response

$$R_t = R_{t-1} + k_c \cdot d_t \cdot |d_t|^{p-1}$$

Introduces nonlinearity into the control loop.

Characteristics:

• Near target, adjustments remain minimal

• Large deviations trigger disproportionately stronger corrections

• Prevents persistent drift

Parameters:

• Sensitivity k_c​: 0.02

• Power p: 2 (quadratic)

Small deviations (±0.05) get gentle adjustments and large deviations (±0.2) trigger corrections 4x stronger.

Resulting Behavior

By making R reactive to incentive market efficiency:

• Inflation contracts when demand is weak, reducing sell pressure

• Inflation expands only when the market can absorb emissions

• Average incentive rates rise organically during scarcity

• Berachain monetizes block rewards more efficiently without hard caps or manual tuning

Future Improvement: Hybrid Adaptive Model

A more sophisticated system could combine the best qualities of all three models:

Hybrid Model

$$R_t = R_{t-1} + k \cdot d_t \cdot (1 + \alpha |d_t|^p)$$

Uses linear response near target, automatically shifts to convex response for large deviations.

Additional Enhancements:

• EMA Smoothing: Filter out daily volatility by tracking smoothed clearing rates

• Asymmetric Response: React faster when contracting, slower when expanding

• Target Bands: Define acceptable range (e.g., 0.75–0.85) rather than exact target and make no adjustments within band

• Velocity-Aware: Considers rate of change and not just current position

• Soft Bounds: Floor and ceiling on R (e.g., 0.5 to 1.5) with dampening as R approaches bounds

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Inflation Calculator

You can use the following calculator to get a feel of how BGT inflation will move with the new model:

PoL Performance Based Inflation Calculatoraesthetic-stardust-f67f0e.netlify.app