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Stabilizing the Staking Rate, Dynamically Distributed Inflation and Delay Induced Oscillations

Carlo Brunetta, Amit Chaudhary, Stefano Galatolo, Massimiliano Sala

TL;DR

The paper tackles instability in dynamic inflation-based staking mechanisms for Proof-of-Stake systems, where steep yield sensitivity near the target combined with delayed staker responses can produce oscillations around the desired staking rate. It introduces a discrete-time mean-staker model with a seven-day lag and compares three inflation-distribution schemes (constant, Polkadot-like, stability corridor) using simulations and linear stability analysis. The key finding is that a stability corridor, which flattens the yield within a target staking band, dampens oscillations and broadens the stability region for the sensitivity parameter, offering a more robust design for PoS economics. The work provides a control-theoretic perspective on inflation-curve design and highlights practical steps for calibrating parameters with on-chain data, aiming to enhance predictability and security in blockchain economies.

Abstract

Dynamically distributed inflation is a common mechanism used to guide a blockchain's staking rate towards a desired equilibrium between network security and token liquidity. However, the high sensitivity of the annual percentage yield to changes in the staking rate, coupled with the inherent feedback delays in staker responses, can induce undesirable oscillations around this equilibrium. This paper investigates this instability phenomenon. We analyze the dynamics of inflation-based reward systems and propose a novel distribution model designed to stabilize the staking rate. Our solution effectively dampens oscillations, stabilizing the yield within a target staking range.

Stabilizing the Staking Rate, Dynamically Distributed Inflation and Delay Induced Oscillations

TL;DR

The paper tackles instability in dynamic inflation-based staking mechanisms for Proof-of-Stake systems, where steep yield sensitivity near the target combined with delayed staker responses can produce oscillations around the desired staking rate. It introduces a discrete-time mean-staker model with a seven-day lag and compares three inflation-distribution schemes (constant, Polkadot-like, stability corridor) using simulations and linear stability analysis. The key finding is that a stability corridor, which flattens the yield within a target staking band, dampens oscillations and broadens the stability region for the sensitivity parameter, offering a more robust design for PoS economics. The work provides a control-theoretic perspective on inflation-curve design and highlights practical steps for calibrating parameters with on-chain data, aiming to enhance predictability and security in blockchain economies.

Abstract

Dynamically distributed inflation is a common mechanism used to guide a blockchain's staking rate towards a desired equilibrium between network security and token liquidity. However, the high sensitivity of the annual percentage yield to changes in the staking rate, coupled with the inherent feedback delays in staker responses, can induce undesirable oscillations around this equilibrium. This paper investigates this instability phenomenon. We analyze the dynamics of inflation-based reward systems and propose a novel distribution model designed to stabilize the staking rate. Our solution effectively dampens oscillations, stabilizing the yield within a target staking range.

Paper Structure

This paper contains 7 sections, 9 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Oscillations and Dynamical Distribution of the Inflation
  3. A Dynamical Model for the Staking Rate Evolution
  4. Stability and Oscillations near the Equilibrium Points
  5. Analytical Investigation
  6. Applications to the Stability Corridors
  7. Conclusions and Future Directions

Figures (2)

  • Figure 1: The total inflation curve $\IfBlankTF{(}{ {-NoValue-} }{ {-NoValue-}\left((\right) }\xspace {I} \xspace ){\sigma}$ as a function of the staking rate $\sigma$ for the three examples described in the text.
  • Figure 2: The dynamics of the staking rate with the three proposed inflation distribution curves. In blue we plot the staking rate as a function of time, in red the relative yield. The actions taken by the staker (increase, maintain or reduce staking) are plotted as background colour.