Owner: @Alexander Mozeika

Reviewers: 🟢@Marvin Jones 🟢@Álvaro Castro-Castilla 🟢@Marcin Pawlowski

Introduction

The Service Declaration Protocol (SDP) introduces a piece of a priori information: the knowledge that a node's relative stake cannot be less than a known threshold, $\alpha_0$. Our research investigates the significance of the impact of this information on the statistical inference of relative stake. We propose a new estimator which explicitly utilises $\alpha_0$ by setting any estimated stake below this threshold to $\alpha_0$.

Our new estimator works better because it fixes estimation errors at the lower end. When a node's true stake value ($\alpha_i$) is close to the minimum threshold ($\alpha_0$), the standard maximum likelihood (ML) estimator often produces values that are too low. By automatically adjusting these too-low estimates up to the minimum threshold ($α_0$), our new approach reduces errors. This improvement can be measured as a lower mean squared error (MSE) compared to the true stake value ($\alpha_i$). Thus any party, including potential adversaries, performing stake inference gains in accuracy by using the new estimator.

Numerical experiments demonstrate reduction in MSE of the new estimator compared to the ML estimator, particularly for stakes near $α_0$. For example for $\alpha_0=10^{-4}$ used in experiments reduction of MSE by (approx.) factor of at most $1/2$ was observed. Furthermore, the probability, measured in the same experiment, that the inferred stake falls within a desired accuracy interval is higher (by factor of (approx.) $3$ at least) when new estimator is used. While the advantage diminishes for much higher stake values where both estimators converge, the heightened accuracy near the critical $α_0$ threshold presents a meaningful enhancement for any party performing stake inference, including potential adversaries.

Key Findings

• Introduction of a priori information: The Service Declaration Protocol (SDP) introduces the knowledge that a node's relative stake cannot be less than a threshold ($α₀$), which impacts statistical inference of relative stake⁠⁠.
• New estimator proposed: The research introduces a new estimator that explicitly uses α₀ by setting any estimated stake below this threshold to $α₀$⁠⁠.
• Improved accuracy: The new estimator performs better because it corrects estimation errors at the lower end, particularly when a node's true stake value is close to the minimum threshold⁠⁠.
• Measurable improvements: Numerical experiments show:
  • Reduction in Mean Squared Error (MSE) of the new estimator compared to the ML estimator, particularly for stakes near $α₀$⁠⁠.
  • For $α₀=10⁻⁴$, MSE reduction by a factor of approximately $1/2$ was observed⁠⁠.
  • Higher probability (by a factor of approximately 3) that inferred stake falls within desired accuracy intervals⁠⁠.
• Statistical significance: The advantage diminishes for much higher stake values where both estimators converge, but the enhanced accuracy near the critical α₀ threshold presents a meaningful improvement for any party performing stake inference⁠⁠.
• Security implications: This improvement benefits anyone performing stake inference, including potential adversaries⁠⁠.

The research provides mathematical proof and numerical simulations to validate these findings, showing that the proposed estimator is both unbiased and consistent in the limit of large number of observations⁠⁠.

Overview

This document examines the impact of minimum stake threshold, introduced in the SDP, on the statistical inference of relative stake along the following points:

In particular:

1. We consider Leader Election Process where nodes allowed to participate only if their relative stake is no less than some prescribed by SDP threshold.
2. We assume that Adversary observes wins (and loses) of nodes and uses statistical inference to infer relative stake of nodes.