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Reward Schemes and Committee Sizes in Proof of Stake Governance

Georgios Birmpas, Philip Lazos, Evangelos Markakis, Paolo Penna

TL;DR

This work analyzes governance in delegated proof-of-stake systems by modeling a binary decision with ground truth and DReps who invest costly effort to improve voting accuracy and attract delegations. It shows that proportional sharing rewards can yield poor equilibria, and advocates threshold-based mechanisms (Threshold$(k)$) that cap the number of compensated DReps to induce higher effort. The paper derives general bounds and examines how the optimal committee size depends on the cost structure (concave, convex, concave-convex) and on learning-curve-inspired S-shaped costs, finding that often a small number of DReps suffices to maximize the probability of correct outcomes. It provides a framework for designing reward schemes and selecting committee sizes in PoS governance, with implications for efficient, incentive-compatible decentralized decision-making.

Abstract

In this paper, we investigate the impact of reward schemes and committee sizes motivated by governance systems over blockchain communities. We introduce a model for elections with a binary outcome space where there is a ground truth (i.e., a "correct" outcome), and where stakeholders can only choose to delegate their voting power to a set of delegation representatives (DReps). Moreover, the effort (cost) invested by each DRep positively influences both (i) her ability to vote correctly and (ii) the total delegation that she attracts, thereby increasing her voting power. This model constitutes the natural counterpart of delegated proof-of-stake (PoS) protocols, where delegated stakes are used to elect the block builders. As a way to motivate the representatives to exert effort, a reward scheme can be used based on the delegation attracted by each DRep. We analyze both the game-theoretic aspects and the optimization counterpart of this model. Our primary focus is on selecting a committee that maximizes the probability of reaching the correct outcome, given a fixed monetary budget allocated for rewarding the delegates. Our findings provide insights into the design of effective reward mechanisms and optimal committee structures (i.e., how many DReps are enough) in these PoS-like governance systems.

Reward Schemes and Committee Sizes in Proof of Stake Governance

TL;DR

This work analyzes governance in delegated proof-of-stake systems by modeling a binary decision with ground truth and DReps who invest costly effort to improve voting accuracy and attract delegations. It shows that proportional sharing rewards can yield poor equilibria, and advocates threshold-based mechanisms (Threshold) that cap the number of compensated DReps to induce higher effort. The paper derives general bounds and examines how the optimal committee size depends on the cost structure (concave, convex, concave-convex) and on learning-curve-inspired S-shaped costs, finding that often a small number of DReps suffices to maximize the probability of correct outcomes. It provides a framework for designing reward schemes and selecting committee sizes in PoS governance, with implications for efficient, incentive-compatible decentralized decision-making.

Abstract

In this paper, we investigate the impact of reward schemes and committee sizes motivated by governance systems over blockchain communities. We introduce a model for elections with a binary outcome space where there is a ground truth (i.e., a "correct" outcome), and where stakeholders can only choose to delegate their voting power to a set of delegation representatives (DReps). Moreover, the effort (cost) invested by each DRep positively influences both (i) her ability to vote correctly and (ii) the total delegation that she attracts, thereby increasing her voting power. This model constitutes the natural counterpart of delegated proof-of-stake (PoS) protocols, where delegated stakes are used to elect the block builders. As a way to motivate the representatives to exert effort, a reward scheme can be used based on the delegation attracted by each DRep. We analyze both the game-theoretic aspects and the optimization counterpart of this model. Our primary focus is on selecting a committee that maximizes the probability of reaching the correct outcome, given a fixed monetary budget allocated for rewarding the delegates. Our findings provide insights into the design of effective reward mechanisms and optimal committee structures (i.e., how many DReps are enough) in these PoS-like governance systems.
Paper Structure (35 sections, 20 theorems, 66 equations, 3 figures)

This paper contains 35 sections, 20 theorems, 66 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Contribution
  3. Related work
  4. Our delegation model
  5. Equilibria.
  6. Budget Constraint.
  7. Objective.
  8. Optimization benchmark.
  9. Warmup: Equilibrium Analysis of Proportional Sharing
  10. Candidate Mechanisms
  11. Equilibria under the Threshold$(k)$ mechanism
  12. Variants of Threshold$(k)$
  13. Optimal Solutions
  14. General Bounds
  15. Concave (and linear) case: One DRep is optimal
  16. ...and 20 more sections

Key Result

Theorem 1

The only symmetric Nash equilibrium under the proportional reward sharing rule, with a linear cost function, is the strategy profile with $x_i = x = \frac{B(n-1)}{an^2}$ for every $i$, as long as $x\in [0, 1/2]$.

Figures (3)

  • Figure 1: The idea of Theorem \ref{['thm:concave-convex-lb-effort']}.
  • Figure 2: The idea of Theorem \ref{['thm:cocave-convex-tangent']} with the inflection point $x_{inflection}^c$ and point $x_{tangent}^c$.
  • Figure 3: An example of a concave-convex cost function from exponential learning (Definition \ref{['def:exp-learning-costs']}).

Theorems & Definitions (47)

  • Remark 1
  • Theorem 1
  • Theorem 2
  • Remark 2
  • Theorem 3
  • proof : Proof Idea
  • Corollary 1
  • proof
  • Theorem 4
  • Remark 3
  • ...and 37 more