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Optimal Reward Allocation via Proportional Splitting

Lukas Aumayr, Zeta Avarikioti, Dimitris Karakostas, Karl Kreder, Shreekara Shastry

TL;DR

This work introduces a reward allocation mechanism, called Proportional Splitting (PRS), which outperforms existing state of the art reward mechanisms and is shown to be an equilibrium, offering the same theoretical guarantees as the state of the art.

Abstract

Following the publication of Bitcoin's arguably most famous attack, selfish mining, various works have introduced mechanisms to enhance blockchain systems' game theoretic resilience. Some reward mechanisms, like FruitChains, have been shown to be equilibria in theory. However, their guarantees assume non-realistic parameters and their performance degrades significantly in a practical deployment setting. In this work we introduce a reward allocation mechanism, called Proportional Splitting (PRS), which outperforms existing state of the art. We show that, for large enough parameters, PRS is an equilibrium, offering the same theoretical guarantees as the state of the art. In addition, for practical, realistically small, parameters, PRS outperforms all existing reward mechanisms across an array of metrics. We implement PRS on top of a variant of PoEM, a Proof-of-Work (PoW) protocol that enables a more accurate estimation of each party's mining power compared to e.g., Bitcoin. We then evaluate PRS both theoretically and in practice. On the theoretical side, we show that our protocol combined with PRS is an equilibrium and guarantees fairness, similar to FruitChains. In practice, we compare PRS with an array of existing reward mechanisms and show that, assuming an accurate estimation of the mining power distribution, it outperforms them across various well-established metrics. Finally, we realize this assumption by approximating the power distribution via low-work objects called "workshares" and quantify the tradeoff between the approximation's accuracy and storage overhead.

Optimal Reward Allocation via Proportional Splitting

TL;DR

This work introduces a reward allocation mechanism, called Proportional Splitting (PRS), which outperforms existing state of the art reward mechanisms and is shown to be an equilibrium, offering the same theoretical guarantees as the state of the art.

Abstract

Following the publication of Bitcoin's arguably most famous attack, selfish mining, various works have introduced mechanisms to enhance blockchain systems' game theoretic resilience. Some reward mechanisms, like FruitChains, have been shown to be equilibria in theory. However, their guarantees assume non-realistic parameters and their performance degrades significantly in a practical deployment setting. In this work we introduce a reward allocation mechanism, called Proportional Splitting (PRS), which outperforms existing state of the art. We show that, for large enough parameters, PRS is an equilibrium, offering the same theoretical guarantees as the state of the art. In addition, for practical, realistically small, parameters, PRS outperforms all existing reward mechanisms across an array of metrics. We implement PRS on top of a variant of PoEM, a Proof-of-Work (PoW) protocol that enables a more accurate estimation of each party's mining power compared to e.g., Bitcoin. We then evaluate PRS both theoretically and in practice. On the theoretical side, we show that our protocol combined with PRS is an equilibrium and guarantees fairness, similar to FruitChains. In practice, we compare PRS with an array of existing reward mechanisms and show that, assuming an accurate estimation of the mining power distribution, it outperforms them across various well-established metrics. Finally, we realize this assumption by approximating the power distribution via low-work objects called "workshares" and quantify the tradeoff between the approximation's accuracy and storage overhead.

Paper Structure

This paper contains 55 sections, 2 theorems, 29 equations, 7 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and Cryptographic Primitives
  4. Execution Model
  5. Network
  6. Parties
  7. Proof-of-Work
  8. Cryptographic Model
  9. Adversary
  10. Randomized Execution
  11. Restrictions on the Environment
  12. Game Theoretic Model
  13. PoEM
  14. Property Definitions
  15. Chain Growth
  16. ...and 40 more sections

Key Result

theorem 1

For any constant $0 < \delta < 1$, and any $p, p_f$, let $R = 17$, $k_f = 2qRk$, and $T_0 = 5\frac{k_f}{\delta}$. Then, in $\Gamma_{\mathsf{PoEM^2}}^{p,p_f,R}$-environments, the Workshare protocol denoted $\Pi_{\mathsf{PoEM^2}}(p, p_f, R)$ satisfies

Figures (7)

  • Figure 1: Uncles can be referenced up to $R$ blocks away. No blocks that are a fork of more than $k$ blocks can be referenced. At every height $h_i + R$, the block rewards at height $h_i$ are distributed, proportional to the work of the valid blocks/uncles at height $h_i$.
  • Figure 2: Comparison of incentive compatibility for various reward mechanisms. The eligibility window for fruits (in FruitChains) and objects (in Proportional Reward Splitting) is set to $w=6$. The fork eligibility window for FruitChains, Reward Splitting, and Proportional Reward Splitting is set to $k=6$. Larger values indicate worse performance.
  • Figure 3: Comparison of subversion gain for various reward mechanisms. The eligibility window for fruits (in FruitChains) and objects (in Proportional Reward Splitting) is set to $w=6$. The fork eligibility window for FruitChains, Reward Splitting, and Proportional Reward Splitting is set to $k=6$. Larger values indicate worse performance.
  • Figure 4: Comparison of censorship susceptibility for various reward mechanisms. The eligibility window for fruits (in FruitChains) and objects (in Proportional Reward Splitting) is set to $w=6$. The fork eligibility window for FruitChains, Reward Splitting, and Proportional Reward Splitting is set to $k=6$. Larger values indicate worse performance.
  • Figure 5: Comparison of incentive compatibility for various values of the workshare eligibility window between Reward Splitting and Proportional Reward Splitting. The fork window for objects is set to $k=6$. Larger values indicate worse performance.
  • ...and 2 more figures

Theorems & Definitions (9)

  • definition 1: $\epsilon$-Nash equilibrium
  • definition 2: Chain growth
  • definition 3: Chain quality
  • definition 4: Consistency
  • definition 5: Fairness
  • theorem 1: Security of $\mathsf{PoEM^2}$
  • lemma 1
  • proof
  • proof