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BAR Nash Equilibrium and Application to Blockchain Design

Maxime Reynouard, Rida Laraki, Olga Gorelkina

TL;DR

This work introduces BAR Nash Equilibrium (BARNE), a generalization of Nash equilibrium to settings with Byzantine, altruistic/honest, and rational agents (BAR). It proves BARNE existence under mild conditions and defines local and global stability refinements, enabling robust assessments of incentive mechanisms in distributed algorithms. The authors apply BARNE to the Verifier's Dilemma in quorum-based blockchain protocols, showing that the standard protocol rarely yields a BARNE and that free-riding is typically globally stable; they propose two amendments—monetary fines and trap blocks—that can render honest behavior globally (or locally) stable BARNE under realistic parameter regimes. The analysis demonstrates that such amendments are practically implementable and provide a structured, predictive framework for designing incentive-compatible distributed systems with mixed agent types. Overall, BARNE and its stability refinements offer a principled approach to resilience against selfish, faulty, and adversarial behavior in distributed consensus and related areas.

Abstract

This paper presents a novel solution concept, called BAR Nash Equilibrium (BARNE) and apply it to analyse the Verifier's dilemma, a fundamental problem in blockchain. Our solution concept adapts the Nash equilibrium (NE) to accommodate interactions among Byzantine, altruistic and rational agents, which became known as the BAR setting in the literature. We prove the existence of BARNE in a large class of games and introduce two natural refinements, global and local stability. Using this equilibrium and its refinement, we analyse the free-rider problem in the context of byzantine consensus. We demonstrate that by incorporating fines and forced errors into a standard quorum-based blockchain protocol, we can effectively reestablish honest behavior as a globally stable BARNE.

BAR Nash Equilibrium and Application to Blockchain Design

TL;DR

This work introduces BAR Nash Equilibrium (BARNE), a generalization of Nash equilibrium to settings with Byzantine, altruistic/honest, and rational agents (BAR). It proves BARNE existence under mild conditions and defines local and global stability refinements, enabling robust assessments of incentive mechanisms in distributed algorithms. The authors apply BARNE to the Verifier's Dilemma in quorum-based blockchain protocols, showing that the standard protocol rarely yields a BARNE and that free-riding is typically globally stable; they propose two amendments—monetary fines and trap blocks—that can render honest behavior globally (or locally) stable BARNE under realistic parameter regimes. The analysis demonstrates that such amendments are practically implementable and provide a structured, predictive framework for designing incentive-compatible distributed systems with mixed agent types. Overall, BARNE and its stability refinements offer a principled approach to resilience against selfish, faulty, and adversarial behavior in distributed consensus and related areas.

Abstract

This paper presents a novel solution concept, called BAR Nash Equilibrium (BARNE) and apply it to analyse the Verifier's dilemma, a fundamental problem in blockchain. Our solution concept adapts the Nash equilibrium (NE) to accommodate interactions among Byzantine, altruistic and rational agents, which became known as the BAR setting in the literature. We prove the existence of BARNE in a large class of games and introduce two natural refinements, global and local stability. Using this equilibrium and its refinement, we analyse the free-rider problem in the context of byzantine consensus. We demonstrate that by incorporating fines and forced errors into a standard quorum-based blockchain protocol, we can effectively reestablish honest behavior as a globally stable BARNE.
Paper Structure (30 sections, 2 theorems, 16 equations, 13 figures, 5 tables, 1 algorithm)

This paper contains 30 sections, 2 theorems, 16 equations, 13 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our contributions
  3. BAR Model and Equilibria
  4. Model Primitives
  5. The BAR model: three types of agents
  6. BAR-Strong Equilibrium
  7. BAR-Nash Equilibrium
  8. Locally and Globally Stable BARNE
  9. Analysing some QBCPs using BARNE
  10. A Classical QBCP
  11. The endorsement extensive-form game
  12. Rewards, losses and costs
  13. Strategies in the endorsement game
  14. The Byzantine-rational simplex
  15. Amending the classical QBCP
  16. ...and 15 more sections

Key Result

Theorem 1

For some given $F$ and $G$, two disjoint subsets of $N$, noting $H=N \setminus (F \cup G)$, if (1) $T$ is a convex compact subset of a topological vector space, (2) any $i\in G$, $u_i$ is continuous and (3) $t_i \mapsto u_i(s_F,(t_i,s_{G \setminus \{i\}}),s_H)$ is concave for any strategy profile $s

Figures (13)

  • Figure 1: Decision Tree in the Endorsement Game
  • Figure 2: The Byzantine-rational simplex, special areas
  • Figure : Base protocol, only for $Q \le \frac{n+1}{2}$
  • Figure : $\sigma_{0}$ BARNE, base protocol
  • Figure : Base protocol, only for $Q \le \frac{n+1}{2}$
  • ...and 8 more figures

Theorems & Definitions (9)

  • Definition 1
  • Definition 2
  • Theorem 1
  • proof
  • Example 1
  • Lemma 1
  • Remark 1
  • Definition 3
  • Definition 4