From Grieving to Stability in Blockchain Mining Economies: A Game-Theoretic Analysis

Table of Contents

1 Introduction

With over 4,000 circulating cryptocurrencies valued above $1 trillion and numerous decentralized applications running on blockchain technologies, understanding the stability and long-term sustainability of these systems is crucial for wider adoption. The critical actors in blockchain ecosystems are miners who provide costly resources to secure consensus through Proof of Work (PoW) or Proof of Stake (PoS) protocols.

Miners operate in a self-interested, decentralized manner and can enter or leave networks at any time. They receive rewards in proportion to their contributed resources, but their incentives for resource allocation across different blockchains remain poorly understood. This paper addresses this gap through a game-theoretic analysis of mining economies.

2 Model and Framework

2.1 Mining Economy Model

We study a game-theoretic model of blockchain mining economies comprising single or multiple co-existing blockchains. The model builds on the work of [3], which derived unique Nash Equilibrium allocations under proportional reward schemes common in PoW and PoS protocols.

The fundamental insight is that at predicted NE levels, active miners still have incentives to deviate by increasing their resources to achieve higher relative payoffs, even when this behavior is sub-optimal for absolute payoffs.

2.2 Grieving Factors

Grieving is defined as the practice where network participants harm other participants at some lesser cost to themselves. We quantify this through grieving factors, which measure network losses relative to the deviator's own losses:

$$GF_i = \frac{\sum_{j \neq i} \Delta u_j}{\Delta u_i}$$

where $GF_i$ is the grieving factor for miner $i$, $\Delta u_j$ represents the utility loss for other miners, and $\Delta u_i$ is the utility loss for the deviating miner.

3 Theoretical Results

3.1 Nash Equilibrium Analysis

Theorem 1 establishes the existence and uniqueness of Nash Equilibrium allocations. However, our analysis reveals that these equilibria are vulnerable to grieving attacks where individual miners can profit by deviating from equilibrium strategies.

Theorem 6 and Corollary 7 demonstrate that the loss a deviating miner incurs to themselves is overcompensated by larger market share and greater losses inflicted on other miners and the network as a whole.

3.2 Evolutionary Stability

Our main technical contribution connects grieving to evolutionary game theory. We show that grieving behavior relates directly to evolutionary stability concepts, providing a formal argument for the dissipation of resources, consolidation of power, and high entry barriers observed in practice.

4 Proportional Response Protocol

4.1 Algorithm Design

As networks grow larger, miner interactions resemble distributed production economies or Fisher markets. For this scenario, we derive a Proportional Response (PR) update protocol:

// Proportional Response Algorithm
for each miner i in network:
    current_allocation = get_current_allocation(i)
    expected_reward = calculate_expected_reward(i, current_allocation)
    
    for each blockchain j:
        new_allocation[i][j] = current_allocation[i][j] * 
                              (expected_reward[j] / total_expected_reward)
    
    normalize(new_allocation[i])
    update_allocation(i, new_allocation[i])

4.2 Convergence Properties

The PR protocol converges to market equilibria where grieving becomes irrelevant. Convergence holds for wide ranges of miner risk profiles and various degrees of resource mobility between blockchains with different mining technologies.

5 Empirical Analysis

5.1 Case Study Methodology

We conducted a case study with four mineable cryptocurrencies to validate our theoretical findings. The study examined how risk diversification, restricted resource mobility, and network growth contribute to ecosystem stability.

5.2 Results and Findings

Our empirical results demonstrate that all three factors—risk diversification, restricted mobility, and network growth—significantly contribute to the stability of the inherently volatile blockchain ecosystem. The convergence behavior of the PR protocol was validated across different network conditions.

6 Technical Implementation

6.1 Mathematical Framework

The core mathematical model builds on evolutionary game theory with non-homogeneous populations. The grieving factor formulation extends traditional stability analysis:

$$\max_{x_i} u_i(x_i, x_{-i}) = \frac{x_i}{\sum_j x_j} R - c_i x_i$$

where $x_i$ represents miner $i$'s resources, $R$ is the total reward, and $c_i$ is the cost coefficient.

6.2 Code Implementation

The Proportional Response algorithm can be implemented in Python for simulation purposes:

import numpy as np

class ProportionalResponseMiner:
    def __init__(self, initial_allocation, risk_profile):
        self.allocation = initial_allocation
        self.risk_profile = risk_profile
    
    def update_allocation(self, market_conditions):
        expected_returns = self.calculate_expected_returns(market_conditions)
        total_return = np.sum(expected_returns)
        
        if total_return > 0:
            new_allocation = self.allocation * (expected_returns / total_return)
            self.allocation = new_allocation / np.sum(new_allocation)
        
        return self.allocation
    
    def calculate_expected_returns(self, market_conditions):
        # Implementation depends on specific market model
        returns = np.zeros_like(self.allocation)
        for i, alloc in enumerate(self.allocation):
            returns[i] = market_conditions[i]['reward'] * alloc / \
                        market_conditions[i]['total_hashrate']
        return returns

7 Future Applications

The Proportional Response protocol and grieving analysis have significant implications for blockchain design and regulation. Future applications include:

• Improved Consensus Mechanisms: Designing PoW/PoS protocols that inherently resist grieving attacks
• Cross-Chain Resource Allocation: Optimizing miner resources across multiple blockchains
• Regulatory Frameworks: Informing policies that promote healthy mining competition
• DeFi Protocol Design: Applying similar stability analysis to decentralized finance systems

Future research should explore how these concepts apply to emerging technologies like proof-of-space, proof-of-stake variants, and hybrid consensus mechanisms.

8 References

1. Cheung, Y. K., Leonardos, S., Piliouras, G., & Sridhar, S. (2021). From Grieving to Stability in Blockchain Mining Economies. arXiv:2106.12332
2. Nakamoto, S. (2008). Bitcoin: A Peer-to-Peer Electronic Cash System
3. Eyal, I., & Sirer, E. G. (2014). Majority is not enough: Bitcoin mining is vulnerable. Financial Cryptography
4. Buterin, V. (2014). Ethereum: A Next-Generation Smart Contract and Decentralized Application Platform
5. Nisan, N., Roughgarden, T., Tardos, E., & Vazirani, V. V. (2007). Algorithmic Game Theory
6. Goodfellow, I., et al. (2014). Generative Adversarial Networks. Neural Information Processing Systems