A Deep Dive into Blockchain Selfish Mining: Multi-Pool Dynamics and Profitability

2.1 Core Insight: The Fragility of the 25% Myth

The most striking takeaway is the demolition of a comforting security heuristic. The blockchain community has clung to the "25% threshold" from Eyal and Sirer as a stable red line. This paper shows that line is porous. When multiple entities engage in selfish mining—a realistic scenario in today's concentrated mining landscape—the effective barrier to entry for this attack lowers significantly (to 21.48% in the symmetric case). This isn't just an incremental finding; it's a paradigm shift. It suggests that the security of major PoW chains is more precarious than widely assumed. The existence of large, opaque mining pools makes the assumption of a single adversary naive. As noted in the IEEE Security & Privacy In community discussions, attack surfaces often expand when moving from idealized to realistic multi-party models.

2.2 Logical Flow: From Single-Actor to Multi-Actor Game Theory

The authors' logical progression is sound and necessary. They start by acknowledging the established single-pool model, then correctly identify its critical limitation: it ignores strategic interaction between malicious actors. Their move to model two selfish pools (unaware of each other's nature) as a Markov game is the right methodological choice. The state space elegantly captures the lengths of public and private chains, and the transitions model the stochastic discovery of blocks. This approach mirrors the advancement in adversarial ML research, such as moving from single-attacker models in CycleGAN training to more complex, multi-adversarial environments. The derivation of closed-form thresholds from this complex model is a notable technical achievement, providing a concrete metric for risk assessment.

2.3 Strengths & Flaws: A Model's Merit and Blind Spots

Strengths: The paper's primary strength is its formalization of a more realistic threat model. The inclusion of transient analysis is particularly praiseworthy. Most analyses focus on steady-state profitability, but miners operate in finite time horizons. Showing that selfish mining is initially unprofitable and requires waiting for difficulty adjustment adds a crucial layer of practical risk, making pools more "cautious." The mathematical rigor is commendable.

Flaws & Blind Spots: The model, while sophisticated, still rests on significant simplifications. The assumption that selfish pools are "unaware" of each other is a major one. In reality, large pools are highly observant; strange chain dynamics would quickly signal the presence of other selfish miners, leading to a more complex, adaptive game. The model also sidesteps the real-world possibility of collusion, which would drastically change the dynamics and lower thresholds even further. Furthermore, it doesn't fully account for network propagation delays and the "mining gap" effect, which are known to influence selfish mining outcomes, as discussed in follow-up work to the original Eyal and Sirer paper.

2.4 Actionable Insights: For Miners, Pools, and Protocol Designers

• For Mining Pools & Monitor: This research is a clarion call for enhanced monitoring. Security teams must look for anomalies indicative of multiple competing selfish miners, not just one. The profitability threshold is lower than you think.
• For Protocol Designers (Ethereum, Bitcoin Cash, etc.): The urgency for post-PoS transition or robust PoW modifications (like GHOST or other chain selection rules) is amplified. Defenses designed for a single adversary may be insufficient.
• For Investors & Analysts: Hashrate concentration in a few pools is not just a decentralization concern; it's a direct security risk multiplier. Evaluate chains not just on the 51% metric, but on the resilience of their consensus to multi-actor selfish mining.
• For Academia: The next step is to model aware and potentially colluding selfish pools. Research should also integrate this with other known attacks (e.g., Bribery Attacks) for a holistic threat assessment.

3.1 State Transition Model

The system state is defined by the lead of the selfish pools' private chains over the public chain. Let $L_1$ and $L_2$ represent the lead of selfish pool 1 and 2, respectively. The public chain is always the longest published chain known to the honest miners. Transitions occur based on stochastic block discovery events:

• Honest pool finds a block: The public chain advances, potentially reducing the relative lead of selfish pools.
• Selfish pool S1 (or S2) finds a block: It adds to its private chain, increasing its lead $L_1$ (or $L_2$).
• Publishing decision: A selfish pool may publish part of its private chain to overtake the public chain when strategically advantageous, resetting its lead and potentially causing a chain reorganization.

The Markov chain captures all possible $(L_1, L_2)$ states and the probabilities of moving between them, determined by the relative Hashrates $\alpha_1$, $\alpha_2$ (for S1 and S2) and $\beta = 1 - \alpha_1 - \alpha_2$ (for the honest pool).

3.2 Key Mathematical Formulations

The analysis solves for the steady-state distribution $\pi_{(L_1, L_2)}$ of the Markov chain. The key metric, relative revenue $R_i$ for selfish pool $i$, is derived from this distribution. It represents the fraction of all blocks eventually included in the canonical chain that were mined by pool $i$.

Profitability Condition: Selfish mining is profitable for pool $i$ if its relative revenue exceeds its proportional Hashrate: $$R_i(\alpha_1, \alpha_2) > \alpha_i$$ The paper derives the minimum $\alpha_i$ (or $\alpha$ in symmetric case) that satisfies this inequality.

Symmetric Case Result: When $\alpha_1 = \alpha_2 = \alpha$, the threshold $\alpha^*$ is found by solving: 21.48%.

4.1 Profitability Thresholds

The paper presents two key numerical findings:

Interpretation: The 21.48% figure is lower than the canonical ~25% threshold. However, if one selfish pool is larger, the smaller Selfish pool needs an even higher Hashrate to compete profitably, as it now battles both the honest network and a dominant selfish rival. This creates a "selfish mining oligarchy" effect where being the dominant malicious actor is advantageous.

4.2 Transient Analysis & Profitable Delay

The paper emphasizes that profitability is not instantaneous. Because selfish mining involves withholding blocks, it initially reduces the pool's short-term reward rate compared to honest mining. Profitability only emerges after the Bitcoin network's difficulty adjustment (every 2016 blocks), which lowers the puzzle difficulty because the observed block rate (slowed by withholding) is lower.

Key Finding: The number of difficulty adjustment periods ("epochs") $D$ a selfish miner must wait to become profitable increases as its Hashrate $\alpha$ decreases. Formally, $D(\alpha)$ is a decreasing function. For a pool just above the threshold (e.g., 22%), the wait could be several epochs, representing weeks or months, during which capital is tied up and strategy risk is high. This delay acts as a natural deterrent for smaller pools considering the attack.

Chart Description (Conceptual): A line chart would show the "Profitable Delay (Epochs)" on the Y-axis against "Selfish Miner Hashrate (α)" on the X-axis. The curve starts very high for α just above 0.2148, sharply decreasing and asymptotically approaching zero as α increases towards 0.5. This visually reinforces that higher hashrate selfish miners reap rewards faster.