Where does the 33.33% threshold for selfish mining come from?

The concept of selfish mining, a strategy where miners aim to contribute more blocks to the blockchain than their fair share relative to their total network hashrate, is critically examined through an analysis of the mathematics underpinning the 2013 research paper "Majority is not enough." This strategy involves deliberately causing competitors' blocks to become stale by selectively revealing mined blocks, which can increase the revenue of the selfish miner beyond what would be expected based on their proportion of the network's hashrate.

Selfish mining becomes profitable when a miner controls a significant portion of the network's hashrate. The threshold at which this strategy becomes beneficial depends not only on the miner's hashrate but also on their ability to propagate their own blocks faster than those found by other miners. Interestingly, even without this propagation advantage, a miner controlling more than a third of the total network hashrate can still profit from selfish mining. The paper's analysis focuses on this scenario as a worst-case basis, assuming that all other miners will mine on top of a competing block in a one-block race situation.

Using a Markov Chain model, the paper outlines the state transitions for a selfish miner, with each state representing the miner's current lead in unpublished blocks over the rest of the network. The transitions between these states are governed by probabilities dependent on the miner's hashrate (α). From the constructed Markov Chain, a system of equations is derived and solved to find the stationary distribution of the states, which in turn allows for the calculation of the expected revenue from selfish mining.

The analysis further delves into the profitability of selfish mining by comparing the proportion of blocks found by the selfish miner to their share of the network hashrate. The findings confirm that selfish mining becomes advantageous when the miner's hashrate exceeds one-third of the total network hashrate. This conclusion is supported by both theoretical calculations and a simulation of the Bitcoin mining process, which accounts for block propagation delays and adopts the worst-case strategy for selfish mining (γ = 0). The simulation results align with the paper's conclusion, suggesting that the threshold for profitability lies slightly above one-third, specifically between 36% and 37% of the network hashrate.

This comprehensive analysis provides valuable insights into the dynamics of selfish mining and its implications for the security and fairness of blockchain networks. The detailed mathematical examination elucidates the conditions under which selfish mining strategies yield a higher return than honest mining, highlighting the importance of considering both hashrate distribution and block propagation advantages in assessing the vulnerability of blockchain protocols to such strategic behavior. For further exploration of the simulation and its findings, interested readers can visit the GitHub repository available at this link.