On profitability of stubborn mining

Abstract

We compute and compare profitabilities of stubborn mining strategies that are variations of selfish mining. These are deviant mining strategies violating Bitcoin's network protocol rules. We apply the foundational set-up from our previous companion article on the profitability of selfish mining, and the new martingale techniques to get a closed-form computation for the revenue ratio, which is the correct benchmark for profitability. Catalan numbers and Catalan distributions appear in the closed-form computations. This marks the first appearance of Catalan numbers in the Mathematics of the Bitcoin protocol.

Bitcoin Selfish Mining and Dyck Words

“…In this strategy, the attacker never tries to override the official blockchain but when it is possible, he broadcasts the part of his secret fork sharing the same height as the official blockchain as soon as the honest miners publish a new block. The attack cycle ends when the attacker has been caught up and overtaken by the honest miners by one block [5,7]. We show that the distribution of L − 1 is what we have called a (p, q)-Catalan distribution of first type in [5].…”

“…The attack cycle ends when the attacker has been caught up and overtaken by the honest miners by one block [5,7]. We show that the distribution of L − 1 is what we have called a (p, q)-Catalan distribution of first type in [5].…”

“…By repeating the same argument as in the proof of Theorem 3.3 for the computation of E[Z], we obtain the following theorem [5].…”

“…Once the competition is resolved, a new attack cycles starts. In this case, the distribution of L − 1 is what we have called a (p, q)-Catalan distribution of second type [5]. Proof.…”

On Profitability of Trailing Mining

“…τ ] This is the correct benchmark for the profitability of the strategy. This analysis was also carried out previously by the authors in [4] to the "Selfish Mining" (SM) strategy from [1] and in [5] to "Lead-Stubborn Mining" (LSM) and "Equal Fork Stubborn Mining" (EFSM) strategies from [8]. In these articles we found for these strategies the exact mathematical formula for the revenue ratios and we compared its profitability in parameter space.…”