  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.
