  For a mining strategy we define the notion of "profit lag" as the minimum
time it takes to be profitable after that moment. We compute closed forms for
the profit lag and the revenue ratio for the strategies "selfish mining" and
"intermittent selfish mining". This confirms some earlier numerical simulations
and clarifies misunderstandings on profitability in the literature. We also
study mining pairs of PoW cryptocurrencies, often coming from a fork, with the
same mining algorithm. This represents a vector of attack that can be exploited
using the "alternate network mining" strategy that we define. We compute closed
forms for the profit lag and the revenue ratiofor this strategy that is more
profitable than selfish mining and intermittent selfish mining. It is also
harder to counter since it does not rely on a flaw in the difficulty adjustment
formula that is the reason for profitability of the other strategies.
