  This paper studies a fundamental problem regarding the security of blockchain
PoW consensus on how the existence of multiple misbehaving miners influences
the profitability of selfish mining. Each selfish miner (or attacker
interchangeably) maintains a private chain and makes it public
opportunistically for acquiring more rewards incommensurate to his Hash power.
We first establish a general Markov chain model to characterize the state
transition of public and private chains for Basic Selfish Mining (BSM), and
derive the stationary profitable threshold of Hash power in closed-form. It
reduces from 25% for a single attacker to below 21.48% for two symmetric
attackers theoretically, and further reduces to around 10% with eight symmetric
attackers experimentally. We next explore the profitable threshold when one of
the attackers performs strategic mining based on Partially Observable Markov
Decision Process (POMDP) that only half of the attributes pertinent to a mining
state are observable to him. An online algorithm is presented to compute the
nearly optimal policy efficiently despite the large state space and high
dimensional belief space. The strategic attacker mines selfishly and more
agilely than BSM attacker when his Hash power is relatively high, and mines
honestly otherwise, thus leading to a much lower profitable threshold. Last, we
formulate a simple model of absolute mining revenue that yields an interesting
observation: selfish mining is never profitable at the first difficulty
adjustment period, but replying on the reimbursement of stationary selfish
mining gains in the future periods. The delay till being profitable of an
attacker increases with the decrease of his Hash power, making blockchain
miners more cautious on performing selfish mining.
