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.
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