This paper studies a fundamental problem regarding the security of blockchain on how the existence of different numbers of misbehaving pools with different selfish mining strategies influences the profitability. Each selfish miner maintains a private chain and makes it public opportunistically for the purpose of acquiring more rewards incommensurate to his Hashrate. We establish a novel Markov chain model to characterize all the state transitions of public and private chains under basic selfish mining. The minimum requirement of Hashrate together with the minimum delay of being profitable is derived in close-form. The former reduces to 21.48% with the symmetric selfish miners, while the profitable threshold decreases with the increases of the number of attackers. The profitable delay increases with the decrease of the Hashrate of selfish miners, making the mining pools more cautious on performing selfish mining. We further investigate the profitability of selfish mining when one of the selfish miners performs the optimal attack based on Partially Observable Markov Decision Process (POMDP). An online search method is presented to compute the approximate optimal policy efficiently. Experimental results show that the optimal strategy significantly improves the revenue of the attacker.
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