  This paper studies a fundamental problem regarding the security of blockchain
on how the existence of multiple misbehaving pools influences the profitability
of selfish mining. 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. 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 their competition with asymmetric Hashrates
puts forward a higher requirement of the profitable threshold. The profitable
delay increases with the decrease of the Hashrate of selfish miners, making the
mining pools more cautious on performing selfish mining.
