  Bitcoin and Ethereum are the top two blockchain-based cryptocurrencies
whether from cryptocurrency market cap or popularity. However, they are
vulnerable to selfish mining and stubborn mining due to that both of them adopt
Proof-of-Work consensus mechanism. In this paper, we develop a novel Markov
model, which can study selfish mining and seven kinds of stubborn mining in
both Bitcoin and Ethereum. The formulas are derived to calculate several key
metrics, including relative revenue of miners, blockchain performance in terms
of stale block ratio and transactions per second, and blockchain security in
terms of resistance against double-spending attacks. Numerical analysis is
conducted to investigate the quantitative relationship between the
relative-revenue-optimal mining strategy for malicious miners and two miner
features in Bitcoin and Ethereum, respectively. The quantitative analysis
results can assist honest miners in detecting whether there is any malicious
miner in the system and setting the threshold of mining node's hash power in
order to prevent malicious miners from making profit through selfish and
stubborn mining.
