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.

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