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In proof-of-work-based (PoW-based) blockchain networks, the miners participate in a crypto-puzzle solving competition to win the reward by publishing a new block. Open mining pools attract a large number of miners for solving difficult problems together. Although the open strategy is likely to be more efficient, it makes pools susceptible to attack at the same time. In this paper, we present a gametheoretic analysis of mining pool strategy selection in order to explore the trade-off between the efficiency of openness and the vulnerability of attacks in a PoW-based blockchain network. We first model the pool mining process as a two-stage game, wherein the pools might decide whether to open or not and to attack or not. Based on the two-stage game model, we analyze the Nash equilibrium and the evolutionary stability of the mining games among pools, which uncovers the pool selection dynamics of PoW-based blockchain networks. In particular, we find that the attack behavior is the norm for a weak pool and triggers lower expected utilities when punishing the attacks more severely. Numerical simulations also support our theoretical findings as well as demonstrate the stability of the pools' strategy selection.INDEX TERMS Blockchain, block withholding attack, game theory, Nash equilibrium, evolutionary stability.

Blockchain technology, has the characteristics of decentralization, openness and transparency, so that everyone can participate in database recording. Therefore, blockchain technology has a good application prospect in various industries. As the most successful application of blockchain technology, the Bitcoin system applies the Proof of Work (PoW) consensus mechanism. Under the PoW consensus mechanism, each miner competes through his own power to solve a SHA256 mathematical problem together, so as to gain profits. Due to the difficulty of the cryptography puzzle, miners tend to join the mining pool to obtain stable income. And the block withholding attacks will be carried out between the mining pools, so as to maximize his own income by controlling the infiltration rate dispatched to other mining pools. In this paper, we build a game model between mining pools based on the PoW consensus algorithm, and analyze its Nash equilibrium from two perspectives. The influence of the mining pools' power, the ratio of the power to be infiltrated, and the betrayed rate of dispatched miners on the mining pool's infiltration rate selection and income were explored, and the results were obtained through numerical simulations.