# Mining Pool Selection under Block WithHolding Attack +
The mining pool selection problem without considering the BWH attack was investigated in [24] based on the evolutionary game theory. The mining pool selection issue in the presence of BWH attack has also been addressed in [25] from the perspective of evolutionary game as well. The authors solved the problem from the viewpoint of pool managers and investigated how pool managers can change strategies to attract miners, which differs from the objective of this paper. In addition, the authors neglected the revenues from attacked pools, which may lead to less accurate and insightful findings. Motivated by these observations, our research objective is to investigate the mining pool selection issue under the BWH attack from the perspective of miners. Like [24; 25], this paper also applies the evolutionary game theory to model the mining pool selection process and find the Evolutionarily Stable States (ESSs) (i.e., stable population states) of the game as the solutions. To achieve this goal, we first derive the expected revenue densities of all pools to determine the expected payoffs of miners in the pools. Based on the expected payoffs, we formulate replicator dynamics to represent the growth rates of the populations in all pools. With the help of the replicator dynamics, we obtain the Nash Equilibria (NE) of the game, i.e., rest points where the population growth rates are zeros, and discuss their stability to identify the ESSs of the game. To demonstrate the process of solving the game, we consider the scenario with two mining pools (Note that the two-pool scenario has been widely adopted in the analysis of mining pool selection [24; 25], because this scenario is easy to analyze yet powerful enough to reveal the fundamental findings. In addition, the analysis and results obtained from this scenario can serve as the building blocks for those of more general scenarios with more mining pools.) and further investigate the case where one pool attacks the other and the case where the two pools attack each other. Simulation and numerical results are also provided to corroborate our analysis and to illustrate the theoretical findings. The results in this paper show that pools can attract more miners to join by launching the BWH attack and the attack power significantly affects the stable pool populations. The conference version of this paper was published in [26], which focused only on the case where one pool attacks the other. This paper extends [26] by including the analysis and numerical results for the case where the two pools attack each other. Compared with the previous work in [25], this paper has the following two main contributions. First, the previous work studied the problem from the viewpoint of pool managers and focused on the payoffs of mining pools, while this paper investigates the mining pool selection problem from the viewpoint of miners and formulates the payoffs of individual miners. To do this, we propose a new concept called _revenue density_ to characterize the revenue per unit hash power of a pool. Second, this paper carefully takes the revenues from attacked pools (i.e., attacking revenue) into consideration, while the previous work neglected the attacking revenue to simplify the analysis. The results in this paper show that neglecting the attacking power may give us inaccurate results and less insightful findings. The rest of the paper is organized as follows. Section 2 introduces the related work. In Section 3, we model the mining pool selection problem in the presence of BWH attack as an evolutionary game and conduct analysis to solve the game. The case study with two mining pools is presented in Section 4. We provide simulations and numerical results in Section 5 and finally conclude this paper in Section 6.