Mining Pool Selection Problem in the Presence of Block Withholding Attack

Mining, the process where multiple miners compete to add blocks to Proof-of-Work (PoW) blockchains, is of great importance to maintain the tamper-resistance feature of blockchains. In current blockchain networks, miners usually form groups, called mining pools, to improve their revenues. When multiple pools exist, a fundamental mining pool selection problem arises: which pool should each miner join to maximize its revenue? In addition, the existence of mining pools also leads to another critical issue, i.e., Block WithHolding (BWH) attack, where a pool sends some of its miners as spies to another pool to gain extra revenues without contributing to the mining of the infiltrated pool. This paper therefore aims to investigate the mining pool selection issue (i.e., the stable population distribution of miners in the pools) in the presence of BWH attack from the perspective of evolutionary game theory. We first derive the expected revenue density of each pool to determine the expected payoff of miners in that pool. Based on the expected payoffs, we formulate replicator dynamics to represent the growth rates of the populations in all pools. Using the replicator dynamics, we obtain the rest points of the growth rates and discuss their stability to identify the Evolutionarily Stable States (ESSs) (i.e., stable population distributions) of the game. Simulation and numerical results are also provided to corroborate our analysis and to illustrate the theoretical findings.