  In a PoW-based blockchain network, mining pools (the solo miner could be
regarded as a mining pool containing one miner) compete to successfully mine
blocks to pursue rewards. Generally, the rewards include the fixed block
subsidies and time-varying transaction fees. The transaction fees are offered
by the senders whose transactions are packaged into blocks and is increasing
with the block size. However, the larger size of a block brings the longer
latency, resulting in a smaller probability of successfully mining. Therefore,
finding the optimal block size to trade off these two factors is a complex and
crucial problem for the mining pools. In this paper, we model a repeated mining
competition dynamics in blockchain system as an evolutionary game to study the
interactions among mining pools. In this game, each pool has two strategies: to
follow the default size $\bar{B}$, i.e., the upper bound of a block size, or
not follow. Because of the bounded rationality, each mining pool pursues its
evolutionary stable block size (ESS) according to the mining pools' computing
power and other factors by continuous learning and adjustments during the whole
mining process. A study framework is built for the general evolutionary game,
based on which we then theoretically explore the existence and stability of the
ESSs for a case of two mining pools. Numerical experiments with real Bitcoin
data are conducted to show the evolutionary decisions of mining pools and to
demonstrate the theoretical findings in this paper.
