  We propose a mean field game model to study the question of how
centralization of reward and computational power occur in Bitcoin-like
cryptocurrencies. Miners compete against each other for mining rewards by
increasing their computational power. This leads to a novel mean field game of
jump intensity control, which we solve explicitly for miners maximizing
exponential utility, and handle numerically in the case of miners with power
utilities. We show that the heterogeneity of their initial wealth distribution
leads to greater imbalance of the reward distribution, and increased wealth
heterogeneity over time, or a "rich get richer" effect. This concentration
phenomenon is aggravated by a higher bitcoin mining reward, and reduced by
competition. Additionally, an advantaged miner with cost advantages such as
access to cheaper electricity, contributes a significant amount of
computational power in equilibrium, unaffected by competition from less
efficient miners. Hence, cost efficiency can also result in the type of
centralization seen among miners of cryptocurrencies.
