A Mean Field Game Approach to Bitcoin Mining

We present an analysis of the Proof-of-Work consensus algorithm, used on the Bitcoin blockchain, using a mean field game framework. Using a master equation, we provide an equilibrium characterization of the total computational power devoted to mining the blockchain (hashrate). This class of models allows us to adapt to many different situations. The essential structure of the game is preserved across all the enrichments. In deterministic settings, the hashrate ultimately reaches a steady state in which it increases at the rate of technological progress only. In stochastic settings, there exists a target for the hashrate for every possible random state. As a consequence, we show that in equilibrium the security of the underlying blockchain and the energy consumption either are constant or increase with the price of the underlying cryptocurrency.