Energy Equilibria in Proof-of-Work Mining

The Bitcoin protocol induces miners, through monetary rewards, to expend energy in order to add blocks to the chain. We show that, when energy costs are substantial and taken into account, counterintuitive and unintended strategic behavior results: In a simple bounded-horizon setting with two identical miners there is a unique pure symmetric equilibrium in which both miners first "slow down" in order to decrease the crypto complexity and then take advantage of this decrease. If miners have different energy efficiencies and are restricted to choose the same hash rate for many epochs, there is a unique pure equilibrium in which miners either participate at low levels that depend in intricate ways on all the other miners' efficiencies, or choose to abstain from mining if their efficiency is too low. In the general setting in which miners can adapt their hash rates over time, we show that, unless the number of miners is very small, the only possible pure equilibria are rather chaotic, with miners quitting and starting again periodically --- or there is no pure equilibrium at all. We discuss the implications of these results for the stability of proof-of-work protocols.