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3.2 Mining There are also M miners who compete for updating a blockchain in subperiods n = 0, ..., N ̄ with all the transactions from subperiod 0. In each subperiod, miners perform exactly one costly computa- tional task with a random success rate by investing computing power, q, measured in real balances of cryptocurrency.16 This task is called the Proof-of-Work (PoW). We assume that miners value real balances linearly. As motivated by the Bitcoin protocol (see Property (i) in Section 2), if the computational power of miner i in a subperiod is q(i), then the probability that a particular miner i will win the mining game is given by ρ(i)= q(i) . (3) Mm=1 q(m) In other words, the probability of wining is proportional to the fraction of computational power owned. We take this feature as given here and provide a micro foundation for this result in the Appendix. By winning the competition in any subperiod, a miner can update the Blockchain (i.e., append the nth block to the Blockchain) and receive a reward R in real balances. We assume that miners receive and consume this reward after the period, discounted by the time preference β. Note that the mining games are independent across subperiods. Hence, in any subperiod miner i solves so that max ρiβR − q(i) (4) q(i) M q(m) − q∗(i) m=1 2 βR = 1. (5) i̸=j q(m) + q∗(i) where he takes as given the choice by all miners m ̸= i. Imposing symmetry, q(m) = Q for all m, we obtain as the Nash equilibrium of the mining game that Q = M − 1βR. (6) M2 Consequently, the total computing cost of mining in any subperiod is MQ = M − 1βR (7) M 16One could assume that miners buy computing power q at price α so that z units of real balances yield computing power q = αz. We can, however, simply normalize α = 1 by changing the unit for q. 14PDF Image | The Economics of Cryptocurrencies Bitcoin and Beyond
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