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A Appendix – Proofs and Derivations A.1 A micro-foundation for the proof-of-work problem Miners perform their mining between trading sessions. By investing computing power q(m), the probability that a miner m can solve the computational task within a time interval t is given by an exponential distribution with parameter μm F (t) = 1 − e−μm t where μm = q(m)/D. The parameter D captures the difficulty of the proof-of-work controlled by the system. The expected time needed to solve the problem is thus given by D. q(m) Aggregating over all M miners, the first solution among all miners, min(τ1, τ2..., τM ), is also an ex- ponential random variable with parameter Mm=1 μm. Hence the expected time needed to complete the proof-of-work by the pool of miners is31 D. Mm=1 q(m) Furthermore, any particular miner m will be the first one to solve the proof-of-work problem with probability ρn = q(n) . Mm=1 q(m) 31In practice, the parameter D is often adjusted to maintain a constant time for completing the proof-of-work problem given any changes in the total computational power. 38PDF Image | The Economics of Cryptocurrencies Bitcoin and Beyond
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