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To derive the no-double-spending constraint we work backwards. Given (d, N ) and R, the expected payoff for a double spending buyer in subperiod N having been successful N times already is DN (d, N) = βR(√∆ − 1)2. (14) Define recursively the expected payoff from double spending in subperiod n for n ∈ {0, . . . , N − 1} by Dn (d, N ) = max ρ(qn )Dn+1 (d, N ) − qn , (15) qn which takes into account that the buyer was n times successful, since if he fails once the double spend fails as well. Note that Dn(d,N) can only be positive if Dn+1(d,N) is positive and qˆn > 0 only if Dn(d,N) > 0. The first-order condition describing the optimal investment is thus given by qˆn(d,N)=QM·Dn+1(d,N)−QM. (16) By backward induction, we then obtain the following result. Lemma 2. As M →∞, DN−s(d, N) = βR √∆ − (s + 1)2 , (17) √ ρN −s (d, N ) = √∆ − s , (18) ∆−(s+1) qˆN−s(d, N) = βR √∆ − (s + 1) . (19) It follows immediately that Dn(d,N) is strictly increasing in n and, consequently, the investment by the double spending buyer qˆn is also increasing in n. Hence, if it was optimal to engage in secret mining in subperiod n, it is optimal to continue with secret mining in subperiod n + 1, if one has been successful in subperiod n. Consequently, double spending is not optimal for the buyer whenever βR √∆ − (N + 1) < 0, so that qˆ = 0 and D (d, N ) = 0. This yields the following no double spending constraint.21 21In a setting where a number nb of buyers can coordinate in their double-spending attempts, the condition becomes dnb < R(N + 1)N. This suggests that a cryptocurrency is more secure in a decentralized environment where it is difficult to coordinate a deviation. 00 18PDF Image | The Economics of Cryptocurrencies Bitcoin and Beyond
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