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C Appendix [FOR ONLINE PUBLICATION ONLY] – Proof of Stake We assume that the right for updating of the blockchain is allocated randomly across people. The probability that one can update the chain is proportional to the unused balances in the night market. This is identical to pledging balances that are not used in transactions in order to gain a right to update the chain. For simplicity, we assume that people can pledge balances after they learn whether they have a match or not. Denote the balances chosen in the day market for transaction purposes as d and the additional amount of balances chosen for having a stake in the voting procedure as z′. Whenever, z′ > 0, we have that there is a strictly positive probability that there is a double spend, since the buyer in a transaction will propose a block that undoes the payment whenever he is allowed doing so. Fix N. The problem is given by ′ N max−(z+d)+σ δ εu(x)+ d,z′ β′ z′ N+1 z′ ̄ z+ d+ R(N+1) + μ (1−σ)Z (1−σ)Z (1−σ)β z′ +d+ z′ +d R(N ̄ +1) μ (1−σ)Z When having a transaction, a buyer is only allowed to update the chain if z′ > 0. Given N, he needs to win the right to update N + 1 in order to double spend. If a buyer has no transaction, he will pledge all his balances as a stake. Note that participation of the seller requires x= βd(1−τ)1−E(z′)N+1. μZ Notice that the buyer’s choice of z′ is not observed by the seller, so x depends on seller’s expectation E(z′) instead. Lemma C.1. The objective function is strictly convex in z′ for all N ≥ 1 and linear in z′ for N = 0. Proof. Differentiating the objective function, we obtain −i+ 1 (στ +(μ−1))+ σ (N +1) z′ N d (C.1) 1−σ 1−σ (1−σ)Z Z 57PDF Image | The Economics of Cryptocurrencies Bitcoin and Beyond
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