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Differentiating again, we obtain σ (N + 1)N z′ N −1 d > 0 (C.2) or 0 if N = 0. We are only looking for a sufficient condition so that we can support the policy N = 0, μ = 1 and τ = 0. Interestingly, for N = 0, we only need to check the slope at z′ = 0. This case implies settling of the trade on the spot without delay. The idea is that the buyer has zero probability of updating the chain and, hence, can never double spend.34 Importantly, setting μ = 1 and τ = 0 implies that people have nothing at stake when updating the chain. The do not receive a reward and there are no penalties. In turn, setting positive rewards increase the incentives for traders to acquire more balances. Proposition C.2. Set N = 0, μ = 1 and τ = 0. Then the PoS protocol avoids double-spending for B sufficiently large and σ sufficiently small. Proof. For the policy parameters, (C.1) implies that the objective function is decreasing in z′ if σ d≤i=1−1 1−σZβ which is satisfied for B sufficiently large and σ sufficiently small since d is bounded. Note that, even though a system based on PoS can avoid the double spending attacks studied above, it may still be subject to “long range attacks”. For example, if all coins are owned initially by one issuer, then he can easily propose a long fork starting from the initial block to claim all existing balances in any future dates. By assuming that the blockchain cannot be rewritten after the end of the night period, our model has ruled out these types of long-range attacks. This captures the feature of checkpoints introduced in Bitcoin. Interestingly, these are exactly the concerns and proposals raised by practitioners. For example, BitFury Group (2015) pointed out that “In PoW-type systems, this [long range] attack is prevented 34We might need to look at N = 1 since the buyer can still prodcast a different transaction so that the seller at least needs to see one transaction in the chain to hand over the good. This will, however, not influence the analysis, since the condition for z′ = 0 being optimal is identical except for that we do neet need to check the slope, but the value of the function at z′ = (1 − σ)Z. Or we could require a negative derivative at z′ = (1 − σ)Z. 1−σ (1−σ)Z Z 58PDF Image | The Economics of Cryptocurrencies Bitcoin and Beyond
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