The Economics of Cryptocurrencies Bitcoin and Beyond

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buyer keeps his original balances and the goods while the seller will be left empty handed. In response, the seller can choose to postpone the delivery of the goods and wait for N confirmations. This confirmation lag can potentially deter double spending by the buyer. The idea is that, to undo a transaction with a confirmation lag of N subperiods, a dishonest buyer needs to win the mining game N + 1 times in a row. As the number of lags increases, the total PoW required to revoke a transfer is increasing, making it more costly for a buyer to double spend. Furthermore, secret mining is deterred by miners’ investments in computing power MQ which, according to Lemma 1, is increasing in the reward R. We look next into the incentives to double spend and call an offer (x,d,N) double spending proof (DS-proof) if the buyer has no incentive to engage in double spending in subperiod 0 after the acceptance of the offer. 3.4 Double Spending Proof Contracts Consider then a trade with the terms (x,d,N). The buyer will receive the goods in subperiod N when exactly N confirmations of the payment d have been observed in the Blockchain. To double spend, a buyer can secretly mine an alternative history to undo his payment after he has received the goods. This implies that the double spender needs to be the first one who solves the PoW problem for N + 1 consecutive subperiods; i.e., from subperiod 0 to subperiod N . For each of the first N subperiods, the buyer does not broadcast the new block immediately, so that some other miners will update the Blockchain and confirm his payment to the seller. The buyer broadcasts his secretly mined blockchain only after he receives the goods and solves the N + 1th block. When the double spending attack succeeds, the original payment is cancelled and the N + 1 rewards will be given to the buyer (see Figure 3.2). Note first that the quantity of good x being exchanged does not directly matter for a double spending attack, as it neither changes the payoffs, nor the incentives to engage in an attack. We consider first subperiod N where the seller delivers his goods. Conditional on having been successful N times with secret mining20, a buyer can now double spend if successful with mining a new block in subperiod N. His expected payoff from investing qN is given by ρ(qN)β[d+(N +1)R]−qN. (9) 20We implicitly assume here that once a buyer starts a double spending attack, he does not stop after wining a block before subperiod N. As shown in the Appendix, this is without loss of generality because, if a buyer decides to double spend, it is not optimal to quit after winning some consecutive blocks before subperiod N. 16

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