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With probability P (d, N ) a double spending attack is successful leaving the seller without balances from the trade. Furthermore, the seller is required to pay the transaction fee which is a fraction τ of the payment in real balances d. If balances can be used for trading in the night market, we have that in the day market buyers will have a positive demand for balances (z′ > 0). Furthermore, since μ ≥ 1 > β, it is costly to carry extra balances into the night market. This implies that sellers do not carry any balances in the night market, while buyers carry only the amount of balances into the market they will use to make an offer; i.e., the cash-in-advance constraint is binding or z′ = d. Hence, the buyer’s optimal take-it-or-leave-it offer is thus a solution to the problem max −d+(1−σ)βd+σ[δNεu(x)+D0(d,N)] (31) (x,d,N ) μ subject to x= βμd(1−τ)(1−P(d,N)) (33) The values of D0(d,N) and P(d,N) depend on whether the offer is DS-proof or not. The mining game and the buyer’s incentives to engage in double spending are identical to Section 3 except for that discounting of payoffs across periods is given by β/μ < β. Hence, we have that P(d,N) = D0(d,N) = 0 ifd≤R(N+1)N –inotherwords,ifthecontractisDSproof–or P (d, N ) = √ ∆ − (N + 1) β 2 > 0 √ √∆ > 0 and D0 (d, N ) = μ R ∆ − (N + 1) (32) if the contract involves double spending. Note that double spending gives the buyer an additional payoff D0(d,N), but also tightens the seller’s participation constraint thereby reducing the amount being transacted in the night market. The buyer’s problem in general can have multiple solutions. For example, sometimes a buyer can be indifferent between double spendig and a DS-proof contracts. Similarly, the buyer can be indifferent between a contract with a long confirmation lag and large consumption and one with earlier but smaller consumption. For completeness, the seller’s value function is given by since they do not receive any surplus. v(z;0)=βw(μz)= βμz. (34) 24PDF Image | The Economics of Cryptocurrencies Bitcoin and Beyond
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