# Profitable Double-Spending Attacks
## 6 Related Works
The fact is that the computing power invested by the attacker cannot be monitored by the honest network and thus it cannot be reflected in the difficulty control routine. [PERSON] [15] conducted simulations on the profitability of DS attacks only in the cases of \(p_{A}>0.5\). Under the cases, a condition on the value of the target transaction that makes DS attacks not profitable has been given based on the simulations. We give theoretical and numerically-calculable results for any \(p_{A}\in(0,1)\), which do not require massive simulations. [PERSON] et al. [16] and [PERSON] et al. [12] have used a Markov decision process (MDP) to analyze profits from DS attacks. These works differ from our contributions in the following regards:  
First, they did not follow the DS attacks scenario considered by [PERSON] [1] and [PERSON] [6]. Instead, the scenario in [12] was a special case of the pre-mining strategy which was introduced in [17; 18]. We show that the success of DS attack under this scenario is even more difficult to occur than the success of the DS attack under the scenario of [PERSON] and [PERSON] (see Appendix D for details). Also, the attack scenario in [16] went even further by modifying the condition for block confirmation in Definition 1. Specifically, under our definition, it is required for the honest chain to have added \(N_{BC}\) blocks, while under their condition it was fraudulent for the chain to do so (see Section 3 of [16]). Thus, it was not ensured that the potential victim has shipped the goods or service, and an attack success did not guarantee for the attacker to obtain the benefit of attacking. Second, one important new advance in this paper is the derivation of the time analysis \(f_{T_{AS}}\) given in Proposition 1. When one uses the MDP framework, one can obtain similar information such as the estimations for the attack success time \(E_{T_{AS}}\), the future profit \(P\) that an attacker will earn in the end, and the minimum value of target transaction \(C_{\text{Req.}}\). However, using MDP to make such estimations would have required extensive Monte Carlo simulations. Using our mathematical results, such estimations can be obtained without Monte Carlo simulations. In addition, we believe that our mathematical results can be utilized in the MDP frameworks to improve the reliability of analyses. Conventionally, a rational user of an MDP will make a decision at every state whether to stop or to continue the process by comparing the rewards that will be incurred in the future by his/her decision. The rewards for stop actions are clear because such actions are either an attack success or a give-up. The reward for the continue action is complex because it needs to consider all the actions in all future possible states as well. In [12; 16], the rewards for the continue action were over-simplified as they were evaluated only for the very next state and did not include the estimation of the profits in further future actions. To improve the reliability, the PDF \(f_{T_{AS}}\) in Proposition 1 can be used at any intermediate Markov state to estimate the future profits. Specifically, the conditional expectation of the time left for an attack success \(T_{AS}\) given \(T_{AS}>\tau\) can be calculated using \(f_{T_{AS}}\), where \(\tau\) is the observable time elapsed for reaching the current state. Once the time left is estimated, the estimation of future profits can be updated by substituting it into Equation (33). That is to say, at each state, the estimation of profits can be updated and used as the rewards resulting from the continue action. [PERSON] [19] and [PERSON] et al. [20] have derived the PDFs of attack success time, but none of their DS attack scenarios matched with ours in Definition 1. In [19], [PERSON] derived the PDF of catch-up time spent for the fraudulent chain to catch up with the honest chain given that the length of honest chain is initially ahead by several blocks.