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Optimal dynamic mining policy of blockchain selfish mining through sensitivity-based optimization

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Abstract

The security and fairness of blockchain are always threatened by selfish mining attacks. To study such selfish mining attacks, some necessary and useful methods need to be developed sufficiently. In this paper, we provide an interesting method for analyzing dynamic decision of blockchain selfish mining by applying the sensitivity-based optimization. Our goal is to find the optimal dynamic blockchain-pegged mining policy of the dishonest mining pool. To this end, we consider a blockchain system with two mining pools: the honest and the dishonest mining pools, where the honest mining pool follows a two-block leading competitive criterion, while the dishonest mining pool follows a modification of two-block leading competitive criterion. To find the optimal blockchain-pegged mining policy, we develop the sensitivity-based optimization to study dynamic decision of blockchain system through setting up a policy-based Poisson equation, and provide an expression for the unique solution of performance potentials. Based on this, we can characterize monotonicity and optimality of the long-run average profit with respect to the blockchain-pegged mining reward. Also, we prove the structure of the optimal blockchain-pegged mining policy. The methodology and results derived in this paper significantly reduce the large search space of finding the optimal policy, thus they can shed light on the optimal dynamic decision research on the selfish mining attacks of blockchain systems.

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References

  • Cao XR (2007) Stochastic learning and optimization–a sensitivity-based approach. Springer, New York

    Book  MATH  Google Scholar 

  • Carlsten M, Kalodner HA, Weinberg SM, et al. (2016) On the instability of bitcoin without the block reward. In: ACM SIGSAC conference on computer and communications security. Association for Computing Machinery, New York pp 154–167

  • Chang D, Hasan M, Jain P (2019) Spy based analysis of selfish mining attack on multi-stage blockchain. IACR Cryptol ePrint 2019(1327):1–34

    Google Scholar 

  • Eyal I, Sirer EG (2014) Majority is not enough: bitcoin mining is vulnerable. International conference on financial cryptography and data security. Springer, Berlin pp 436–454

  • Fauziah Z, Latifah H, Omar X et al (2020) Application of blockchain technology in smart contracts: a systematic literature review. Aptisi Trans Technopreneurship 2(2):160–166

    Article  Google Scholar 

  • Feng C, Niu J (2019) Selfish mining in Ethereum. The 39th international conference on distributed computing systems (ICDCS). IEEE, Dallas, pp 1306–1316

    Google Scholar 

  • Gao S, Li Z, Peng Z et al (2019) Power adjusting and bribery racing: novel mining attacks in the bitcoin system. ACM SIGSAC conference on Computer and Communications Security. Association for Computing Machinery, New York, pp 833–850

    Google Scholar 

  • Gervais A, Karame GO, Wüst K et al (2016) On the security and performance of Proof of Work blockchains. ACM SIGSAC conference on computer and communications security. Association for Computing Machinery, New York, pp 3–16

    Google Scholar 

  • Göbel J, Keeler HP, Krzesinski AE et al (2016) Bitcoin blockchain dynamics: the selfish-mine strategy in the presence of propagation delay. Perform Eval 104:23–41

    Article  Google Scholar 

  • Jain P (2019) Revenue generation strategy through selfish mining focusing multiple mining pools. Bachelor Thesis, Computer Science & Applied Mathematics, Indraprastha Institute of Information Technology, India

  • Jaoude JA, Saade RG (2019) Blockchain applications–Usage in different domains. IEEE Access 7:45360–45381

    Article  Google Scholar 

  • Javier K, Fralix B (2020) A further study of some Markovian bitcoin models from Göbel et al. Stoch Models 36(2):223–250

    Article  MathSciNet  MATH  Google Scholar 

  • Li QL (2010) Constructive computation in stochastic models with applications: the RG factorizations. Springer, Heidelberg

    Book  MATH  Google Scholar 

  • Li QL, Ma JY, Chang YX (2018) Blockchain queue theory. International conference on computational social networks. Springer, Cham, pp 25–40

    Google Scholar 

  • Li QL, Ma JY, Chang YX et al (2019) Markov processes in blockchain systems. Comput Soc Netw 6(1):1–28

    Article  Google Scholar 

  • Li QL, Li YM, Ma JY et al (2022) The optimal dynamic rationing policy in the stock-rationing queue. In: The 16th international conference on algorithmic aspects in information and management (AAIM). Springer, Guangzhou, pp 66–82

  • Li QL, Chang YX, Wu X, et al (2021) A new theoretical framework of pyramid Markov processes for blockchain selfish mining. J Syst Sci Syst Eng 1–45

  • Li QL, Chang YX, Zhang C (2022) Tree representation, growth rate of blockchain and reward allocation in Ethereum with multiple mining pools. IEEE Trans Netw Serv 1–19

  • Ma JY, Li QL (2021) Sensitivity-based optimization for blockchain selfish mining. arXiv: 2111.07070

  • Ma JY, Li QL (2021) Sensitivity-based optimization for blockchain selfish mining. The 15th international conference on algorithmic applications in management (AAIM). Springer, Dallas, pp 329–343

    Google Scholar 

  • Ma JY, Xia L, Li QL (2019) Optimal energy-efficient policies for data centers through sensitivity-based optimization. Discrete Event Dyn Syst 29(4):567–606

    Article  MathSciNet  MATH  Google Scholar 

  • Ma JY, Li QL, Xia L (2021) Optimal asynchronous dynamic policies in energy-efficient data centers. Systems 10(2):27

    Article  Google Scholar 

  • Nayak K, Kumar S, Miller A et al (2016) Stubborn mining: generalizing selfish mining and combining with an eclipse attack. IEEE European symposium on security and privacy. IEEE, Saarbruecken, pp 305–320

    Google Scholar 

  • Puterman ML (2014) Markov decision processes: discrete stochastic dynamic programming. Wiley, New York

    MATH  Google Scholar 

  • Sapirshtein A, Sompolinsky Y, Zohar A (2016) Optimal selfish mining strategies in Bitcoin. The 20th international conference on financial cryptography and data security. Springer, Berlin, pp 515–532

    Google Scholar 

  • Sharma P, Jindal R, Borah MD (2020) Blockchain technology for cloud storage: a systematic literature review. ACM Comput Surv 53(4):1–32

    Article  Google Scholar 

  • Wüst K (2016) Security of blockchain technologies. Master Thesis, Department of Computer Science, ETH Zürich, Switzerland

  • Xia L, Zhang ZG, Li QL A (2021) \(c/\mu \)-rule for job assignment in heterogeneous group-server queues. Prod Oper Manag 31(3):1191–1215

  • Xia L, Shihada B (2015) A Jackson network model and threshold policy for joint optimization of energy and delay in multi-hop wireless networks. Eur J Oper Res 242(3):778–787

    Article  MathSciNet  MATH  Google Scholar 

  • Xia L (2020) Risk-sensitive Markov decision processes with combined metrics of mean and variance. Prod Oper Manag 29(12):2808–2827

    Article  Google Scholar 

  • Zur RB, Eyal I, Tamar A (2020) Efficient MDP analysis for selfish-mining in blockchains. The 2nd ACM conference on advances in financial technologies. Association for Computing Machinery, New York, pp 113–131

    Chapter  Google Scholar 

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Acknowledgements

Some parts of this work were presented at the 15th International Conference on Algorithmic Aspects in Information and Management in 2021: Sensitivity-based Optimization for Blockchain Selfish Mining. See Ma and Li (2021a, 2021b). This paper was supported by the National Natural Science Foundation of China under grant No. 71932002 and by the Beijing Social Science Foundation Research Base Project under grant No. 19JDGLA004. In addition, this paper is based on a series of conference talks: AAIM 2021, CSIAM-BTAF 2021 and 2022, author Jing-Yu Ma thanks Professor Ding-Zhu Du at University of Texas, Dallas for recommending this research work enthusiastically, also thanks Professor Jian Li at Beijing University of Technology for the important help in presentation.

Funding

This paper was supported by the National Natural Science Foundation of China under grant No. 71932002 and the Beijing Social Science Foundation Research Base Project under grant No. 19JDGLA004.

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  1. Bussiness School, Xuzhou University of Technology, Xuzhou, 221018, China

    Jing-Yu Ma

  2. School of Economics and Management, Beijing University of Technology, Beijing, 100124, China

    Quan-Lin Li

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Appendices

Appendix A: two special cases

In this appendix, we provide two special cases of the policy-based continuous-time Markov process \(\{\mathbf {X}^{(p)}\left( t\right) :t\ge 0\}\) for \(m=3\) and \(m=4,\) respectively.

Case (a): \(m=3\)

In this case, we obtain the state transition relation in Fig. 3. It is easy to see that there is only one state correspond to Part A-1 in Fig. 2. Thus, the infinitesimal generator of the Markov process is given by

$$\begin{aligned} \mathbf {Q}^{(p)}=\left( \begin{array}{cccccc} Q_{0,0} &{} B_{0} &{} &{} &{} &{} \\ Q_{1,0} &{} Q_{1,1} &{} B_{1} &{} &{} &{} \\ Q_{2,0} &{} &{} Q_{2,2} &{} B_{2} &{} &{} \\ Q_{3,0} &{} &{} &{} Q_{3,3} &{} B_{3} &{} \\ Q_{4,0} &{} &{} &{} &{} Q_{4,4} &{} B_{4}\\ Q_{5,0} &{} &{} &{} &{} &{} Q_{5,5} \end{array} \right) . \end{aligned}$$
Fig. 3
figure 3

The state transition relation for the case of \(m=3\)

Full size image

For Level 0, it is easy to see that

$$\begin{aligned} Q_{0,0}=\left( \begin{array}{cccc} -\left( \lambda _{1}+\lambda _{2}\right) &{} \lambda _{2} &{} &{} \\ 0 &{} -\left( \lambda _{1}+\lambda _{2}\right) &{} \lambda _{2} &{} \\ \mu p &{} &{} -v\left( p\right) &{} \lambda _{2}\\ \mu &{} &{} &{} -\left( \mu +\lambda _{1}\right) \end{array} \right) , B_{0}=\left( \begin{array}{cccc} \lambda _{1} &{} &{} &{} \\ &{} \lambda _{1} &{} &{} \\ &{} &{} \lambda _{1} &{} \\ &{} &{} &{} \lambda _{1} \end{array} \right) ; \end{aligned}$$

for Level 1, we have

$$\begin{aligned} Q_{1,0}= & {} \left( \begin{array}{cccc} &{} &{} &{} \\ &{} &{} &{} \\ &{} &{} &{} \\ \mu &{} &{} &{} \end{array} \right) , B_{1}=\left( \begin{array}{cccc} \lambda _{1} &{} &{} &{} \\ &{} \lambda _{1} &{} &{} \\ &{} &{} \lambda _{1} &{} \\ &{} &{} &{} \lambda _{1} \end{array} \right) , \\ Q_{1,1}= & {} \left( \begin{array}{cccc} -\left( \lambda _{1}+\lambda _{2}\right) &{} \lambda _{2} &{} &{} \\ &{} -\left( \lambda _{1}+\lambda _{2}\right) &{} \lambda _{2} &{} \\ &{} &{} -\left( \lambda _{1}+\lambda _{2}\right) &{} \lambda _{2}\\ &{} &{} &{} -\left( \mu +\lambda _{1}\right) \end{array} \right) ; \end{aligned}$$

for Level 2, we obtain

$$\begin{aligned} Q_{2,0}= & {} \left( \begin{array}{cccc} \mu &{} &{} &{} \\ &{} &{} &{} \\ &{} &{} &{} \\ &{} &{} &{} \end{array} \right) , B_{2}=\left( \begin{array}{ccc} &{} &{} \\ \lambda _{1} &{} &{} \\ &{} \lambda _{1} &{} \\ &{} &{} \lambda _{1} \end{array} \right) , \\ Q_{2,2}= & {} \left( \begin{array}{cccc} -\left( \mu +\lambda _{2}\right) &{} \lambda _{2} &{} &{} \\ &{} -\left( \lambda _{1}+\lambda _{2}\right) &{} \lambda _{2} &{} \\ &{} &{} -\left( \lambda _{1}+\lambda _{2}\right) &{} \lambda _{2}\\ &{} &{} &{} -\lambda _{1} \end{array} \right) ; \end{aligned}$$

for Level 3,

$$\begin{aligned} Q_{3,0}=\left( \begin{array}{cccc} \mu &{} &{} &{} \\ &{} &{} &{} \\ &{} &{} &{} \end{array} \right) ,B_{3}=\left( \begin{array}{cc} &{} \\ \lambda _{1} &{} \\ &{} \lambda _{1} \end{array} \right) , Q_{3,3}=\left( \begin{array}{ccc} -\left( \mu +\lambda _{2}\right) &{} \lambda _{2} &{} \\ &{} -\left( \lambda _{1}+\lambda _{2}\right) &{} \lambda _{2}\\ &{} &{} -\lambda _{1} \end{array} \right) ; \end{aligned}$$

for Level 4, we have

$$\begin{aligned} Q_{4,0}=\left( \begin{array}{cccc} \mu &{} &{} &{} \\ &{} &{} &{} \end{array} \right) ,B_{4}=\left( \begin{array}{c} \\ \lambda _{1} \end{array} \right) , Q_{4,4}=\left( \begin{array}{cc} -\left( \mu +\lambda _{2}\right) &{} \lambda _{2}\\ &{} -\lambda _{1} \end{array} \right) ; \end{aligned}$$

for Level 5, we have

$$\begin{aligned} Q_{5,0}=\left( \begin{array}{llll} \mu&\,&\end{array} \right) , Q_{5,5}=-\mu . \end{aligned}$$

It is easy to see that these matrices with Level 2 to 5 have a similar structure but different dimensions.

Fig. 4
figure 4

The state transition relation for the case of \(m=4\)

Full size image

Case (b): \(m=4\)

In this case, we obtain the state transition relation in Fig. 4. It is seen that there are three states (i.e., a delta-shaped region) correspond to Part A-1 in Fig. 2. The infinitesimal generator of the Markov process is given by

$$\begin{aligned} \mathbf {Q}^{(p)}=\left( \begin{array}{ccccccc} Q_{0,0} &{} B_{0} &{} &{} &{} &{} &{} \\ Q_{1,0} &{} Q_{1,1} &{} B_{1} &{} &{} &{} &{} \\ Q_{2,0} &{} &{} Q_{2,2} &{} B_{2} &{} &{} &{} \\ Q_{3,0} &{} &{} &{} Q_{3,3} &{} B_{3} &{} &{} \\ Q_{4,0} &{} &{} &{} &{} Q_{4,4} &{} B_{4} &{} \\ Q_{5,0} &{} &{} &{} &{} &{} Q_{5,5} &{} B_{5}\\ Q_{6,0} &{} &{} &{} &{} &{} &{} Q_{6,6} \end{array} \right) . \end{aligned}$$

For Level 0, we have

$$\begin{aligned} Q_{0,0}=\left( \begin{array}{ccccc} -\left( \lambda _{1}+\lambda _{2}\right) &{} \lambda _{2} &{} &{} &{} \\ 0 &{} -\left( \lambda _{1}+\lambda _{2}\right) &{} \lambda _{2} &{} &{} \\ \mu p &{} &{} -v\left( p\right) &{} \lambda _{2} &{} \\ \mu p &{} &{} &{} -v\left( p\right) &{} \lambda _{2}\\ \mu &{} &{} &{} &{} -\left( \mu +\lambda _{1}\right) \end{array} \right) , B_{0}=\left( \begin{array}{ccccc} \lambda _{1} &{} &{} &{} &{} \\ &{} \lambda _{1} &{} &{} &{} \\ &{} &{} \lambda _{1} &{} &{} \\ &{} &{} &{} \lambda _{1} &{} \\ &{} &{} &{} &{} \lambda _{1} \end{array} \right) ; \end{aligned}$$

for Level 1, we have

$$\begin{aligned} Q_{1,0}= & {} \left( \begin{array}{ccccc} &{} &{} &{} &{} \\ &{} &{} &{} &{} \\ &{} &{} &{} &{} \\ \mu p &{} &{} &{} &{} \\ \mu &{} &{} &{} &{} \end{array} \right) , B_{1}=\left( \begin{array}{ccccc} \lambda _{1} &{} &{} &{} &{} \\ &{} \lambda _{1} &{} &{} &{} \\ &{} &{} \lambda _{1} &{} &{} \\ &{} &{} &{} \lambda _{1} &{} \\ &{} &{} &{} &{} \lambda _{1} \end{array} \right) , \\ Q_{1,1}= & {} \left( \begin{array}{ccccc} -\left( \lambda _{1}+\lambda _{2}\right) &{} \lambda _{2} &{} &{} &{} \\ &{} -\left( \lambda _{1}+\lambda _{2}\right) &{} \lambda _{2} &{} &{} \\ &{} &{} -\left( \lambda _{1}+\lambda _{2}\right) &{} \lambda _{2} &{} \\ &{} &{} &{} -v\left( p\right) &{} \lambda _{2}\\ &{} &{} &{} &{} -\left( \mu +\lambda _{1}\right) \end{array} \right) ; \end{aligned}$$

for Level 2, we have

$$\begin{aligned} Q_{2,0}= & {} \left( \begin{array}{ccccc} \mu &{} &{} &{} &{} \\ &{} &{} &{} &{} \\ &{} &{} &{} &{} \\ &{} &{} &{} &{} \\ \mu &{} &{} &{} &{} \end{array} \right) , B_{2}=\left( \begin{array}{cccc} &{} &{} &{} \\ \lambda _{1} &{} &{} &{} \\ &{} \lambda _{1} &{} &{} \\ &{} &{} \lambda _{1} &{} \\ &{} &{} &{} \lambda _{1} \end{array} \right) , \\ Q_{2,2}= & {} \left( \begin{array}{ccccc} -\left( \mu +\lambda _{2}\right) &{} \lambda _{2} &{} &{} &{} \\ &{} -\left( \lambda _{1}+\lambda _{2}\right) &{} \lambda _{2} &{} &{} \\ &{} &{} -\left( \lambda _{1}+\lambda _{2}\right) &{} \lambda _{2} &{} \\ &{} &{} &{} -\left( \lambda _{1}+\lambda _{2}\right) &{} \lambda _{2}\\ \mu &{} &{} &{} &{} -\left( \mu +\lambda _{1}\right) \end{array} \right) ; \end{aligned}$$

for Level 3, we have

$$\begin{aligned} Q_{3,0}= & {} \left( \begin{array}{ccccc} \mu &{} &{} &{} &{} \\ &{} &{} &{} &{} \\ &{} &{} &{} &{} \\ &{} &{} &{} &{} \end{array} \right) ,B_{3}=\left( \begin{array}{ccc} &{} &{} \\ \lambda _{1} &{} &{} \\ &{} \lambda _{1} &{} \\ &{} &{} \lambda _{1} \end{array} \right) , \\ Q_{3,3}= & {} \left( \begin{array}{cccc} -\left( \mu +\lambda _{2}\right) &{} \lambda _{2} &{} &{} \\ &{} -\left( \lambda _{1}+\lambda _{2}\right) &{} \lambda _{2} &{} \\ &{} &{} -\left( \lambda _{1}+\lambda _{2}\right) &{} \lambda _{2}\\ &{} &{} &{} -\lambda _{1} \end{array} \right) ; \end{aligned}$$

for Level 4, we have

$$\begin{aligned} Q_{4,0}=\left( \begin{array}{ccccc} \mu &{} &{} &{} &{} \\ &{} &{} &{} &{} \\ &{} &{} &{} &{} \end{array} \right) ,B_{4}=\left( \begin{array}{cc} &{} \\ \lambda _{1} &{} \\ &{} \lambda _{1} \end{array} \right) , Q_{4,4}=\left( \begin{array}{ccc} -\left( \mu +\lambda _{2}\right) &{} \lambda _{2} &{} \\ &{} -\left( \lambda _{1}+\lambda _{2}\right) &{} \lambda _{2}\\ &{} &{} -\lambda _{1} \end{array} \right) ; \end{aligned}$$

for Level 5, we have

$$\begin{aligned} Q_{5,0}=\left( \begin{array}{ccccc} \mu &{} &{} &{} &{} \\ &{} &{} &{} &{} \end{array} \right) ,B_{5}=\left( \begin{array}{c} \\ \lambda _{1} \end{array} \right) , Q_{5,5}=\left( \begin{array}{cc} -\left( \mu +\lambda _{2}\right) &{} \lambda _{2}\\ &{} -\lambda _{1} \end{array} \right) ; \end{aligned}$$

for Level 6, we have

$$\begin{aligned} Q_{6,0}=\left( \begin{array}{lllll} \mu&\,&\,&\end{array} \right) , Q_{6,6}=-\mu . \end{aligned}$$

Appendix B: two different optimal policies

In this appendix, we provide the state transition relation of the policy-based continuous-time Markov process \(\{\mathbf {X}^{(p)}\left( t\right) :t\ge 0\}\) for the optimal blockchain-pegged mining policy either \(p^{*}=1\) or \(p^{*}=0\).

Case (a): \(p^{*}=1\)

In this case, compared with Fig. 2, we easily check that the state transition diagram has changed as Fig. 5.

Fig. 5
figure 5

The state transition relation for \(p^{*}=1\)

Full size image

Thus the infinitesimal generator of the Markov process is given by

$$\begin{aligned} \mathbf {Q}^{(p)}=\left( \begin{array}{cccccc} Q_{0,0} &{} B_{0} &{} &{} &{} &{} \\ Q_{1,0} &{} Q_{1,1} &{} B_{1} &{} &{} &{} \\ Q_{2,0} &{} &{} Q_{2,2} &{} B_{2} &{} &{} \\ \vdots &{} &{} &{} \ddots &{} \ddots &{} \\ Q_{m+1,0} &{} &{} &{} &{} Q_{m+1,m+1} &{} B_{m+1}\\ Q_{m+2,0} &{} &{} &{} &{} &{} Q_{m+2,m+2} \end{array} \right) . \end{aligned}$$

For Level 0,

$$\begin{aligned} Q_{0,0}=\left( \begin{array}{ccc} -\left( \lambda _{1}+\lambda _{2}\right) &{} \lambda _{2} &{} \\ &{} -\left( \lambda _{1}+\lambda _{2}\right) &{} \lambda _{2}\\ \mu &{} &{} -\left( \lambda _{1}+\mu \right) \end{array} \right) , \ B_{0}=\left( \begin{array}{cccc} \lambda _{1} &{} 0 &{} &{} \\ &{} \lambda _{1} &{} 0 &{} \\ &{} &{} \lambda _{1} &{} 0 \end{array} \right) ; \end{aligned}$$

for Level 1,

$$\begin{aligned} Q_{1,0}= & {} \left( \begin{array}{ccc} &{} &{} \\ &{} &{} \\ &{} &{} \\ \mu &{} &{} \end{array} \right) ,\ \ B_{1}=\left( \begin{array}{ccccc} \lambda _{1} &{} 0 &{} &{} &{} \\ &{} \lambda _{1} &{} 0 &{} &{} \\ &{} &{} \lambda _{1} &{} 0 &{} \\ &{} &{} &{} \lambda _{1} &{} 0 \end{array} \right) , \\ Q_{1,1}= & {} \left( \begin{array}{cccc} -\left( \lambda _{1}+\lambda _{2}\right) &{} \lambda _{2} &{} &{} \\ &{} -\left( \lambda _{1}+\lambda _{2}\right) &{} \lambda _{2} &{} \\ &{} &{} -\left( \lambda _{1}+\lambda _{2}\right) &{} \lambda _{2}\\ &{} &{} &{} -\left( \mu +\lambda _{1}\right) \end{array} \right) ; \end{aligned}$$

for Level \(k=2,3,\ldots ,m-3,\)

$$\begin{aligned} Q_{k,0}= & {} \left( \begin{array}{cccc} \mu &{} &{} &{} \\ 0 &{} &{} &{} \\ 0 &{} &{} &{} \\ 0 &{} &{} &{} \\ \mu &{} &{} &{} \end{array} \right) , B_{k}=\left( \begin{array}{ccccc} 0 &{} &{} &{} &{} \\ \lambda _{1} &{} 0 &{} &{} &{} \\ &{} \lambda _{1} &{} 0 &{} &{} \\ &{} &{} \lambda _{1} &{} 0 &{} \\ &{} &{} &{} \lambda _{1} &{} 0 \end{array} \right) , \\ Q_{k,k}= & {} \left( \begin{array}{ccccc} -\left( \mu +\lambda _{2}\right) &{} \lambda _{2} &{} &{} &{} \\ &{} -\left( \lambda _{1}+\lambda _{2}\right) &{} \lambda _{2} &{} &{} \\ &{} &{} -\left( \lambda _{1}+\lambda _{2}\right) &{} \lambda _{2} &{} \\ &{} &{} &{} -\left( \lambda _{1}+\lambda _{2}\right) &{} \lambda _{2}\\ &{} &{} &{} &{} -\left( \lambda _{1}+\mu \right) \end{array} \right) ; \end{aligned}$$

for Level \(m-2\),

$$\begin{aligned} Q_{m-2,0}= & {} \left( \begin{array}{cccc} \mu &{} &{} &{} \\ 0 &{} &{} &{} \\ 0 &{} &{} &{} \\ 0 &{} &{} &{} \\ \mu &{} &{} &{} \end{array} \right) , B_{m-2}=\left( \begin{array}{cccc} &{} &{} &{} \\ \lambda _{1} &{} &{} &{} \\ &{} \lambda _{1} &{} &{} \\ &{} &{} \lambda _{1} &{} \\ &{} &{} &{} \lambda _{1} \end{array} \right) , \\ Q_{m-2,m-2}= & {} \left( \begin{array}{ccccc} -\left( \mu +\lambda _{2}\right) &{} \lambda _{2} &{} &{} &{} \\ &{} -\left( \lambda _{1}+\lambda _{2}\right) &{} \lambda _{2} &{} &{} \\ &{} &{} -\left( \lambda _{1}+\lambda _{2}\right) &{} \lambda _{2} &{} \\ &{} &{} &{} -\left( \lambda _{1}+\lambda _{2}\right) &{} \lambda _{2}\\ &{} &{} &{} &{} -\left( \lambda _{1}+\mu \right) \end{array} \right) ; \end{aligned}$$

for Level \(m-1\),

$$\begin{aligned} Q_{m-1,0}= & {} \left( \begin{array}{cccc} \mu &{} &{} &{} \\ &{} &{} &{} \\ &{} &{} &{} \\ &{} &{} &{} \end{array} \right) ,B_{m-1}=\left( \begin{array}{ccc} &{} &{} \\ \lambda _{1} &{} &{} \\ &{} \lambda _{1} &{} \\ &{} &{} \lambda _{1} \end{array} \right) , \\ Q_{m-1,m-1}= & {} \left( \begin{array}{cccc} -\left( \mu +\lambda _{2}\right) &{} \lambda _{2} &{} &{} \\ &{} -\left( \lambda _{1}+\lambda _{2}\right) &{} \lambda _{2} &{} \\ &{} &{} -\left( \lambda _{1}+\lambda _{2}\right) &{} \lambda _{2}\\ &{} &{} &{} -\lambda _{1} \end{array} \right) ; \end{aligned}$$
Fig. 6
figure 6

The state transition relation for \(p^{*}=0\)

Full size image

for Level m,

$$\begin{aligned} Q_{m,0}=\left( \begin{array}{cccc} \mu &{} &{} &{} \\ &{} &{} &{} \\ &{} &{} &{} \end{array} \right) ,B_{m}=\left( \begin{array}{cc} &{} \\ \lambda _{1} &{} \\ &{} \lambda _{1} \end{array} \right) , \ Q_{m,m}=\left( \begin{array}{ccc} -\left( \mu +\lambda _{2}\right) &{} \lambda _{2} &{} \\ &{} -\left( \lambda _{1}+\lambda _{2}\right) &{} \lambda _{2}\\ &{} &{} -\lambda _{1} \end{array} \right) ; \end{aligned}$$

for Level \(m+1\),

$$\begin{aligned} Q_{m+1,0}=\left( \begin{array}{ccc} \mu &{} &{} \\ &{} &{} \end{array} \right) ,B_{m+1}=\left( \begin{array}{c} \\ \lambda _{1} \end{array} \right) , Q_{m+1,m+1}=\left( \begin{array}{cc} -\left( \mu +\lambda _{2}\right) &{} \\ &{} -\lambda _{1} \end{array} \right) ; \end{aligned}$$

for Level \(m+2\),

$$\begin{aligned} Q_{m+2,0}=\left( \begin{array}{ccc} \mu&\,&\end{array} \right) , Q_{m+2,m+2}=-\mu . \end{aligned}$$

Case (b): \(p^{*}=0\)

For the case of \(p^{*}=0\), we obtain the the state transition relation, see Fig. 6. Compared with Fig. 2, the shape of such state transition diagram is not changed. The infinitesimal generator of the Markov process is omitted here.

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Ma, JY., Li, QL. Optimal dynamic mining policy of blockchain selfish mining through sensitivity-based optimization. J Comb Optim 44, 3663–3700 (2022). https://doi.org/10.1007/s10878-022-00910-w

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