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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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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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  1. Jing-Yu Ma
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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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  • DOI: https://doi.org/10.1007/s10878-022-00910-w

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