In this paper, we provide a new theoretical framework of pyramid Markov processes to solve some open and fundamental problems of blockchain selfish mining under a rigorous mathematical setting. We first describe a more general model of blockchain selfish mining with both a two-block leading competitive criterion and a new economic incentive mechanism. Then we establish a pyramid Markov process and show that it is irreducible and positive recurrent, and its stationary probability vector is matrix-geometric with an explicitly representable rate matrix. Also, we use the stationary probability vector to study the influence of many orphan blocks on the waste of computing resource. Next, we set up a pyramid Markov reward process to investigate the long-run average profits of the honest and dishonest mining pools, respectively. As a by-product, we build three approximative Markov processes and provide some new interesting interpretation on the Markov chain and the revenue analysis reported in the seminal work by Eyal and Sirer (2014). Note that the pyramid Markov (reward) processes can open up a new avenue in the study of blockchain selfish mining. Thus we hope that the methodology and results developed in this paper shed light on the blockchain selfish mining such that a series of promising research can be developed potentially.
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