  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.
