In this paper, we provide a new theoretical framework of pyramid Markov processes to solve some open and fundamental problems of blockchain selfish mining. To this end, we first describe a more general blockchain selfish mining with both a two-block leading competitive criterion and a new economic incentive, and establish a pyramid Markov process to express the dynamic behavior of the selfish mining from both consensus protocol and economic incentive. Then we show that the pyramid Markov process is stable and so is the blockchain, and its stationary probability vector is matrix-geometric with an explicitly representable rate matrix. Furthermore, we use the stationary probability vector to be able to analyze the waste of computational resource due to generating a lot of orphan (or stale) blocks. Nextly, we set up a pyramid Markov reward process to investigate the long-run average profits of the honest and dishonest mining pools, respectively. Specifically, we show that the long-run average profits are multivariate linear such that we can measure the improvement of mining efficiency of the dishonest mining pool comparing to the honest mining pool. As a by-product, we build three approximative Markov processes when the system states are described as the block-number difference of two forked block branches. Also, by using their special cases with non network latency, we can further provide some useful interpretation for both the Markov chain (Figure 1) and the revenue analysis ((1) to (3)) of the seminal work by Eyal and Sirer (2014). Finally, we use some numerical examples to verify the correctness and computability of our theoretical results. 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 produced potentially.

中文翻译：

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用于区块链自私挖矿的金字塔马尔可夫过程的新理论框架

在本文中，我们提供了一个新的金字塔马尔可夫过程理论框架来解决区块链自私挖矿的一些开放性和基础性问题。为此，我们首先描述了一个更通用的区块链自私挖矿，同时具有双区块领先竞争标准和新的经济激励，并建立金字塔马尔可夫过程从共识协议和经济激励两个方面来表达自私挖矿的动态行为. 然后我们证明金字塔马尔可夫过程是稳定的，区块链也是稳定的，它的平稳概率向量是矩阵几何的，具有明确可表示的速率矩阵。此外，我们使用平稳概率向量来分析由于生成大量孤立（或陈旧）块而导致的计算资源浪费。接下来，我们建立了一个金字塔马尔可夫奖励过程来分别调查诚实和不诚实矿池的长期平均利润。具体来说，我们表明长期平均利润是多元线性的，因此我们可以衡量与诚实矿池相比不诚实矿池的挖矿效率的提高。作为副产品，当系统状态被描述为两个分叉块分支的块数差异时，我们构建了三个近似马尔可夫过程。此外，通过使用他们的非网络延迟的特殊情况，我们可以进一步为 Eyal 和 Sirer 开创性工作的马尔可夫链（图 1）和收入分析（（1）至（3））提供一些有用的解释（ 2014）。最后，我们通过一些数值例子来验证我们的理论结果的正确性和可计算性。