Despite the popularity and practical applicability of blockchains, there is very limited work on the theoretical foundation of blockchains: The lack of rigorous theory and analysis behind the curtain of blockchains has severely staggered its broader applications. This paper attempts to lay out a theoretical foundation for a specific type of blockchains---the ones requiring basic authenticity from the participants, also called \textit{permissioned blockchain}. We formulate permissioned blockchain systems and operations into a game-theoretical problem by incorporating constraints implied by the wisdom from distributed computing and Byzantine systems. We show that in a noncooperative blockchain game (NBG), a Nash equilibrium can be efficiently found in a closed-form even though the game involves more than two players. Somewhat surprisingly, the simulation results of the Nash equilibrium implies that the game can reach a stable status regardless of the number of Byzantine nodes and trustworthy players. We then study a harder problem where players are allowed to form coalitions: the coalitional blockchain game (CBG). We show that although the Shapley value for a CBG can be expressed in a more succinct form, its core is empty.

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重新审视许可区块链：拜占庭博弈论视角

尽管区块链的普及和实际适用性，但在区块链的理论基础上的工作非常有限：区块链幕后缺乏严谨的理论和分析，严重阻碍了其更广泛的应用。本文试图为特定类型的区块链奠定理论基础——要求参与者具有基本真实性的区块链，也称为\textit{许可区块链}。我们通过结合分布式计算和拜占庭系统的智慧所隐含的约束，将许可的区块链系统和操作公式化为一个博弈论问题。我们表明，在非合作区块链游戏 (NBG) 中，即使游戏涉及两个以上的玩家，也可以在封闭形式中有效地找到纳什均衡。有点意外，纳什均衡的模拟结果意味着无论拜占庭节点和可信赖玩家的数量如何，游戏都可以达到稳定状态。然后我们研究了一个更难的问题，允许玩家组成联盟：联盟区块链游戏（CBG）。我们表明，虽然 CBG 的 Shapley 值可以用更简洁的形式表达，但其核心是空的。