We analyze Bitcoin mining from the perspective of a game and propose an optimal mining model that maximizes profits of pools and miners. The model is a two-stage Stackelberg game in which each stage forms a sub-game. In stage I, pools are the leaders who assign a computing power to be consumed by miners. In stage II, miners decide of their power consumption and distribution. They find themselves in a social dilemma in which they must choose between mining in solo, therefore prioritizing their individual preferences, and participating in a pool for the collective interest. The model relies on a pool protocol based on a simulated game in which the miners compete for the reward won by the pool. The solutions for the stage I sub-game and the simulated protocol game are unique and stable Nash equilibriums while the stage II sub-game leads to a stable cooperative equilibrium only when miners choose their strategies according to certain criteria. We conclude that the cooperative optimal mining model has the potential to favor Bitcoin decentralization and stability. Mainly, the social dilemma faced by miners together with the balance of incentives ensure a certain distribution of the network computing power between pools and solo miners, while equilibriums in the game solutions provide stability to the system.

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比特币的合作优化挖掘模型

我们从博弈的角度分析比特币挖矿，提出矿池和矿工利润最大化的最优挖矿模型。该模型是一个两阶段的 Stackelberg 博弈，其中每个阶段形成一个子博弈。在第一阶段，矿池是指派矿工消耗的计算能力的领导者。在第二阶段，矿工决定他们的电力消耗和分配。他们发现自己处于一种社会困境，他们必须在单独挖矿（因此优先考虑他们的个人偏好）和为了集体利益而参与矿池之间做出选择。该模型依赖于基于模拟游戏的矿池协议，在该游戏中，矿工争夺矿池赢得的奖励。第一阶段子博弈和模拟协议博弈的解是唯一且稳定的纳什均衡，而第二阶段子博弈只有在矿工根据一定的标准选择策略时才会导致稳定的合作均衡。我们得出结论，合作最优挖掘模型有可能有利于比特币的去中心化和稳定性。主要是矿工面临的社会困境以及激励的平衡保证了矿池和单独矿工之间的网络算力的一定分布，而博弈解决方案中的平衡则为系统提供了稳定性。