自2008年比特币被提出以来，区块链技术和加密货币一直备受各界的关注。共识算法是区块链技术的核心，用以驱使网络中的各节点对链上数据持续地达成一致。工作量证明是最为广泛使用的区块链共识算法，其创新性地将经济激励和博弈等概念引入分布式系统共识。节点投入算力以换取贡献区块链数据的机会，而正确数据的诚实贡献者可以获得数字代币作为奖励，投入算力越高，获得的数字代币奖励金额也越高。长久以来，人们相信工作量证明共识是激励相容的：诚实节点都应该尽可能地投入全部算力，并专注于一种加密货币，以便最大化自己能获得的奖励金额，同时这样的行为也降低了恶意节点的算力占比，维护了区块链系统的安全。然而本文使用博弈论方法对工作量证明共识进行了更为细致的刻画，否定了这一信念。我们使用成比例分配博弈对工作量证明共识中节点的竞争进行了建模。博弈考虑了任意多个节点。节点们具备不同的算力成本和最大算力配额，他们可以将算力投入到不止一种数字代币之中。我们首先分析了该博弈的纯策略纳什均衡。结果表明，在纯策略纳什均衡中，最佳的策略并不是投入所有的算力配额，也不是只关注一种数字代币。算力配额较大而算力成本较低的节点会抑制一部分算力配额。如果其算力成本相同，则这些节点投入的算力大小是相同的。此外，节点们应该将算力投入每一种数字代币，投入的算力大小与代币的奖励金额成正比。我们给出了纯策略纳什均衡的解析形式并且证明了其唯一性。我们之后分析了该博弈的斯塔克尔伯格均衡，并同样给出了其解析形式。结果显示，与纯策略纳什均衡相似，在斯塔克尔伯格均衡中，节点也不应投入全部算力。而由于斯塔克尔伯格领导节点的加入，网络中的总算力有所增加。接着，通过多组仿真实验，我们展示了区块链系统的安全性和能耗在均衡解中的表现：如果节点的算力配额分布得越不均匀，那么网络中诚实节点的算力投入就越低，从而会进一步削弱区块链系统的安全性，与此同时，区块链系统的能源消耗也会随之降低。最后，我们收集了现实中多种主要代币的历史网络算力和奖励数据，并与我们的理论进行了比对。结果表明二者之间存在着高度的契合性。这进一步验证了本文所提出模型和所得到结论的实际意义。
Since the invention of Bitcoin in 2008, blockchain technology and cryptocurrencies have attracted wide attention. As the core of blockchain technology, consensus is aimed to drive all nodes in the network to reach agreement on some data constantly. Proof-of-work is the most commonly used consensus of blockchain, which is innovative in that it incorporates game theory into the distributed system consensus. Nodes invest computational power in exchange to commit blockchain data, and honest committers are rewarded with digital tokens. Intuitively, nodes investing more power can acquire more reward. There is a long-held belief in the incentive compatibility of proof-of-work: To maximize the reward, honest nodes should invest all of the computational power in one token. Such behavior works to reduce the ratio of malicious computational power and thus guards the blockchain.However, in this thesis, we study the consensus with a more delicate and precise model and disprove this theory. We formalize the competition among nodes using the proportional allocation game, where any number of nodes with different costs and capacities distribute their computational power into multiple cryptocurrency tokens. We first analyze the pure Nash-equilibrium, which states that the optimal strategy is neither to invest full power nor to focus on a single token. Instead, nodes with high capacities and low costs should withhold some computational power, while they invest the same amount of power if their costs are the same. In addition, nodes should invest in all tokens, and the amount of computational power is proportional to the value of reward. We derive the closed-form of the pure Nash-equilibrium and prove its existence and uniqueness. Then we study the Stackelberg-equilibrium and also show its closed-form. Much like the pure Nash-equilibrium, nodes in the Stackelberg-equilibrium do not invest full power, either. Whereas due to the Stackelberg leader, the network computational power increases. Through simulation, we showcase the security and energy consumption of blockchain in the equilibrium. Our results indicate that with a higher disparity of nodes’ capacities, the honest computational power is lower, which, while serving to lower energy consumption, also undermines security. Finally, we collect the practical data of major tokens and find that they match our theory well, further showing the practicality of our models and results.
