Majority Is Not Required: A Rational Analysis of the Private Double-Spend Attack from a Sub-Majority Adversary

We study the incentives behind double-spend attacks on Nakamoto-style Proof-of-Work cryptocurrencies. In these systems, miners are allowed to choose which transactions to reference within their block, and a common strategy for selecting transactions is to greedily choose those with the highest fees. This can be problematic if these transactions originate from an adversary with substantial computational power (even if it is less than 50% of the total), as high-value transactions are targets for double-spend attacks. The most common mechanism for deterring double-spend attacks is for the recipients of large transactions to wait for additional block confirmations beyond the number suggested by the protocol (effectively increasing the attack cost). We argue that this defense mechanism is not satisfactory, as the security of the system is contingent on the actions of its users . Instead, defending against double-spend attacks should be the responsibility of the miners ; to this end, we propose a protocol rule under which miners limit the amount of transaction value in a block (i.e., reduce the attack reward). To demonstrate the efficacy of our proposed rule, we model cryptocurrency mining as a mean-field game in which the standard mining reward function is augmented to simulate the presence of a rational, double-spending adversary. We design and implement an algorithm which characterizes the behavior of miners at equilibrium, and we show that miners who respond to the adversary-aware reward function accumulate more wealth than those who do not. Under this reward function, the optimal strategy for honest miners is to limit the value transferred in each block such that the adversary’s expected profit is 0. Additionally, we examine Bitcoin’s resilience to double-spend attacks under our model. Assuming a six-block confirmation time, we find that a Bitcoin miner with 26% of the network mining power expects to profit from a double-spend attack.