A predicament in securing blockchain consensus via controlling cryptopuzzle difficulty

Blockchain systems often employ proof-of-work consensus protocols to validate and add transactions into hashchains. These protocols involve competition among miners in solving cryptopuzzles (e.g. SHA-256 hash computation in Bitcoin) in exchange for a monetary reward. Here, we model mining as an all-pay auction, where miners' computational efforts are interpreted as bids, and the allocation function is the probability of solving the cryptopuzzle in a single attempt with unit (normalized) computational capability. Such an allocation function captures how blockchain systems control the difficulty of the cryptopuzzle as a function of miners' computational abilities (bids). In an attempt to reduce mining costs, we investigate designing a mining auction mechanism which induces a logit equilibrium amongst the miners with choice distributions that are unilaterally decreasing with costs at each miner. We show it is impossible to design a lenient allocation function that does this. Specifically, we show that there exists no allocation function that discourages miners to bid higher costs at logit equilibrium, if the rate of change of difficulty with respect to each miner's cost is bounded by the inverse of the sum of costs of all the miners. Additionally, we also show that it is necessary to have allocation functions that decrease with increasing number of players. As a result, it is difficult to achieve decentralization and accomplish secure blockchain systems with global block difficulty which relies only on total hash rate.