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Blockchain systems often employ proof-of-work consensus protocols to validateand add transactions into hashchains. These protocols stimulate competitionamong miners in solving cryptopuzzles (e.g. SHA-256 hash computation inBitcoin) in exchange for a monetary reward. Here, we model mining as an all-payauction, where miners' computational efforts are interpreted as bids, and theallocation function is the probability of solving the cryptopuzzle in a singleattempt with unit (normalized) computational capability. Such an allocationfunction captures how blockchain systems control the difficulty of thecryptopuzzle as a function of miners' computational abilities (bids). In anattempt to reduce mining costs, we investigate designing a mining auctionmechanism which induces a logit equilibrium amongst the miners with choicedistributions that are unilaterally decreasing with costs at each miner. Weshow it is impossible to design a lenient allocation function that does this.Specifically, we show that there exists no allocation function that discouragesminers to bid higher costs at logit equilibrium, if the rate of change ofdifficulty with respect to each miner's cost is bounded by the inverse of thesum of costs at all the miners.

A Difficulty in Controlling Blockchain Mining Costs via Cryptopuzzle Difficulty