  Blockchain systems often employ proof-of-work consensus protocols to validate
and add transactions into hashchains. These protocols stimulate 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 at all the miners.
