Urns Filled with Bitcoins: New Perspectives on Proof-of-Work Mining

Abstract

The probability of a miner finding a valid block in the bitcoin blockchain is assumed to follow the Poisson distribution. However, simple, descriptive, statistical analysis reveals that blocks requiring a lot of time to find-long blocks-are won only by miners with a relatively higher hash power per second. This suggests that relatively bigger miners might have an advantage with regard to winning long blocks, which can be understood as a sort of "within block learning". Modelling the bitcoin mining problem as a race, and by means of a multinomial logit model, we can reject that the time spent mining a particular block does not affect the probability of a miner finding a valid version of this block in a manner that is proportional to her size. Further, we postulate that the probability of a miner finding a valid block is governed by the negative hypergeometric distribution. This would explain the descriptive statistics that emerge from the data and be aligned with the technical aspects of bitcoin mining. We draw an analogy between bitcoin mining and the classical "urn problem" in statistics to sustain our theory. This result can have important consequences for the miners of proof-of-work cryptocurrencies in general, and for the bitcoin mining community in particular.

The economic dependency of bitcoin security

“…Our estimates suggest that negative network externality dominates positive network externality. Our results are in line with those of Parra-Moyano, Reich, and Schmedders (2019), who found that the probability of winning a mining contest increases with the miner size. This motivates miners to join mining pools to increase their probability of winning the mining contest and receiving the reward.…”

“…Cocco and Marchesi 2016; Thum 2018; Schilling and Uhlig 2019). However, Parra-Moyano, Reich and Schmedders (2019) show that the probability of relatively bigger miners winning the mining contest is higher than that of relatively smaller miners because there is a "learning" effect when mining a particular block with larger mining computers learning faster than smaller mining computers. To account for the learning by mining, we assume the following transformation of the probability for a miner winning a block: ), where 𝛾 is a transformation parameter (with 0 < 𝛾 ≤ 0), which implies that the ratio of odds between big and small miners (mining computers) of winning a block increases with the miners' size, m it , while keeping the ratio of miner' size between miners fixed.…”

“…Cocco and Marchesi 2016;Thum 2018;Schilling andUhlig 2019). However, Parra-Moyano, Reich, andSchmedders (2019) show that the probability of relatively bigger miners winning the mining contest is higher than that of relatively smaller miners because there is a 'learning' effect when mining a particular block with larger mining computers learning faster than smaller mining computers. To account for the learning by mining, we assume the following transformation of the probability for a miner winning a block:…”

Confirmed or Dropped? Reliability Analysis of Transactions in PoW Blockchains

“…PoW is a memoryless process, i.e., for each miner the probability of solving the puzzle in a certain time slot is independent of how long he/she has been working on it. This has two consequences [29]. First, the time between two consecutive block consolidations is independent and exponentially distributed, since the exponential distribution is the only nonnegative continuous distribution that satisfies the memoryless property.…”