Abstract
Mining pools decrease the variance in the income of cryptocurrency miners (compared to solo mining) by distributing rewards to participating miners according to the shares submitted over a period of time. The most common definition of a “share” is a proof-of-work for a difficulty level lower than that required for block authorization—for example, a hash with at least 65 leading zeroes (in binary) rather than at least 75.
The first contribution of this paper is to investigate more sophisticated approaches to pool reward distribution that use multiple classes of shares—for example, corresponding to differing numbers of leading zeroes—and assign different rewards to shares from different classes. What’s the best way to use such finer-grained information, and how much can it help? We prove that the answer is not at all: using the additional information can only increase the variance in rewards experienced by every miner.
Our second contribution is to identify variance-optimal reward-sharing schemes. Here, we first prove that pay-per-share rewards simultaneously minimize the variance of all miners over all reward-sharing schemes with long-run rewards proportional to miners’ hash rates. We then show that, if we impose natural restrictions including a no-deficit condition on reward-sharing schemes, then the pay-per-last-N-shares method is optimal.
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Notes
- 1.
Technically, in Bitcoin this is defined by finding a hash that is smaller than a number that gets adjusted over time. For ease of discussion we will continue to refer to the number of leading zeros.
- 2.
- 3.
Variance-minimization has been regarded as a key objective function for mining pool design since Rosenfeld’s seminal analysis of Bitcoin mining pools [16].
- 4.
Our results remain the same if each miner has its own subset \(A_i\) of acceptable messages, provided the \(A_i\)’s all have the same size.
- 5.
The Poisson assumption is for convenience. The important property is that the identity of the sender of a new signed message is distributed proportionally to the hashrate distribution, independent of the past.
- 6.
The constant \(c > 0\) would typically be chosen so that the rate at which rewards are granted to miners equals the rate at which the pool accrues block rewards (and possibly transaction fees), less a commission.
- 7.
Other schemes with future-dependent rewards can be similarly modeled. The key requirement is that the probability distribution over the reward associated with a share (with respect to future samples from the message distribution) is independent of the hashrate distribution \(\mathbf {h}\). This is the case for most of the well-studied RSSes (including e.g. the geometric reward schemes studied in [7]).
- 8.
All of the common RSSes that motivate this work are also anonymous, meaning that \(\varphi ((s_1,m_1),\ldots ,(s_t,m_t))\) is independent of \(s_1,s_2,\ldots ,s_t\). While anonymity is natural (and arguably unavoidable) in a permissionless blockchain setting, our positive results do not require that assumption. In any case, the RSSes advocated by our results are anonymous.
- 9.
MLEs are deterministic (up to tie-breaking). Thus no randomized RSS (such as \({\text {PPLNS}}\) or the estimator induced by the proportional rule) can be a MLE.
- 10.
A similar result can be found in [14] under the reasonable assumption that the shares follow the Poisson distribution.
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Roughgarden, T., Shikhelman, C. (2021). Ignore the Extra Zeroes: Variance-Optimal Mining Pools. In: Borisov, N., Diaz, C. (eds) Financial Cryptography and Data Security. FC 2021. Lecture Notes in Computer Science(), vol 12675. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-64331-0_12
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