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1. Bitcoin: A Natural Oligopoly

   https://doi.org/10.1287/mnsc.2021.4095  

   Arnosti, Nick; Weinberg, S. Matthew (July 2022, Management Science)

   We argue that the concentrated production and ownership of Bitcoin mining hardware arise naturally from the economic incentives of Bitcoin mining. We model Bitcoin mining as a two-stage competition; miners compete in prices to sell hardware while competing in quantities for mining rewards. We characterize equilibria in our model and show that small asymmetries in operational costs result in highly concentrated ownership of mining equipment. We further show that production of mining equipment will be dominated by the miner with the most efficient hardware, who will sell hardware to competitors while possibly also using it to mine. This paper was accepted by Kay Giesecke, finance. 

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2. Ignore the Extra Zeroes: Variance-Optimal Mining Pools

   https://doi.org/10.1007/978-3-662-64331  

   Roughgarden, Tim; Shikhelman, Clara (January 2021, International Conference on Financial Cryptography and Data Security)

   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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3. Dynamic power control for rational cryptocurrency mining

   https://doi.org/10.1145/3410699.3413797  

   Chang, Sang-Yoon; Wuthier, Simeon (September 2020, the 3rd Workshop on Cryptocurrencies and Blockchains for Distributed Systems)

   In blockchain and cryptocurrency, miners participate in a proof-of-work-based distributed consensus protocol to find and generate a valid block, process transactions, and earn the corresponding reward. Because cryptocurrency is designed to adapt to the dynamic miner network size, a miner's participation affects the block difficulty which sets the expected amount of work to find a valid block. We study the dependency between the mining power control and the block difficulty and study a rational miner utilizing such dependency to dynamically control its mining power over a longer horizon than just the impending block. More specifically, we introduce I-O Mining strategy where a miner takes advantage of the block difficulty adjustment rule and toggles between mining with full power and power off between the difficulty adjustments. In I-O Mining, the miner influences the block difficulty and mines only when the difficulty is low, gaming and violating the design integrity of the mining protocol for its profit gain. We analyze the I-O Mining's incentive/profit gain over the static-mining strategies and its negative impact on the rest of the blockchain mining network in the block/transaction scalability. Our results show that I-O Mining becomes even more effective and profitable as there are greater competitions for mining and the reward and the cost difference becomes smaller, which are the trends in cryptocurrencies. 

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4. Coprime automorphisms of finite groups

   https://doi.org/10.1090/tran/8553  

   Acciarri, Cristina; Guralnick, Robert; Shumyatsky, Pavel (July 2022, Transactions of the American Mathematical Society)

   Let G G be a finite group admitting a coprime automorphism α \alpha of order e e . Denote by I G ( α ) I_G(\alpha ) the set of commutators g − 1 g α g^{-1}g^\alpha , where g ∈ G g\in G , and by [ G , α ] [G,\alpha ] the subgroup generated by I G ( α ) I_G(\alpha ) . We study the impact of I G ( α ) I_G(\alpha ) on the structure of [ G , α ] [G,\alpha ] . Suppose that each subgroup generated by a subset of I G ( α ) I_G(\alpha ) can be generated by at most r r elements. We show that the rank of [ G , α ] [G,\alpha ] is ( e , r ) (e,r) -bounded. Along the way, we establish several results of independent interest. In particular, we prove that if every element of I G ( α ) I_G(\alpha ) has odd order, then [ G , α ] [G,\alpha ] has odd order too. Further, if every pair of elements from I G ( α ) I_G(\alpha ) generates a soluble, or nilpotent, subgroup, then [ G , α ] [G,\alpha ] is soluble, or respectively nilpotent. 

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5. Selfish Mining Under General Stochastic Rewards

   https://doi.org/10.4230/lipics.aft.2025.20  

   Bahrani, Maryam; Neuder, Michael; Weinberg, S Matthew (January 2025, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Avarikioti, Zeta; Christin, Nicolas (Ed.)

   Selfish miners selectively withhold blocks to earn disproportionately high revenue. The vast majority of the selfish mining literature focuses exclusively on block rewards. [Carlsten et al., 2016] is a notable exception, observing that similar strategic behavior is profitable in a zero-block-reward regime (the endgame for Bitcoin’s quadrennial halving schedule) if miners are compensated with transaction fees alone. Neither model fully captures miner incentives today. The block reward remains 3.125 BTC, yet some blocks yield significantly higher revenue. For example, congestion during the launch of the Babylon protocol in August 2024 caused transaction fees to spike from 0.14 BTC to 9.52 BTC, a 68× increase in fees within two blocks. Our results are both practical and theoretical. Of practical interest, we study selfish mining profitability under a combined reward function that more accurately models miner incentives. This analysis enables us to make quantitative claims about protocol risk (e.g., the mining power at which a selfish strategy becomes profitable is reduced by 22% when optimizing over the combined reward function versus block rewards alone) and qualitative observations (e.g., a miner considering both block rewards and transaction fees will mine more or less aggressively respectively than if they cared about either alone). These practical results follow from our novel model and methodology, which constitute our theoretical contributions. We model general, time-accruing stochastic rewards in the Nakamoto Consensus Game, which requires explicit treatment of difficult adjustment and randomness; we characterize reward function structure through a set of properties (e.g., that rewards accrue only as a function of time since the parent block). We present a new methodology to analytically calculate expected selfish miner rewards under a broad class of stochastic reward functions and validate our method numerically by comparing it with the existing literature and simulating the combined reward sources directly. 

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