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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. 

Paleoclimate records are rich sources of information about the past history of the Earth system. Information theory provides a new means for studying these records. We demonstrate that weighted permutation entropy of water-isotope data from the West Antarctica Ice Sheet (WAIS) Divide ice core reveals meaningful climate signals in this record. We find that this measure correlates with accumulation (meters of ice equivalent per year) and may record the influence of geothermal heating effects in the deepest parts of the core. Dansgaard-Oeschger and Antarctic Isotope Maxima events, however, do not appear to leave strong signatures in the information record, suggesting that these abrupt warming events may actually be predictable features of the climate’s dynamics. While the potential power of information theory in paleoclimatology is significant, the associated methods require well-dated and high-resolution data. The WAIS Divide core is the first paleoclimate record that can support this kind of analysis. As more high-resolution records become available, information theory could become a powerful forensic tool in paleoclimate science. 

Permutation entropy techniques can be useful for identifying anomalies in paleoclimate data records, including noise, outliers, and post-processing issues. We demonstrate this using weighted and unweighted permutation entropy with water-isotope records containing data from a deep polar ice core. In one region of these isotope records, our previous calculations (See Garland et al. 2018)revealed an abrupt change in the complexity of the traces: specifically, in the amount of new information that appeared at every time step. We conjectured that this effect was due to noise introduced by an older laboratory instrument. In this paper, we validate that conjecture by reanalyzing a section of the ice core using a more advanced version of the laboratory instrument. The anomalous noise levels are absent from the permutation entropy traces of the new data. In other sections of the core, we show that permutation entropy techniques can be used to identify anomalies in the data that are not associated with climatic or glaciological processes, but rather effects occurring during field work, laboratory analysis, or data post-processing. These examples make it clear that permutation entropy is a useful forensic tool for identifying sections of data that require targeted reanalysis—and can even be useful for guiding that analysis.