Absorption and Inertness in Coarse-Grained Arithmetic: A Heuristic Application to the St. Petersburg Paradox

[2507.12475] Absorption and Inertness in Coarse-Grained Arithmetic: A Heuristic Application to the St. Petersburg Paradox Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Economics > Theoretical Economics arXiv:2507.12475 (econ) [Submitted on 5 Jul 2025 ( v1 ), last revised 10 Apr 2026 (this version, v2)] Title: Absorption and Inertness in Coarse-Grained Arithmetic: A Heuristic Application to the St. Petersburg Paradox Authors: Takashi Izumo View a PDF of the paper titled Absorption and Inertness in Coarse-Grained Arithmetic: A Heuristic Application to the St. Petersburg Paradox, by Takashi Izumo View PDF HTML (experimental) Abstract: The St. Petersburg paradox presents a longstanding challenge in decision theory: its classical expected value diverges, yet no correspondingly large finite stake is typically regarded as rational. Traditional responses introduce auxiliary assumptions, such as diminishing marginal utility, temporal discounting, or extended number systems. This paper explores a different approach based on a modified operation of addition defined over coarse-grained partitions of the underlying numerical scale. In this framework, exact values are grouped into ordered grains, each grain is assigned an internal representative, and addition proceeds by repeated projection to those representatives. On this basis, the paper defines coarse representative addition and coarse cell addition, and studies several of their structural properties, including absorption, inertness, and non-associativity. In particular, repeated additions may eventually cease to change the coarse state, a phenomenon called inertness. The paper then applies this framework heuristically to the St. Petersburg setting by considering a rescaled sequence corresponding to its equal expected increments, and shows that this sequence can become inert under a suitably chosen countable partition and representative map. The claim is not that the paradox is resolved within standard decision theory, nor that the classical expectation becomes finite in the ordinary probabilistic sense. Rather, the contribution is structural and heuristic: it exhibits an explicit mathematical mechanism through which a divergent reward structure may fail to produce unbounded growth once aggregation itself is made coarse. More broadly, the framework may be relevant to the study of bounded numerical cognition and behavioral models of aggregation. Comments: 20 pages, no figure Subjects: Theoretical Economics (econ.TH) ; Artificial Intelligence (cs.AI); Optimization and Control (math.OC) Cite as: arXiv:2507.12475 [econ.TH] (or arXiv:2507.12475v2 [econ.TH] for this version) https://doi.org/10.48550/arXiv.2507.12475 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Takashi Izumo [ view email ] [v1] Sat, 5 Jul 2025 07:34:46 UTC (46 KB) [v2] Fri, 10 Apr 2026 14:46:04 UTC (44 KB) Full-text links: Access Paper: View a PDF of the paper titled Absorption and Inertness in Coarse-Grained Arithmetic: A Heuristic Application to the St. Petersburg Paradox, by Takashi Izumo View PDF HTML (experimental) TeX Source view license Current browse context: econ.TH < prev | next > new | recent | 2025-07 Change to browse by: cs cs.AI econ math math.OC References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... 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