arXiv subjects

Takashi Izumo

Publications and source records attributed to Takashi Izumo.

3 recordsLinked to original sources

arXiv2026

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

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.

econ.TH↗

arXiv2026

Quantifying Information Loss under Coarse-Grained Partitions: A Discrete Framework for Explainable Artificial Intelligence

As artificial intelligence (AI) systems are increasingly used in ethically sensitive domains such as education, healthcare, and transportation, balancing accuracy and interpretability has become a central concern. Coarse ethics (CE) motivates coarse-grained evaluations under cognitive, institutional, and contextual constraints, but it still lacks a simple mathematical formalization of admissible coarse-graining and its informational consequences. This paper introduces coarse-grained partitions (CGPs) as a discrete framework for modeling coarse evaluation on a finite totally ordered score scale. A CGP represents coarse evaluation as a partition into grains with an index assignment, and induces a coarse-grained distribution by pushforward. To compare admissible coarse-grainings, we introduce categorical unification (CU), which constructs a canonical fine-scale reconstruction from the coarse representation under minimal assumptions. On this basis, we define a KL-based measure of information loss, $D_{\mathrm{KL\text{-}CU}}$, as the divergence between the original fine-grained distribution and its CU-based reconstruction. We prove that $D_{\mathrm{KL\text{-}CU}}=0$ if and only if the original distribution is already uniform within each grain. This shows that zero loss, in the sense of the proposed measure, is a highly exceptional limiting case rather than a realistic benchmark for ordinary evaluative practice. We also show that the framework leads naturally to an optimization problem for comparing alternative admissible CGPs. Applications to educational grading and explainable AI (XAI) illustrate how the framework clarifies trade-offs among informational fidelity, interpretability, and coarsening cost.

cs.AI↗

arXiv2025

Coarse-Grained Games: A Framework for Bounded Perception in Game Theory

In everyday life, we frequently make coarse-grained judgments. When we say that Olivia and Noah excel in mathematics, we disregard the specific differences in their mathematical abilities. Similarly, when we claim that a particular automobile manufacturer produces high-quality cars, we overlook the minor variations among individual vehicles. These coarse-grained assessments are distinct from erroneous or deceptive judgments, such as those resulting from student cheating or false advertising by corporations. Despite the prevalence of such judgments, little attention has been given to their underlying mathematical structure. In this paper, we introduce the concept of coarse-graining into game theory, analyzing games where players may perceive different payoffs as identical while preserving the underlying order structure. We call it a Coarse-Grained Game (CGG). This framework allows us to examine the rational inference processes that arise when players equate distinct micro-level payoffs at a macro level, and to explore how Nash equilibria are preserved or altered as a result. Our key findings suggest that CGGs possess several desirable properties that make them suitable for modeling phenomena in the social sciences. This paper demonstrates two such applications: first, in cases of overly minor product updates, consumers may encounter an equilibrium selection problem, resulting in market behavior that is not driven by objective quality differences; second, the lemon market can be analyzed not only through objective information asymmetry but also through asymmetries in perceptual resolution or recognition ability.

econ.TH↗