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

TL;DR

This paper introduces a coarse-grained arithmetic framework with absorption and inertness properties, offering a heuristic perspective on the St. Petersburg paradox by showing how divergence can be mitigated through aggregation.

Contribution

It develops a novel coarse-grained addition method with structural properties like inertness, providing a heuristic approach to understanding the paradox without resolving it in standard decision theory.

Findings

Repeated coarse addition can become inert, ceasing to change the state.

The framework demonstrates how divergent expected values can be bounded through aggregation.

Application to the St. Petersburg paradox shows divergence can be mitigated heuristically.

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.