Uncountably many conditionally inaccessible decisions exist in every finite probability space

TL;DR

This paper proves that in every finite probability space, there are uncountably many probability measures and utility function pairs leading to decisions that are inaccessible when conditioned on partial evidence, highlighting decision-making limitations.

Contribution

It establishes the existence of uncountably many conditionally inaccessible decisions in any finite probability space, confirming a conjecture from prior work.

Findings

Uncountably many probability measures exist for each finite space.

Uncountably many utility pairs lead to conditionally inaccessible decisions.

The result links subjective and objective probabilities in decision-making.

Abstract

In a recent paper \cite{Redei-Jing2026} the notion of conditional $p$-inaccessibility of a decision based on utility maximization was defined and examples of conditionally $p$-inaccessible decisions were given. The conditional inaccessibility of a decision based on maximizing utility calculated by a probability measure $p^*$ expresses that the decision cannot be obtained if the expectation values of the utility functions are calculated using the (Jeffrey) conditional probability measure obtained by conditioning $p$ on partial evidence about the probability $p^*$ that determines the decision. The paper \cite{Redei-Jing2026} conjectured that conditionally $p$-inaccessible decisions exist in some probability spaces having arbitrary large finite number of elementary events. In this paper we prove that for any $p$ in any finite probability space there exist an uncountable number of probability measures $p^*$ for each of which there exist an uncountable number of pairs of utility functions that represent conditionally $p$-inaccessible decisions. If $p^*$ is an objective probability determining objectively good decisions and $p$ is the subjective probability determining a rational decision of a decision making Agent, the result says that there is an enormous number of decision situations in which the Agent's subjective probability prohibits the Agent's informed rational decision to be objectively good.