📈 economics

Uncountably many conditionally inaccessible decisions exist in every finite probability space

This paper proves that in any finite probability space, there exist uncountably many objective probability measures and utility function pairs that render decisions conditionally inaccessible, demonstrating that an agent's subjective probability can systematically prevent them from making objectively optimal decisions.

Original authors: Zalán Gyenis, Miklós Rédei, Leszek Wroński

Published 2026-05-05

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Zalán Gyenis, Miklós Rédei, Leszek Wroński

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: The "Blind Spot" of Rationality

Imagine you are a captain steering a ship (the Agent). You have a map in your head representing what you think is true about the ocean currents and weather (your Subjective Probability, or $p$ ). You also have a "God's Eye View" of the ocean that represents the actual, physical laws of the water (the Objective Probability, or $p^*$ ).

In a perfect world, if you look at the real weather, update your map, and then make a decision, you will always make the best possible choice. This paper argues that this is not always true.

The authors prove that there are countless situations where, no matter how much partial information you gather about the real world, your updated map will never lead you to the objectively best decision. In fact, your rational updates might actively steer you away from the truth.

The Core Concepts

1. The Decision Context (The Menu)

Imagine you are choosing between two actions, like taking Route A or Route B.

The Objective Truth ( $p^*$ ): The actual traffic data. Based on this, Route A is clearly faster.
Your Belief ( $p$ ): Your current guess about the traffic.
The Decision: You choose the route that looks fastest based on your belief.

2. The Update Strategy (Jeffrey Conditioning)

You don't know the full traffic report. Instead, you get "partial evidence."

Example: You hear a rumor that "there is an accident on the north side," but you don't know the exact location.
The Process: You take your current belief ( $p$ ) and update it based on this partial rumor. This creates a new, "upgraded" belief ( $q$ ).
The Goal: You want this upgraded belief to tell you to take Route A (the objectively good choice).

3. The Problem: "Conditionally Inaccessible" Decisions

The paper defines a decision as "conditionally inaccessible" if, no matter what partial evidence you get, your upgraded belief never tells you to take the objectively best route.

It's like having a compass that is magnetically distorted. No matter which direction you look (what partial evidence you gather), the needle never points North. It might point East, West, or South, but never North.

The Main Discovery: It's Everywhere, Not Rare

In a previous paper, the authors found a few examples where this "broken compass" happened. They guessed it might happen often in complex situations.

This paper proves they were wrong to be cautious. They show that:

It happens in almost every situation: If you have a decision with at least 3 options (like 3 routes), this problem exists.
It's infinite: For any way you currently think ( $p$ ), there are an uncountably infinite number of real-world scenarios ( $p^*$ ) where your rational updates will fail to find the best decision.
It's robust: It's not a fluke. There are infinite pairs of choices (Route A vs. Route B) where this failure occurs.

The Metaphor:
Imagine you are trying to find a specific key in a giant, dark room. You have a flashlight (your partial evidence). The paper proves that for almost every possible location of the key, there is a way your flashlight is broken such that no matter how you sweep the beam, you will never shine light on the key. You might think you are being rational by sweeping the floor, but the physics of your flashlight guarantees you will miss the target.

The "Blind Spot" Explanation

The authors use a mathematical concept called the "Bayes Blind Spot."

Think of your current belief ( $p$ ) as a pair of glasses.
The "Blind Spot" is the set of all possible realities ( $p^*$ ) that you cannot reach just by looking through those glasses and adjusting them based on partial clues.
The paper proves that this "Blind Spot" is massive. It covers almost the entire universe of possibilities.
If the "real world" falls into this blind spot, your rational updates will always fail to align with reality.

The Degree of Failure

The paper also introduces a way to measure how broken the compass is.

Degree 0: Your updates always work perfectly.
Maximum Degree: Your updates never work, no matter what evidence you get.
The Finding: The authors prove that you can find situations where the failure happens for any specific number of evidence types you might encounter. You can engineer a scenario where your compass fails exactly 5 times out of 10 possible clues, or 99 times out of 100.

The "Surprising" Result: Learning Makes It Worse?

One might think: "If I start with a bad guess, but I learn the truth, surely I get better?"
The paper proves a specific, counter-intuitive fact:
If your rational updates (using partial evidence) lead you to a wrong decision, it implies that your original guess (before any updates) was also wrong.

Translation: You cannot start with a "good" guess, learn some partial truth, and end up with a "bad" decision. If the update leads you astray, you were already astray.
However: The paper's main point is that there are so many "bad" starting points (subjective beliefs) that correspond to "good" realities, that it is statistically overwhelming to find yourself in a situation where your rational updates fail to find the truth.

Summary in One Sentence

The paper proves that for any rational person with a specific set of beliefs, there is a vast, infinite ocean of real-world scenarios where, no matter how much partial information they gather and process, their rational updates will systematically prevent them from making the objectively best choice.

Technical Summary: Uncountably Many Conditionally Inaccessible Decisions in Finite Probability Spaces

Problem Statement
This paper addresses a problem in decision theory concerning the relationship between an Agent's subjective probability (credence) and objective probability. Specifically, it investigates the phenomenon of "conditionally $p$ -inaccessible decisions." A decision to choose action $f_1$ over $f_2$ is considered objectively good if the expected utility of $f_1$ exceeds that of $f_2$ under an objective probability measure $p^*$ . However, an Agent makes decisions based on a subjective probability $p$ . If the Agent updates their prior $p$ using Jeffrey conditioning on partial evidence regarding $p^*$ , the resulting posterior probability $q_\Pi$ might lead the Agent to make a different choice (or remain indifferent) than the one dictated by $p^*$ .

The central question is whether there exist decision contexts where, regardless of the partial evidence (represented by a proper non-trivial partition $\Pi$ ) the Agent acquires about $p^*$ , the Agent's updated rational decision never aligns with the objectively good decision. Previous work [16] established the existence of such decisions in specific finite probability spaces and conjectured their existence in spaces with $n \ge 3$ elementary events.

Methodology
The authors employ a rigorous mathematical framework within finite probability spaces $(X, S)$ , where $X = \{1, \dots, n\}$ with $n \ge 3$ . The methodology relies on:

Jeffrey Conditioning: Utilizing the definition of the Jeffrey posterior $q_\Pi$ derived from a prior $p$ and a partition $\Pi$ , given objective probabilities $p^*$ .
Radon-Nikodym Derivatives: Analyzing the function $r(i) = p^*(i)/p(i)$ to characterize the relationship between the objective and subjective measures.
Kullback-Leibler Divergence: Employing information-theoretic inequalities to establish strict bounds on the expected values of utility functions under different probability measures.
Constructive Proofs: Explicitly constructing utility functions (random variables) that satisfy the conditions for conditional inaccessibility by leveraging the properties of the "Bayes $p$ -Blind Spot" (the set of measures not obtainable by conditioning $p$ ).
Combinatorial Analysis: Investigating the cardinality of partitions and the multiplicity of posteriors to determine the "degree" of inaccessibility.

Key Contributions and Results

Proof of Existence in All Finite Spaces (Theorem 3.2):
The paper proves a theorem stronger than the conjecture in [16]. It establishes that for any probability measure $p$ and any objective measure $p^*$ in a finite space with $n \ge 3$ elements, if $p^*$ lies in the Bayes $p$ -Blind Spot (i.e., $p^*$ cannot be obtained by conditioning $p$ on any proper non-trivial partition), then there exist uncountably many pairs of utility functions representing conditionally $p$ -inaccessible decisions.
- Mechanism: The proof demonstrates that if the Radon-Nikodym derivative $r$ is injective, one can construct a function $g$ based on $\log r$ such that $E_{p^*}[g] > E_{q_\Pi}[g]$ for all partitions $\Pi$ . By shifting this function, a decision $d$ is created where $E_{p^*}[d] > 0$ but $E_{q_\Pi}[d] \le 0$ for all $\Pi$ .
Cardinality of Inaccessible Decisions:
The authors show that the set of such "bad" decision situations is not merely non-empty but "enormous." Specifically, for any fixed $p$ , there is a continuum (uncountable) number of objective measures $p^*$ in the Bayes $p$ -Blind Spot. For each such $p^*$ , there is a continuum of utility function pairs yielding conditionally inaccessible decisions.
Theorem on Informed Rationality (Theorem 2.4):
The paper proves that if a decision is conditionally $p$ -inaccessible (meaning the updated decision is never objectively good), then the Agent's initial decision based on the unconditioned prior $p$ must also be objectively wrong ( $E_p[d] < 0$ ). This rules out the counter-intuitive scenario where learning objective truth via conditioning leads to a wrong decision, whereas the uninformed decision was correct.
Characterization of Degrees of Inaccessibility (Theorem 4.4):
The paper resolves Conjecture 7.2 from [16] regarding the "degree" of conditional inaccessibility. It defines the degree as the number of partitions $\Pi$ for which the updated decision fails to be objectively good. The authors prove that for a fixed decision context, the set of achievable degrees depends on the multiplicity of the posteriors generated by different partitions. If the mapping from partitions to posteriors is injective, then every integer degree from $0$ to the total number of proper non-trivial partitions ($Bell(n)-2$) is achievable.

Significance
The paper claims that conditionally $p$ -inaccessible decisions are not rare anomalies but a robust and substantial feature of decision theory in finite spaces.

Interpretation: If $p^*$ represents objective reality and $p$ represents an Agent's subjective belief, the results imply that there is an "enormous number" of decision situations where the Agent's subjective probability structure fundamentally prohibits them from reaching an objectively good decision, even when they update their beliefs with true but partial information about the objective probabilities.
Implication for Rationality: The existence of the Bayes $p$ -Blind Spot suggests that for a wide class of objective probabilities, no amount of partial evidence (conditioning) can bridge the gap between the Agent's subjective rationality and objective optimality. The paper concludes that the Agent's subjective probability can systematically block the realization of objectively good decisions, regardless of the specific partial evidence acquired.