Recognition: unknown
Uncountably many conditionally inaccessible decisions exist in every finite probability space
Pith reviewed 2026-05-08 02:39 UTC · model grok-4.3
The pith
In every finite probability space, uncountably many objective probabilities p* exist for which uncountably many utility pairs yield decisions that remain inaccessible under conditioning from any subjective p.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The central claim is that for any probability p on a finite space there exist uncountably many probabilities p* for each of which there exist uncountably many pairs of utility functions representing decisions that are conditionally p-inaccessible. A decision is conditionally p-inaccessible when the action that maximizes expected utility calculated with p* cannot be recovered as the maximizer when the same utilities are evaluated with the conditional probability obtained by conditioning p on partial evidence about the value of p*.
What carries the argument
Conditional p-inaccessibility: the property that a utility-maximizing decision under p* cannot be reproduced when expectations are recomputed with the Jeffrey conditional probability obtained by conditioning the agent's subjective p on partial information about p*.
Load-bearing premise
The definition of conditional p-inaccessibility is well-posed and the constructions used to produce the uncountable families remain valid inside standard finite probability spaces equipped with real-valued utility functions.
What would settle it
Exhibit one finite probability space together with a probability p such that the set of p* admitting even a single pair of utilities that produce a conditionally p-inaccessible decision is at most countable.
read the original 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.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proves that for any probability measure p on any finite sample space, there exist uncountably many probability measures p* such that for each p* there exist uncountably many pairs of real-valued utility functions (u, u*) realizing conditionally p-inaccessible decisions. Conditional p-inaccessibility means that the argmax of expected utility under p* differs from the argmax obtained by taking the Jeffrey conditional of p on partial information about p*. The proof supplies an explicit parametrization of p* by a real perturbation parameter in an open interval inside the probability simplex, and for each such p* a corresponding continuum of utility pairs.
Significance. If the result holds, it establishes a strong, uniform existence statement: every finite decision problem admits continuum-many objective probabilities p* and continuum-many utility pairs for which subjective updating via Jeffrey conditioning on evidence about p* blocks access to the objectively optimal decision. The explicit, finite-dimensional construction (perturbations within the standard simplex, no non-standard analysis or heavy choice principles) is a notable strength, as it yields concrete, falsifiable examples and directly confirms the conjecture from the cited 2026 work. This has potential implications for decision theory, highlighting systematic gaps between subjective rationality and objective optimality in finite settings.
major comments (2)
- [§3] §3 (main construction): the argument that the Jeffrey conditional of p on the chosen partial information about p* produces a strictly different argmax relies on the specific form of the perturbation and the support of the evidence event; a fully expanded verification that this difference holds for all utility pairs in the parametrized family (rather than for generic utilities) would make the load-bearing step fully transparent.
- [§2] §2 (application of the definition): although the construction is said to satisfy the definition of conditional p-inaccessibility exactly as stated in Rede i-Jing 2026, the manuscript does not re-state the definition or verify the key property (different argmax under the conditional) inside the present text; because the entire existence claim rests on this imported notion, a self-contained recap of the definition plus a one-paragraph check that the constructed pairs meet it would eliminate any dependence on external reading.
minor comments (3)
- [References] The citation to Rede i-Jing 2026 should include the full title, journal or arXiv identifier, and year to allow readers to locate the source definition without ambiguity.
- [§3] Notation for the perturbation parameter (e.g., ε or δ) and the partial-information event should be introduced once in §3 and used consistently; currently the same symbol appears to be reused for different quantities in the utility-pair construction.
- [§4 (proof)] A short remark on why the construction requires only countable choice (or none beyond the reals) would be useful for readers concerned with foundational assumptions in probability theory.
Simulated Author's Rebuttal
We thank the referee for the careful reading, the positive evaluation of the result, and the recommendation of minor revision. The two comments identify opportunities to improve self-containment and transparency; we agree to incorporate both clarifications.
read point-by-point responses
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Referee: [§3] §3 (main construction): the argument that the Jeffrey conditional of p on the chosen partial information about p* produces a strictly different argmax relies on the specific form of the perturbation and the support of the evidence event; a fully expanded verification that this difference holds for all utility pairs in the parametrized family (rather than for generic utilities) would make the load-bearing step fully transparent.
Authors: We agree that an expanded, uniform verification strengthens the exposition. In the revised manuscript we will insert, immediately after the definition of the parametrized family, a direct calculation that compares the expected-utility argmax under p* with the argmax under the Jeffrey conditional for an arbitrary pair (u, u*) in the family. The calculation proceeds by substituting the explicit linear perturbation into the conditional probabilities, evaluating the two linear forms, and showing that their maximizers differ for every choice of the perturbation parameter in the open interval and for every admissible utility pair; no genericity assumption is required. revision: yes
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Referee: [§2] §2 (application of the definition): although the construction is said to satisfy the definition of conditional p-inaccessibility exactly as stated in Rede i-Jing 2026, the manuscript does not re-state the definition or verify the key property (different argmax under the conditional) inside the present text; because the entire existence claim rests on this imported notion, a self-contained recap of the definition plus a one-paragraph check that the constructed pairs meet it would eliminate any dependence on external reading.
Authors: We accept the suggestion. The revised §2 will open with a one-paragraph restatement of the definition of conditional p-inaccessibility taken verbatim from Redei-Jing 2026, followed by a short verification paragraph that confirms the constructed pairs satisfy the differing-argmax condition. The verification will be phrased in terms of the objects already introduced in §2 and will point forward to the explicit computation supplied in the expanded §3, thereby removing any need for the reader to consult the earlier paper for the core claim. revision: yes
Circularity Check
No significant circularity
full rationale
The paper imports the definition of conditional p-inaccessibility and a related conjecture from a 2026 citation with author overlap, but the central result—an explicit construction proving uncountably many p* and utility pairs exist for any p in any finite space—is developed independently in the present manuscript using only standard finite probability spaces and real-valued functions. No step in the derivation reduces by construction to the cited definition, no parameter is fitted and relabeled as a prediction, and no uniqueness theorem or ansatz is smuggled via self-citation to force the outcome. The argument remains self-contained within elementary measure theory on finite sets.
Axiom & Free-Parameter Ledger
axioms (2)
- domain assumption The definition of conditional p-inaccessibility as introduced in Rédei-Jing2026.
- standard math Finite probability spaces are discrete measurable spaces equipped with a probability measure p.
Reference graph
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