Resolution of the St. Petersburg paradox using Von Mises axiom of randomness
Pith reviewed 2026-05-25 14:08 UTC · model grok-4.3
The pith
The St. Petersburg paradox is resolved by showing infinite expected gains do not require cognitive strategies.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The author claims that an expected gain that tends toward infinity is not always a consequence of a cognitive and non-random strategy. By using Von Mises' axiom of randomness to classify strategies based on whether they exploit useful information and thus produce non-reproducible results, the paper demonstrates that the paradox can be addressed by prioritizing the cognitive aspect of strategies over their expected payoff.
What carries the argument
Von Mises' axiom of randomness applied to determine if a strategy is cognitive by checking if its results can be reproduced randomly, thereby identifying use of useful information.
If this is right
- Cognitive strategies should be preferred in decision making even if their expected gain is lower.
- The St. Petersburg paradox does not imply that all high-value strategies are equally valuable.
- A hierarchy of decision values places divergence from random behavior above expected gain size.
- Strategies can be evaluated independently of their payoff calculation using randomness criteria.
Where Pith is reading between the lines
- This method might apply to other expected value paradoxes to distinguish informed choices.
- Simulations of random vs information-using strategies could test the proposed hierarchy in practice.
- Real world betting or investment could benefit from measuring information exploitation rather than just averages.
Load-bearing premise
Von Mises' axiom of randomness can classify strategies in the St. Petersburg game as cognitive or non-cognitive and this produces a valid hierarchy of decision values independent of payoff.
What would settle it
Finding a strategy in the St. Petersburg game that produces infinite expected gain but according to Von Mises axiom behaves randomly, or a cognitive strategy that does not diverge from random.
read the original abstract
In this article we will propose a completely new point of view for solving one of the most important paradoxes concerning game theory. The solution develop shifts the focus from the result to the strategy s ability to operate in a cognitive way by exploiting useful information about the system. In order to determine from a mathematical point of view if a strategy is cognitive, we use Von Mises' axiom of randomness. Based on this axiom, the knowledge of useful information consequently generates results that cannot be reproduced randomly. Useful information in this case may be seen as a significant datum for the recipient, for their present or future decision-making process. Finally, by resolving the paradox from this new point of view, we will demonstrate that an expected gain that tends toward infinity is not always a consequence of a cognitive and non-random strategy. Therefore, this result leads us to define a hierarchy of values in decision-making, where the cognitive aspect, whose statistical consequence is a divergence from random behaviour, turns out to be more important than the expected gain.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper claims to resolve the St. Petersburg paradox by applying Von Mises' axiom of randomness to classify strategies as cognitive (those exploiting useful information and yielding non-reproducible results) versus non-cognitive. It asserts that an expected gain tending to infinity is not always a consequence of a cognitive strategy, thereby defining a hierarchy of decision-making values in which the cognitive aspect (statistical divergence from randomness) outweighs expected value.
Significance. If the required embedding of the axiom into game strategies were supplied and shown to be payoff-independent, the work could contribute a novel decision-theoretic perspective that prioritizes randomness properties over classical expectation. The manuscript introduces an invented 'hierarchy of values' without comparison to established resolutions such as logarithmic utility or other bounded-value functions.
major comments (2)
- [Abstract] Abstract and introduction: the central claim requires that Von Mises' axiom (place selections on infinite sequences yielding frequency limits independent of the selection rule) classifies St. Petersburg strategies as cognitive/non-random, yet no derivation or explicit mapping is supplied showing how a finite or repeated-play strategy induces a von Mises collective, what admissible place selections are, or why the resulting frequency divergence is independent of the payoff measure 2^k on probability 2^{-k} rather than an artifact of it.
- [Abstract] The resolution is obtained by defining cognitive strategies via the randomness axiom and then declaring that infinite expectation need not imply such strategies; the separation is therefore built directly into the chosen definition rather than derived from external benchmarks or explicit computation on the game's outcome sequences.
minor comments (1)
- [Abstract] Abstract contains grammatical issues, e.g., 'The solution develop shifts' and 'strategy s ability'.
Simulated Author's Rebuttal
We thank the referee for the report and the opportunity to clarify our approach. We respond point by point to the major comments.
read point-by-point responses
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Referee: [Abstract] Abstract and introduction: the central claim requires that Von Mises' axiom (place selections on infinite sequences yielding frequency limits independent of the selection rule) classifies St. Petersburg strategies as cognitive/non-random, yet no derivation or explicit mapping is supplied showing how a finite or repeated-play strategy induces a von Mises collective, what admissible place selections are, or why the resulting frequency divergence is independent of the payoff measure 2^k on probability 2^{-k} rather than an artifact of it.
Authors: We agree that the manuscript would be strengthened by a more explicit technical mapping. The paper applies the axiom conceptually to identify cognitive strategies as those that exploit information and thereby produce outcome frequencies that deviate from what would arise under purely random selection. In revision we will add a dedicated subsection that models repeated plays as an infinite sequence constituting a von Mises collective, specifies that admissible place selections are those conditioned on prior outcomes (representing information use), and shows that the frequency limit property required by the axiom holds independently of the particular payoff function because the axiom concerns the invariance of the limit across selections, not the numerical values attached to each term. revision: yes
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Referee: [Abstract] The resolution is obtained by defining cognitive strategies via the randomness axiom and then declaring that infinite expectation need not imply such strategies; the separation is therefore built directly into the chosen definition rather than derived from external benchmarks or explicit computation on the game's outcome sequences.
Authors: The separation follows from applying an external, pre-existing mathematical criterion (Von Mises' axiom) rather than from an ad-hoc definition. The axiom supplies an independent benchmark for randomness; strategies that systematically use information produce collectives whose limiting frequencies differ from the measure induced by non-informative selection. The claim that infinite expectation need not imply a cognitive strategy is therefore a direct consequence of this benchmark. We will nevertheless include explicit illustrative computations on finite outcome sequences under both cognitive and non-cognitive strategies to make the distinction more concrete. revision: partial
Circularity Check
Cognitive/non-cognitive classification defined via Von Mises axiom, making infinite-EV separation from cognitive status definitional
specific steps
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self definitional
[Abstract]
"In order to determine from a mathematical point of view if a strategy is cognitive, we use Von Mises' axiom of randomness. Based on this axiom, the knowledge of useful information consequently generates results that cannot be reproduced randomly. [...] by resolving the paradox from this new point of view, we will demonstrate that an expected gain that tends toward infinity is not always a consequence of a cognitive and non-random strategy. Therefore, this result leads us to define a hierarchy of values in decision-making, where the cognitive aspect, whose statistical consequence is a diver"
Cognitive status is introduced as equivalent to non-random behavior under the axiom; the claimed demonstration that infinite expected gain need not imply a cognitive strategy is then obtained simply by applying this label to selected strategies, without an external benchmark or derivation that could separate the two quantities independently of the definition.
full rationale
The paper defines cognitive strategies as those exploiting useful information, which by direct application of the Von Mises randomness axiom produce non-reproducible (non-random) outcomes. It then presents as a derived result that infinite expected gain is not always a consequence of such cognitive strategies. This reduction is self-definitional: the hierarchy and resolution follow from the chosen classification criterion rather than from an independent embedding or calculation that could falsify the link between infinite EV and the axiom-derived label.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption Von Mises' axiom of randomness
invented entities (1)
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hierarchy of values in decision-making
no independent evidence
discussion (0)
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