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Summary of the Research

This paper explores whether the two core rules used in neoclassical economic theory are consistent with each other. The first rule is the expected utility formula used for decisions under risk. The second rule is exponential discounting used for decisions over time. The author uses probability discounting models from behavioral psychology, especially those developed by Rachlin, to show that when risk is converted into an equivalent waiting time, expected utility ends up matching the form of a hyperbolic discounting function rather than an exponential one. This means that expected utility theory becomes equivalent to a model that produces dynamically inconsistent choices. The author argues that if we assume risk is processed as a time like quantity, then expected utility is no longer a fully normative rule. The paper then discusses how this insight may affect work in behavioral economics, neuroeconomics, and quantum epistemics.

Major Issues

1. The central argument depends on a strong assumption that is not fully defended

The paper treats risk as waiting time by using Rachlin’s probability discounting model. This is the basis for the claim that expected utility theory becomes mathematically equivalent to a special case of hyperbolic discounting. However, this assumption is very strong. It assumes that all risky choices behave as if the decision maker is effectively playing a repeated game where the probability of winning translates to expected delay. In many situations, risky choices are not repeated. For example, decisions about insurance, medical treatment, or large investments often involve a single event. The paper would be stronger if it compared one shot risk and repeated risk more carefully, and showed exactly when the probability to delay transformation is valid and when it is not.

2. The key mathematical equivalence needs more detail to be convincing

The main result is the claim that p times u(x) in expected utility theory can be rewritten as u(x) divided by a linear function of odds against. This requires that the hyperbolic discounting model uses a specific parameter value of one. The paper shows this quickly, but the steps are not fully laid out. A clearer walk through would be helpful. For example, readers would benefit from seeing how the odds against formula O(p) equals 1 divided by p minus one, and how this maps cleanly into the hyperbolic discounting function. This is the main foundation of the incompatibility claim, so it deserves a more complete derivation.

3. Exponential discounting is treated as the only valid benchmark for rationality

The paper states that exponential discounting is the normative rule for intertemporal choices. But modern economics and psychology do not treat exponential discounting as the only candidate for rational time preferences. Several models in the literature argue that exponential discounting may not reflect real human behavior and that rationality may need a different form. Since the main conclusion is about incompatibility with exponential discounting, the paper needs to address why exponential discounting must be considered normative in the first place.

4. The paper frequently introduces connections to neuroscience and gambling but does not show their exact relevance

The discussion of dopamine, cognitive ability, gambling behavior, and computational psychiatry is interesting, but the links to the main mathematical result are not fully explained. These topics appear abruptly, and the reader is expected to infer the connection. For example, the paper mentions that pathological gamblers show a mix of delay and probability discounting issues, but it does not show how the incompatibility between expected utility and exponential discounting helps us understand these results. It would help to explain these links more clearly.

5. The paper expands into quantum epistemics without providing enough structure

The extension into QBism and quantum probability is intriguing, but it is not clear how the incompatibility between normative principles in economics would affect the interpretation of probabilities in quantum theory. The paper would benefit from clearer examples or a focused explanation of how these two areas overlap.

Minor Issues

1. Some concepts need clearer early definitions

Terms such as ergodicity, dynamic consistency, subjective probability, and odds against are introduced quickly. Many readers will not be familiar with these ideas. A short glossary or a short explanation before using them would improve readability.

2. Some background sections are repetitive

The descriptions of hyperbolic discounting, delay discounting, and probability discounting overlap several times. Streamlining these sections would help keep the argument focused.

3. Some paragraphs are too long and contain several ideas mixed together

For example, the section that transitions from behavioral psychology to neuroscience and then to gambling research contains three different themes without clear separation. Breaking this into shorter sections would improve clarity.

4. More concrete examples would help readers understand the implications

The paper does not give simple numerical examples of how expected utility maps onto the hyperbolic form under the probability delay transformation. A short numeric example would make the argument more intuitive.

5. The discussion of limitations could be more complete

The author notes that the argument relies on Rachlin’s model but does not fully explore how sensitive the conclusions are to alternative formulations of probability discounting. For example, if a different psychological model of risk were used, would the same incompatibility appear?

Competing interests

The author declares that they have no competing interests.

Use of Artificial Intelligence (AI)

The author declares that they did not use generative AI to come up with new ideas for their review.