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A duality between utility transforms and probability distortions
Christopher P Chambers1, Peng Liu2, Ruodu Wang3
1Department of Economics, Georgetown University, Washington, USA.
Theory and Decision
|November 24, 2025
Summary
This study reveals a mathematical duality between utility transforms and probability distortions, fundamental to decision-making under risk. These findings simplify the axiomatic foundations of decision theories.
Area of Science:
- Decision Theory
- Mathematical Economics
- Risk Analysis
Background:
- Utility transforms and probability distortions are foundational to decision under risk theories.
- Classic theories include expected utility, dual utility, and rank-dependent utility.
Purpose of the Study:
- Establish a mathematical duality between utility transforms and probability distortions.
- Simplify the axiomatic characterization of these transforms.
Main Methods:
- Mathematical analysis to establish duality.
- Axiomatic characterization of transforms.
- Exploration of commutation properties.
Main Results:
- A direct mathematical duality exists between utility transforms and probability distortions.
- Probability distortions commute with utility transforms, and vice versa.
- These characterizations require no additional conditions, simplifying axiomatization.
Conclusions:
- The established duality provides a unified framework for understanding decision under risk.
- Rank-dependent utility transforms can be characterized by set commutation under monotonicity.
- This work offers a more parsimonious axiomatic basis for decision theories.
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