TLDR: This research paper introduces a novel theory of coherent preference that significantly relaxes traditional assumptions like transitivity, Archimedeanness, boundedness, and continuity. It demonstrates that any preference system satisfying a coherence requirement can be extended to a complete system and represented by utility in an ordered field extension of real numbers, utilizing formal power series. This framework unifies numerical probability and expected utility, extending classical theorems and providing a more flexible foundation for decision theory.
A groundbreaking new theory of preference is set to redefine how we understand decision-making and utility. This research, detailed in the paper “FORMAL POWER SERIES REPRESENTATIONS IN PROBABILITY AND EXPECTED UTILITY THEORY” by Arthur Paul Pedersen and Samuel Allen Alexander, challenges long-held restrictions in orthodox economic and decision theories.
Traditionally, models of preference and utility have relied on strict assumptions such as transitivity (if A is preferred to B, and B to C, then A must be preferred to C), Archimedeanness (no quantity is infinitely large or small compared to another), boundedness, and continuity. However, this new theory introduces a more general framework that surrenders these limitations.
The core innovation is the demonstration that any preference system, provided it meets a specific “coherence” requirement (analogous to de Finetti’s foundational concept for probability), can be extended to a complete system of preferences. Crucially, this complete system can then be represented by a form of “utility” that exists not just in the familiar real numbers, but in a richer mathematical structure known as an “ordered field extension of the reals.” This extension is achieved through the use of “formal power series,” which are essentially infinite sums that allow for the representation of quantities that might be infinitely large or infinitesimally small relative to standard real numbers.
This approach has significant implications, as it extends classical mathematical results like H¨older’s Theorem and strengthens Hahn’s Embedding Theorem. It offers a unified treatment of numerical probability and expected utility, relaxing the stringent standards imposed by the orthodox canon, which includes the staple theories of de Finetti, von Neumann and Morgenstern, Savage, and Anscombe and Aumann.
The paper illustrates its concepts by focusing on de Finetti’s foundations of probability and expected utility, making the application of its central result more accessible without delving into overly complex mathematical detours. The concept of “coherence” is paramount, ensuring that an individual’s preferences are consistent and do not lead to guaranteed net losses. The theory shows that a coherent preference system can be quantified by an “F-valued expectation,” where F is this extended ordered field, allowing for a numerical representation even when traditional real-valued utility functions fall short.
This work not only provides rigorous definitions and proofs for previously asserted claims regarding the Hahn Embedding Theorem and the use of Hahn fields of formal power series but also includes practical examples that illuminate the theory’s broad scope and power, such as extending real-valued expectations and representing comparative probabilities.
Also Read:
- New Algorithmic Tests Uncover Inconsistencies in Multi-Criteria Decision Making
- Beyond Utility: How AI Can Be Designed to Empower Humans
For more in-depth details, you can refer to the full research paper: Formal Power Series Representations in Probability and Expected Utility Theory.
