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Projective Expected Utility

Pierfrancesco La Mura

TL;DR

This paper introduces a projective generalization of expected utility inspired by quantum probability to address classic decision-theoretic paradoxes such as Allais' and Ellsberg's. By representing preferences with a symmetric payoff matrix $U$ and a quadratic form $u(x)=x' U x$ over lotteries on the unit sphere, it allows nontrivial risk interactions while preserving tractable analysis; a subjective extension using states of nature and a mapping to subjective outcomes yields a representation in terms of a mixture $\sum_{s} \pi(s) f_s' U f_s$. The framework accommodates both objective risk and subjective uncertainty, resolves key paradoxes, and guarantees the existence of equilibria in finite games, with extensions to complex Hermitian cases for quantum uncertainty. Overall, it provides a unified, operational approach to decision under risk and ambiguity that integrates quantum-inspired structure with Anscombe–Aumann foundations.

Abstract

Motivated by several classic decision-theoretic paradoxes, and by analogies with the paradoxes which in physics motivated the development of quantum mechanics, we introduce a projective generalization of expected utility along the lines of the quantum-mechanical generalization of probability theory. The resulting decision theory accommodates the dominant paradoxes, while retaining significant simplicity and tractability. In particular, every finite game within this larger class of preferences still has an equilibrium.

Projective Expected Utility

TL;DR

This paper introduces a projective generalization of expected utility inspired by quantum probability to address classic decision-theoretic paradoxes such as Allais' and Ellsberg's. By representing preferences with a symmetric payoff matrix and a quadratic form over lotteries on the unit sphere, it allows nontrivial risk interactions while preserving tractable analysis; a subjective extension using states of nature and a mapping to subjective outcomes yields a representation in terms of a mixture . The framework accommodates both objective risk and subjective uncertainty, resolves key paradoxes, and guarantees the existence of equilibria in finite games, with extensions to complex Hermitian cases for quantum uncertainty. Overall, it provides a unified, operational approach to decision under risk and ambiguity that integrates quantum-inspired structure with Anscombe–Aumann foundations.

Abstract

Motivated by several classic decision-theoretic paradoxes, and by analogies with the paradoxes which in physics motivated the development of quantum mechanics, we introduce a projective generalization of expected utility along the lines of the quantum-mechanical generalization of probability theory. The resulting decision theory accommodates the dominant paradoxes, while retaining significant simplicity and tractability. In particular, every finite game within this larger class of preferences still has an equilibrium.

Paper Structure

This paper contains 12 sections, 4 theorems, 6 equations, 1 figure.

Table of Contents

  1. Introduction
  2. von Neumann - Morgenstern Expected Utility
  3. The "Double-Slit Paradox" of Decision Theory
  4. Ellsberg's Paradox
  5. Projective Expected Utility
  6. Subjective Formulation
  7. Properties of the Representation
  8. Example: Objective Uncertainty
  9. Example: Subjective Uncertainty
  10. Multi-Agent Decisions and Equilibrium
  11. Conclusions
  12. References

Key Result

Theorem 1

(von Neumann and Morgenstern) Axioms $1$ and $2$ are jointly equivalent to the existence of a functional $u : \Delta \rightarrow R$ which represents $\succeq$ such that, for all $p \in \Delta$,

Figures (1)

  • Figure 1: Examples of indifference maps on the probability triangle

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4