A new foundation of quantum decision theory

TL;DR

A new foundation of quantum decision theory is proposed, based on Helland's epistemic foundation for quantum theory, where an observer $C$ interacts with an inaccessible variable $\phi$ and two maximal accessible decision variables $\theta$ and $\eta$ to yield an $r$-dimensional Hilbert space $\mathcal{H}$ with associated self-adjoint operators. The Born rule is derived from a weak likelihood-principle and a rational-epistemic setting, showing that there exists a density operator $\rho$ such that $q(F|\tau)=\mathrm{trace}(\rho(\tau)F)$ for likelihood effects $F$, and in particular $P(\theta^b=u_j^b|\theta^a=u_k^a)=|<a;k|b;j>|^2$. The framework treats complementary maximal decision variables and shows that quantum probabilities can induce non-classical features, including violation of the law of total probability and order-dependent outcomes. The approach is illustrated with a doctor-patient example and discussed in the context of learning, culture, everyday decisions, and decisions by political leaders. The work highlights the epistemic interpretation of quantum probabilities and their potential to illuminate fast, intuition-driven decision making while delineating the scope and limitations of applying QDT in complex, real-world settings.

Abstract

Quantum decision theory is introduced here, and new basis for this theory is proposed. It is first based upon the author's general arguments for the Hilbert space formalism in quantum theory, next on arguments for the Born rule, that is, the basis for calculating quantum probabilities. A basic notion behind the quantum theory foundation is that of theoretical variables, that are divided into accessible and inaccessible ones. This is here specialized to decision variables. It is assumed that all accessible variables can be seen as functions of a specific inaccessible variable. Another assumption is that there exist two different maximal accessible theoretical variables in the given situation. Two basic assumptions behind the Born rule are 1) the likelihood principle, 2) the actor in question has motivations that can be modeled by a hypothetical perfectly rational higher being. The theory is illustrated by a medical example. Finally, a brief discussion of decision processes is given.

Figures (1)

• Figure 1: Geometry of two spaces.