Mathematical Structure of Quantum Decision Theory

Abstract

Following the ideas of Bohr, Von Neumann, and Benioff, we formulate quantum decision theory (QDT) as the quantum-mechanical theory of measurement for probability operators. QDT captures the effect of superposition of composite prospects, including many incorporated intentions. It is based on the hypothesis that the thought processes of real human beings involved in the definition and analysis of alternative prospects and scenarios do not necessarily separate them according to the recipes of standard probability theory and of classical utility theory. Our QDT formalizes systematically a broader class of decision making processes in which prospects can interact, interfere and remain entangled. The mathematical QDT is developed so as to be applicable to real-life decision making processes. We demonstrate that all known anomalies and paradoxes documented in the context of classical utility theory are reducible to just a few mathematical archetypes, all of which finding straightforward explanations in the framework of QDT. The following paradoxes are addressed: the Allais paradox (or compatibility paradox), the independence paradox, the Ellsberg paradox, the inversion paradox, the invariance violation described by Kahneman and Tversky, the certainty effects, the disjunction effect (or violation of the Savage sure-thing principle), the conjunction fallacy, and the isolation effect (or focusing, availability, salience, framing, or elicitation effects). Interference terms, which are essential for resolving the paradoxes, quantify the aversion of human beings to uncertainty and/or to perceived potential loss resulting from their decisions.