Decision Theory with Prospect Interference and Entanglement (1102.2738v1)

Abstract: We present a novel variant of decision making based on the mathematical theory of separable Hilbert spaces. This mathematical structure captures the effect of superposition of composite prospects, including many incorporated intentions, which allows us to describe a variety of interesting fallacies and anomalies that have been reported to particularize the decision making of real human beings. The theory characterizes entangled decision making, non-commutativity of subsequent decisions, and intention interference. We demonstrate how the violation of the Savage's sure-thing principle, known as the disjunction effect, can be explained quantitatively as a result of the interference of intentions, when making decisions under uncertainty. The disjunction effects, observed in experiments, are accurately predicted using a theorem on interference alternation that we derive, which connects aversion-to-uncertainty to the appearance of negative interference terms suppressing the probability of actions. The conjunction fallacy is also explained by the presence of the interference terms. A series of experiments are analysed and shown to be in excellent agreement with a priori evaluation of interference effects. The conjunction fallacy is also shown to be a sufficient condition for the disjunction effect and novel experiments testing the combined interplay between the two effects are suggested.

• The paper introduces Quantum Decision Theory (QDT), which applies the mathematical framework of quantum mechanics to model and explain cognitive biases like the disjunction effect and conjunction fallacy.
• QDT represents decisions in a Hilbert space, generalizing probability and proposing that phenomena like interference and entanglement explain deviations from classical predictions.
• Empirical analysis supports QDT's predictions, showing observed disjunction effect instances align with the model and suggesting implications for understanding cognitive biases and improving computational decision models.

Quantum Decision Theory: A Novel Approach to Decision-Making Under Uncertainty

The paper "Decision Theory with Prospect Interference and Entanglement" by V.I. Yukalov and D. Sornette introduces a novel variant of decision-making through Quantum Decision Theory (QDT). This approach extends the classical probability model by employing the mathematical framework of separable Hilbert spaces, traditionally used in quantum mechanics. QDT addresses the anomalies and fallacies in human decision-making, such as the disjunction effect and the conjunction fallacy, by incorporating concepts like intention interference and entanglement.

Key Concepts and Mathematical Framework

QDT diverges from the classical decision theory's reliance on expected utility models by generalizing probability into the field of nonlinear, subjective probabilities. Decisions are represented within the structure of a separable Hilbert space, facilitating a probabilistic definition similar to quantum mechanics' treatment of quantum states. Prospects, defined as sets of intended actions, exhibit properties of entanglement and interference—features not accounted for in classical theories.

The paper identifies two crucial phenomena: the disjunction effect and conjunction fallacy. The disjunction effect, defined as the deviation from the sure-thing principle, occurs when individuals choose differently under uncertainty compared to when outcomes are known. The conjunction fallacy illustrates the intuitive yet incorrect assumption that specific conditions (e.g., an overlap of events) are more probable than a single general one. QDT posits that these effects result from the interference of cognitive intentions.

Empirical Validation and Theoretical Implications

The authors support their theoretical propositions with empirical analysis. They demonstrate that observed instances of the disjunction effect align with their model, providing quantitative assessments of interference effects through the "interference-quarter law." The paper simulates experimental setups showing how decision-makers' behavior deviates from classical predictions, correlating with the interference effects forecasted by QDT.

The research further argues that the conjunction fallacy serves as a sufficient condition for the disjunction effect, suggesting an intrinsic link between these cognitive phenomena. This dual explanation proposes that interference alternation and uncertainty aversion—core tenets of QDT—jointly influence real-world decision-making. Their model offers predictions that could redefine how researchers understand risk, cognitive biases, and decision-making heuristics in psychologically uncertain scenarios.

Conclusion and Future Directions

By grounding its approach in quantum theoretical methods, QDT represents an innovative and rigorous reconceptualization of decision theory, particularly under uncertainty. While the model doesn't imply individuals possess quantum properties, it offers a sophisticated means of capturing the intricacies of human cognition and decision preference deviations at a nuanced level.

The presented concepts open pathways for further exploration into artificial intelligence's advancement in decision simulation and cognitive science's integration with quantum principles. Future inquiries may explore broader empirical validations across different demographics and contexts, further challenging or substantiating the model's applicability in real-world settings. The potential extension of QDT's framework to enhance predictive algorithms in cognitive modeling also presents an enticing opportunity for artificial intelligence and computational decision-making research.