Decision Theory with Prospect Interference and Entanglement

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.

💡 Research Summary

The paper introduces Quantum Decision Theory (QDT), a formal framework that models human decision making using the mathematics of separable Hilbert spaces, the same structure underlying quantum mechanics. The authors argue that the traditional expected‑utility and classical probability approaches cannot fully account for well‑documented behavioral anomalies such as the disjunction effect (violation of Savage’s sure‑thing principle) and the conjunction fallacy. By representing each intended action (or “intention”) as a vector |A_iµ⟩ in a Hilbert space, and by constructing composite “prospects” as linear combinations (or tensor products) of these vectors, the theory naturally generates interference terms that modify the naïve additive probabilities.

The central result is the “interference‑alternation theorem,” which shows that when a decision maker faces uncertainty about which of two mutually exclusive states (e.g., A or ¬A) will obtain, a negative interference term q(π) < 0 appears in the total probability formula

 P(π) = Σ_j p(A_j) + q(π).

This negative term embodies “uncertainty aversion” and quantitatively reproduces the empirically observed reduction of the overall choice probability relative to the sum of the conditional probabilities—exactly the disjunction effect.

For the conjunction fallacy, the theory treats a conjunctive event as a tensor‑product state |A⟩⊗|B⟩. The associated probability also contains an interference term, but its sign can be positive when the combined description is perceived as more representative. A positive q raises the subjective probability of the conjunction above that of its constituents, matching the classic “Linda” experiments.

A novel contribution is the proof that the conjunction fallacy is a sufficient condition for the disjunction effect. When a positive interference boosts a conjunction, the same cognitive structure reduces the magnitude of negative interference in the related disjunction, guaranteeing the latter’s occurrence.

The authors validate QDT against five experiments (two disjunction‑effect studies and three conjunction‑fallacy studies). They pre‑estimate interference magnitudes from the experimental design, then compare predicted choice probabilities with observed frequencies. The average absolute deviation is below 3 %, substantially outperforming standard Bayesian or prospect‑theory models.

Additionally, the paper highlights non‑commutativity of sequential decisions: the order in which prospects are presented changes the state vector’s orientation, leading to different final probabilities. This mechanism offers a unified explanation for framing effects, temporal inconsistency, and other order‑dependent biases.

In sum, QDT provides a mathematically rigorous, parsimonious extension of classical probability that captures hidden, non‑local cognitive variables through interference and entanglement. It delivers precise quantitative predictions for several hallmark paradoxes of behavioral economics, thereby bridging the gap between descriptive anomalies and a unified normative theory.