Contextuality Derived from Minimal Decision Dynamics: Quantum Tug-of-War Decision Making

TL;DR

This work asks whether contextuality in decision making is an inherent feature of decision dynamics or merely a modeling artifact. It introduces QTOW, a quantum extension of the Tug-of-War model with conservation-based unitary updates and decision-induced disturbance, requiring an auxiliary memory degree of freedom. In a minimal qutrit internal state, QTOW enables KCBS-type contextuality probes, showing that no single non-contextual classical probability space can reproduce all decision-related statistics across contexts; thus quantum probability emerges as the structural minimum for adaptive decision dynamics. The results provide a principled bridge between physical update constraints and quantum cognition, predicting testable KCBS violations in single-system decision processes and offering a foundation for context-aware cognitive modeling without relying on data fitting. Overall, contextuality is presented as a dynamical property of decision learning, not merely a descriptive feature of human behavior, with implications for both theory and experimental design in cognitive and physical decision systems.

Abstract

Decision making often exhibits context dependence that challenges classical probability theory. While quantum cognition has successfully modeled such phenomena, it remains unclear whether quantum probability is merely a convenient assumption or a necessary consequence of decision dynamics. Here we present a theoretical framework in which contextuality arises generatively from physically grounded constraints on decision making. By developing a quantum extension of the Tug-of-War (TOW) model, we show that conservation-based internal state updates and measurement-induced disturbance preclude any non-contextual classical description with a single, unified internal state. Contextuality therefore emerges as a structural consequence of adaptive learning dynamics. We further show that the resulting measurement structure admits Klyachko-Can-Binicioglu-Shumovsky (KCBS)-type contextuality witnesses in a minimal single-system setting. These results indicate that quantum probability is not merely a descriptive convenience, but an unavoidable effective theory for adaptive decision dynamics.

Figures (10)

• Figure 1: Conceptual structure of QTOW. A qutrit internal state evolves by conservation-preserving updates. Decision making is a measurement, and learning is driven by conservation-preserving unitary updates. To sustain adaptive learning under measurement-induced disturbance, an auxiliary degree of freedom enabling incompatible access is structurally required. This necessity leads to contextual statistics independently of any specific contextuality test. KCBS-type configurations provide one explicit diagnostic realization of this contextuality, but are not assumed in the learning dynamics themselves.
• Figure 2: Simplified schematic of the QTOW decision-making apparatus. The single-photon optical path is folded into two rows (top and middle) to avoid horizontal overflow. An interferometer implements the $R_{02}$-type mixing associated with the memory parameter $\mu_t$, while the polarization analyzer (PA) implements the A/B decision. Outcomes and rewards update device parameters for the next trial.
• Figure 3: Time-line of a single QTOW trial. A fresh photon is injected in each trial. The learning history is stored in the device control parameters: the PA angle $\alpha_t$ for A/B decision and the u--d mixing ratio $\mu_t$ encoding environmental strength. The u--d interference (updating $\mu_t$) is applied only after the decision, hence it does not bias the within-trial A/B fairness determined by $\alpha_t$.
• Figure 4: Conceptual architecture of the QTOW decision process. Decision making is performed by measuring polarization on the upper path, while reward-dependent unitary updates are applied via classical feedback to the preparation of the next photon. Optional contextual probes are implemented by path interference between $|u,H\rangle$ and $|d,H\rangle$, introducing incompatible measurement contexts.
• Figure 5: Conceptual structure of QTOW. A qutrit internal state evolves by conservation-preserving updates. Decision making is a measurement, and an optional contextual probe $P_i$ (KCBS) can be inserted before the decision. Because probing and decision measurements are generally incompatible, inserting $P_i$ disturbs the state and changes decision statistics, enabling KCBS contextuality in a single decision-making system. Contextuality disappears if probes are restricted to commuting operations. The auxiliary degree of freedom functions as a quantum memory enabling adaptive estimation under measurement-induced disturbance.

Theorems & Definitions (7)

• Proposition 1: Classical reducibility under full state accessibility
• Definition 1: Quantum TOW internal state
• Proposition 2: Unitary realization of the TOW conservation law
• Theorem 3: Inaccessibility and disturbance of the internal state
• Definition 2: Contextual probing
• Theorem 4: Emergence of contextuality in QTOW
• corollary 1: Impossibility of a global classical description