There has been a strong recent interest in applying quantum theory (QT) outside physics, including in cognitive science. We analyze the applicability of QT to two basic properties in opinion polling. The first property (response replicability) is that, for a large class of questions, a response to a given question is expected to be repeated if the question is posed again, irrespective of whether another question is asked and answered in between. The second property (question order effect) is that the response probabilities frequently depend on the order in which the questions are asked. Whenever these two properties occur together, it poses a problem for QT. The conventional QT with Hermitian operators can handle response replicability, but only in the way incompatible with the question order effect. In the generalization of QT known as theory of positive-operator-valued measures (POVMs), in order to account for response replicability, the POVMs involved must be conventional operators. Although these problems are not unique to QT and also challenge conventional cognitive theories, they stand out as important unresolved problems for the application of QT to cognition. Either some new principles are needed to determine the bounds of applicability of QT to cognition, or quantum formalisms more general than POVMs are needed.

In quantum physics there are well-known situations when measurements of the same property in different contexts (under different conditions) have the same probability distribution, but cannot be represented by one and the same random variable. Such systems of random variables are called contextual. More generally, true contextuality is observed when different contexts force measurements of the same property (in psychology, responses to the same question) to be more dissimilar random variables than warranted by the difference of their distributions. The difference in distributions is itself a form of context-dependence, but of another nature: it is attributable to direct causal influences exerted by contexts upon the random variables. The Contextuality-by-Default (CbD) theory allows one to separate true contextuality from direct influences in the overall context-dependence. The CbD analysis of numerous previous attempts to demonstrate contextuality in human judgments shows that all context-dependence in them can be accounted for by direct influences, with no true contextuality present. However, contextual systems in human behavior can be found. In this paper we present a series of crowdsourcing experiments that exhibit true contextuality in simple decision making. The design of these experiments is an elaboration of one introduced in the "Snow Queen" experiment (Decision 5, 193-204, 2018), where contextuality was for the first time demonstrated unequivocally.

We present a quantum-like model of decision making in games of the Prisoner's Dilemma type. By this model the brain processes information by using representation of mental states in a complex Hilbert space. Driven by the master equation the mental state of a player, say Alice, approaches an equilibrium point in the space of density matrices (representing mental states). This equilibrium state determines Alice's mixed (i.e., probabilistic) strategy. We use a master equation which in quantum physics describes the process of decoherence as the result of interaction with environment. Thus our model is a model of thinking through decoherence of the initially pure mental state. Decoherence is induced by the interaction with memory and the external mental environment. We study (numerically) the dynamics of quantum entropy of Alice's mental state in the process of decision making. We also consider classical entropy corresponding to Alice's choices. We introduce a measure of Alice's diffidence as the difference between classical and quantum entropies of Alice's mental state. We see that (at least in our model example) diffidence decreases (approaching zero) in the process of decision making. Finally, we discuss the problem of neuronal realization of quantum-like dynamics of in the brain; especially roles played by lateral prefrontal cortex or and orbitofrontal cortex.

The recent wave of interest to modeling the process of decision making with the aid of the quantum formalism gives rise to the following question: ‘How can neurons generate quantum-like statistical data?’ (There is a plenty of such data in cognitive psychology and social science). Our model is based on quantum-like representation of uncertainty in generation of action potentials. This uncertainty is a consequence of complexity of electrochemical processes in the brain; in particular, uncertainty of triggering an action potential by the membrane potential. Quantum information state spaces can be considered as extensions of classical information spaces corresponding to neural codes; e.g., 0/1, quiescent/firing neural code. The key point is that processing of information by the brain involves superpositions of such states. Another key point is that a neuronal group performing some psychological function F is an open quantum system. It interacts with the surrounding electrochemical environment. The process of decision making is described as decoherence in the basis of eigenstates of F. A decision state is a steady state. This is a linear representation of complex nonlinear dynamics of electrochemical states. Linearity guarantees exponentially fast convergence to the decision state.

In this paper, we introduce a new model of selection behavior under risk that describes an essential cognitive process for comparing values of objects and making a selection decision. This model is constructed by the quantum-like approach that employs the state representation specific to quantum theory, which has the mathematical framework beyond the classical probability theory. We show that our quantum approach can clearly explain the famous examples of anomalies for the expected utility theory, the Ellsberg paradox, the Machina paradox and the disparity between WTA and WTP. Further, we point out that our model mathematically specifies the characteristics of the probability weighting function and the value function, which are basic concepts in the prospect theory.