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Open Systems & Information Dynamics

Quantum Markov Model for Data from Shafir-Tversky Experiments in Cognitive Psychology

Luigi Accardi, Andrei Yu. Khrennikov, Masanori Ohya

Published Nov 30, 2009• 49 citations
Publisher

Abstract

We analyze, from the point of view of quantum probability, statistical data from two interesting experiments, done by Shafir and Tversky [1, 2] in the domain of cognitive psychology. These are gambling experiments of Prisoner Dilemma type. They have important consequences for economics, especially for the justification of the Savage "Sure Thing Principle" [3] (implying that agents of the market behave rationally). Data from these experiments were astonishing, both from the viewpoint of cognitive psychology and economics and probability theory. Players behaved irrationally. Moreover, all attempts to generate these data by using classical Markov model were unsuccessful. In this note we show (by inventing a simple statistical test — generalized detailed balance condition) that these data are non-Kolmogorovian. We also show that it is neither quantum (i.e., it cannot be described by Dirac-von Neumann model). We proceed towards a quantum Markov model for these data.

DOI
10.1142/s123016120900027x

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arXiv:1603.03079•Feb 11, 2016•1 citation
We analyze, from the point of view of quantum probability, statistical data from two interesting experiments, done by Shafir and Tversky [1, 2] in the domain of cognitive psychology. These are gambling experiments of Prisoner Dilemma type. They have important consequences for economics, especially for the justification of the Savage "Sure Thing Principle" [3] (implying that agents of the market behave rationally). Data from these experiments were astonishing, both from the viewpoint of cognitive psychology and economics and probability theory. Players behaved irrationally. Moreover, all attempts to generate these data by using classical Markov model were unsuccessful. In this note we show (by inventing a simple statistical test — generalized detailed balance condition) that these data are non-Kolmogorovian. We also show that it is neither quantum (i.e., it cannot be described by Dirac-von Neumann model). We proceed towards a quantum Markov model for these data.
Two experimental tasks in psychology, the two-stage gambling game and the Prisoner's Dilemma game, show that people violate the sure thing principle of decision theory. These paradoxical findings have resisted explanation by classical decision theory for over a decade. A quantum probability model, based on a Hilbert space representation and Schrödinger's equation, provides a simple and elegant explanation for this behaviour. The quantum model is compared with an equivalent Markov model and it is shown that the latter is unable to account for violations of the sure thing principle. Accordingly, it is argued that quantum probability provides a better framework for modelling human decision-making.
In cognitive psychology, some experiments of games were reported [1, 2, 3, 4], and these demonstrated that real players did not use the "rational strategy" provided by classical game theory. To discuss probabilities of such "irrational choice", recently, we proposed a decision-making model which is based on the formalism of quantum mechanics [5, 6, 7, 8]. In this paper, we briefly explain the above model and calculate the probability of irrational choice in several prisoner's dilemma (PD) games.
In this paper we present quantum-like (QL) representation of the Shafir-Tversky statistical effect. We apply so called contextual approach. The Shafir-Tversky effect is considered as a consequence of combination of a number of incompatible contexts which are involved e.g. in Prisoner's Dilemma or in more general games inducing the disjunction effect. As a consequence, the law of total probability is violated for experimental data obtained by Shafir and Tversky (1992) as well as Tversky and Shafir (1992). Moreover, we can find a numerical measure of contextual incompatibility (so called coefficient of interference) as well as represent contexts which are involved in Prisoner's Dilemma (PD) by probability amplitudes -- normalized vectors (``mental wave functions''). We remark that statistical data from Shafir and Tversky (1992) and Tversky and Shafir (1992) experiments differ crucially from the point of view of mental interference. The second one exhibits the conventional trigonometric (cos⁡\coscos-type) interference, but the first one exhibits so called hyperbolic (cosh⁡\coshcosh-type) interference. We discuss QL processing of information by cognitive systems, in particular, QL decision making as well as classical and QL rationality.
We show that (in contrast to rather common opinion) the domain of applications of the mathematical formalism of quantum mechanics is not restricted to physics. This formalism can be applied to the description of various quantum-like (QL) information processing. In particular, the calculus of quantum (and more general QL) probabilities can be used to explain some paradoxical statistical data which was obtained in psychology and cognitive science. We consider the QL description of prisoners dilemma (PD) and so called disjunction effect (violation of Savage's sure thing principle which plays the fundamental role in modern economics).
We proceed towards an application of the mathematical formalism of quantum mechanics to cognitive psychology — the problem of decision-making in games of the Prisoners Dilemma type. These games were used as tests of rationality of players. Experiments performed in cognitive psychology by Shafir and Tversky [1, 2], Croson [3], Hofstader [4, 5] demonstrated that in general real players do not use "rational strategy" provided by classical game theory; this psychological phenomenon was called the disjunction effect. We elaborate a model of quantum-like decision making which can explain this effect ("irrationality" of plays). Our model is based on quantum information theory. The main result of this paper is the derivation of Gorini-Kossakowski-Sudarshan-Lindblad equation whose equilibrium solution gives the quantum state used for decision making. It is the first application of this equation in cognitive psychology.
Recently the mathematical formalism of quantum mechanics, especially methods of quantum probability theory, started to be widely used in a variety of applications outside of physics, e.g., cognition and psychology as well as economy and finances. To distinguish such models from genuine quantum physical models, they often called quantum-like (although often people simply speak about, e.g., "quantum cognition"). These novel applications generate a number of foundational questions. Nowadays we can speak about a new science - foundations of quantum-like modeling. At the first stage this science was mainly about comparison of classical and quantum models, mainly in the probabilistic setting. It was found that statistical data from cognitive psychology violate some basic constraints posed on data by classical probability theory (Kolmogorov, 1933); in particular, the constraints given by the formula of total probability and Bell's type inequalities. Recently another question attracted some attention. In spite of real success in applications, there are no reason to believe that the quantum probability would cover completely all problems of, e.g., cognition. May be more general probability models have to be explored. A similar problem attracted a lot of attention in foundations of quantum physics culminating in a series of experiments to check Sorkin's equality for the triple-slit experiment by Weihs' group. In this note we present a similar test in the cognitive experimental setting. Performance of this test would either give further confirmation of the adequacy of the quantum probability model to cognitive applications or rejection of the conventional quantum model. Thus this note opens the door for a series of exciting experimental tests for the quantum-like model of cognition.