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arXiv:0708.2993

Quantum-like Description of Probabilistic Data from Shafir-Tversky Experiments: evidence of trigonometric and hyperbolic (!) interference

Andrei Yu. Khrennikov

Published Aug 22, 2007• 0 citations
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Abstract

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 ($\cos$-type) interference, but the first one exhibits so called hyperbolic ($\cosh$-type) interference. We discuss QL processing of information by cognitive systems, in particular, QL decision making as well as classical and QL rationality.

DOI
10.48550/arxiv.0708.2993

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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 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.
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We consider the following model of decision-making by cognitive systems. We present an algorithm -- quantum-like representation algorithm (QLRA) -- which provides a possibility to represent probabilistic data of any origin by complex probability amplitudes. Our conjecture is that cognitive systems developed the ability to use QLRA. They operate with complex probability amplitudes, mental wave functions. Since the mathematical formalism of QM describes as well (under some generalization) processing of such quantum-like (QL) mental states, the conventional quantum decision-making scheme can be used by the brain. We consider a modification of this scheme to describe decision-making in the presence of two ``incompatible'' mental variables. Such a QL decision-making can be used in situations like Prisoners Dilemma (PD) as well as others corresponding to so called disjunction effect in psychology and cognitive science.
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