Материал: искусственный интеллект

express the transport of probability amplitude in line with cognitive theoretical principles need to be implemented. Formally the Hamiltonian needs to fulfill the Hermitian property H† = H, a property related to its original function as the energy operator, with real-valued energy eigenvalues.

Table A1.

Appendix B. Prospect Theory and formal Utility functions

The motivation for taking the second-stage gamble is quantified by the utility difference between taking the second-stage gamble and stopping the gamble after the first-stage gamble, Eqs. (14), (15), (13). In the Prospect Theory approach by Tversky and Shafir (1992) the utility function is given the formal expression of a power law. The utility expression xa, with a < 1, expresses the diminished utility of monetary value due to risk aversion with respect to gains. Similarly for the utility of losses x b, with b < 1, expressing risk seeking, and with b < a, implementing the principle that losses loom larger than gains. Using this power law utility expression, and anchoring the outcome of the first-stage gamble in Unknown condition to zero, Tversky and Shaffir provided a theoretical argument for how a participant’s choices would produce a violation of the STP.

Besides the observation of both an inflative and deflative violation of the LTP, our study also shows systematic decreasing tendencies to take the second-stage gamble when the payoff increases. In this section we show that this observed decreasing tendency is not only at odds with the power law form of the utility function (Tversky & Shafir, 1992), but also with the logarithmic utility form and the exponential utility form.

Power law Utility.

A gain x is evaluated with utility xa and a loss x at xb (x > 0, 0 b a 1). The motivation to take the second-stage gamble is expressed by the difference of Expected Utility and utility of stopping the gamble, Eqs. (14), (15), (13):

One can easily show that this approach predicts increasing probability to take the second-stage gamble when the payoff x increases (loss x/2 decreases). Moreover this is the case for both Known previous outcome conditions W and L. The derivatives of the utility differences in the W and L condition are

Both expressions are non negative, which indicates increasing motivation to take the second-stage gamble with increasing payoff X. According to the Prospect Theoretic approach (Tversky & Shafir, 1992) the probability to take the second-stage gamble should thus increase with payoff, for the W and L condition.

For the U condition the derivative of the utility motivation is

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J.B. Broekaert, et al. Cognitive Psychology 117 (2020) 101262

Our observations of the gamble probabilities, Figs. 4 and 3, show a persistent trend of diminished playing for higher payoffs.16

Logarithmic utility.

which can produce in principle decreasing probabilities with increasing payoff for larger X.

For a loss x < 0 the logarithmic utility is evaluated at U (x) = log( x + c), with c > 0. For continuity of the utility function over the domains of loss and gain we can set b = c. The motivation to take the second-stage gamble in the Lose condition is evaluated by the difference of Expected Utility and utility of stopping the gamble, Eq. (14),

which is negative and thus shows the utility increases with x , and larger losses should lead to a higher probability to gamble. Thus the logarithmic utility expression contradicts the observed gamble probabilities as well, Fig. 4.

Exponential utility.

A gain x is evaluated at exponential utility U (X ) = (1 e ax)/a, with a > 0. The motivation to take the second-stage gamble in the Win condition is evaluated by the difference of Expected Utility and utility of stopping the gamble, Eq. (14),

The latter expression is a third-degree polynomial in eax/2. It has three real-valued roots, one negative and two positive. The derivative has therefore two zero-points. One can easily see from the sign of the bracketed expression, that the first derivative will be positive for payoffs x > 2/a.

Again we conclude larger payoffs will lead to a higher probability to gamble. The exponential utility expression thus also contradicts the observed gamble probabilities, Fig. 4.

Appendix C. Supplementary material

Supplementary data associated with this article can be found, in the online version, at https://doi.org/10.1016/j.cogpsych.2019. 101262.

16 Our observations show that the probability to take the second-stage gamble on Lose is larger than on Win, Fig. 4. This feature is correctly predicted by Prospect Theory of Kahneman and Tversky (1979), and by Reyna and Brainerd (1991) in their fuzzy-trace theory. One can easily see that by Prospect Theory, Eqs. (B1) and (B2), for a given payoff x the gamble under L has more utility than under W:

since 0 b a 1 both summands are negative. Notice that the gamble probabilities in the original experiment of Tversky and Shafir (1992) do not adhere to this ordering, Table 1.

26

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The Spanish Journal of Psychology (2019), 22, e53, 1–9.

© Universidad Complutense de Madrid and Colegio Oficial de Psicólogos de Madrid doi:10.1017/sjp.2019.51

Primer on quantum cognition

Jerome R. Busemeyer1 and Zheng Wang2

1Indiana University Bloomington (USA)

2The Ohio State University (USA)

Abstract.  Quantum cognition is a new field in psychology, which is characterized by the application of quantum probability theory to human judgment and decision making behavior. This article provides an introduction that presents several examples to illustrate in a simple and concrete manner how to apply these principles to interesting psychological phenomena. Following each simple example, we present the general mathematical derivations and new predictions related to these applications.

Received 28 February 2019; Revised 21 August 2019; Accepted 25 October 2019

Keywords: conjunction fallacy, interference effects, order effects, quantum probability.

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Downloaded from https://www.cambridge.org/core. Auckland University of Technology, on 28 Dec 2019 at 13:01:39, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/sjp.2019.51