AUTHOR=Takemura Kazuhisa , Murakami Hajime 

TITLE=Probability Weighting Functions Derived from Hyperbolic Time Discounting: Psychophysical Models and Their Individual Level Testing

JOURNAL=Frontiers in Psychology

VOLUME=7

YEAR=2016

URL=https://www.frontiersin.org/journals/psychology/articles/10.3389/fpsyg.2016.00778

DOI=10.3389/fpsyg.2016.00778

ISSN=1664-1078

ABSTRACT=<p>A probability weighting function (<italic>w</italic>(<italic>p</italic>)) is considered to be a nonlinear function of probability (<italic>p</italic>) in behavioral decision theory. This study proposes a psychophysical model of probability weighting functions derived from a hyperbolic time discounting model and a geometric distribution. The aim of the study is to show probability weighting functions from the point of view of waiting time for a decision maker. Since the expected value of a geometrically distributed random variable <italic>X</italic> is 1/<italic>p</italic>, we formulized the probability weighting function of the expected value model for hyperbolic time discounting as <italic>w</italic>(<italic>p</italic>) = (1 − <italic>k</italic> log <italic>p</italic>)<sup>−1</sup>. Moreover, the probability weighting function is derived from Loewenstein and Prelec's (<xref ref-type="bibr" rid="B11">1992</xref>) generalized hyperbolic time discounting model. The latter model is proved to be equivalent to the hyperbolic-logarithmic weighting function considered by Prelec (<xref ref-type="bibr" rid="B13">1998</xref>) and Luce (<xref ref-type="bibr" rid="B12">2001</xref>). In this study, we derive a model from the generalized hyperbolic time discounting model assuming Fechner's (<xref ref-type="bibr" rid="B4">1860</xref>) psychophysical law of time and a geometric distribution of trials. In addition, we develop median models of hyperbolic time discounting and generalized hyperbolic time discounting. To illustrate the fitness of each model, a psychological experiment was conducted to assess the probability weighting and value functions at the level of the individual participant. The participants were 50 university students. The results of individual analysis indicated that the expected value model of generalized hyperbolic discounting fitted better than previous probability weighting decision-making models. The theoretical implications of this finding are discussed.</p>