Where V is the subjective value of the reward, A is the amount of the reward, b is a parameter that reflects the rate of discounting (when s is constant), X represents the independent variable, and the exponent s is a nonlinear scaling parameter whose interpretation depends on whether the equation is used to describe delay or probability discounting.
The hyperboloid discounting model favored by Green and Myerson (2004 Myerson and Green, 1995) to describe the discounting of both delayed and probabilistic rewards is of the form: Two such hyperboloid models have been proposed, and the present study examines the theoretical and empirical bases for these models. However, two-parameter hyperboloids often provide an even better fit, and the improvement in variance accounted for by such models is frequently significantly greater than would be expected based merely on the addition of a free parameter ( Myerson and Green, 1995). Considerable research has examined the mathematical form of the discounting function (for a review, see Green and Myerson, 2004), and there is an emerging consensus that an exponential function ( Samuelson, 1937) provides a relatively poor description of discounting data compared to a simple one-parameter hyperbola ( Mazur, 1987). In both cases, the value of a reward is assumed to be discounted with increasing delay or decreasing likelihood. Probability discounting refers to the decrease in the subjective value of a reward as the likelihood of its receipt decreases.
For probability discounting, however, the exponent in both models increased as the probabilistic amount increased-a finding inconsistent with the scaling interpretation.ĭelay discounting refers to the decrease in the subjective value of a reward as the time until its receipt increases. For delay discounting, the exponent in both models did not vary as a function of delayed amount, consistent with the psychophysical scaling interpretation. Both the Rachlin model and the Green and Myerson model provided very good fits to delay and probability discounting of both small and large amounts at both the group and individual levels (all R 2s >. In particular, we examined the effects of amount on the exponents in the two hyperboloid models of delay and probability discounting in order to evaluate key theoretical predictions of the standard psychophysical scaling interpretation of these exponents. Here, we extend this effort by comparing fits of the two-parameter hyperboloid models to data from a larger sample of participants ( N = 171) who discounted probabilistic as well as delayed rewards.
Previously, we ( McKerchar et al., 2009) showed that two-parameter hyperboloid models ( Green and Myerson, 2004 Rachlin, 2006) provide significantly better fits to delay discounting data than simple, one-parameter hyperbolic and exponential models.
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