A comparison of four models of delay discounting in humans

The present study compared four prominent models of delay discounting: a one-parameter exponential decay, a one-parameter hyperbola [Mazur, J.E., 1987. An adjusting procedure for studying delayed reinforcement. In: Commons, M.L., Mazur, J.E., Nevin, J.A., Rachlin, H. (Eds.), Quantitative Analyses of Behavior: The Effect of Delay and of Intervening Events on Reinforcement Value, vol. 5. Erlbaum, Hillsdale, NJ, pp. 55-73], a two-parameter hyperboloid in which the denominator is raised to a power [Green, L., Myerson, J., 2004. A discounting framework for choice with delayed and probabilistic rewards. Psychol. Bull. 130, 769-792], and a two-parameter hyperbola in which delay is raised to a power [Rachlin, H., 2006. Notes on discounting. J. Exp. Anal. Behav. 85, 425-435]. Sixty-four college undergraduates made choices between hypothetical monetary rewards, one immediate and one delayed, and the fit of the four models to their data was assessed. All four equations accounted for a large proportion of the variance at both the group and the individual levels, but the exponents of both two-parameter models were significantly less than 1.0 at the group level, and frequently so at the individual level. Taken together, these results strongly suggest that more than one parameter is needed to accurately describe delay discounting by humans. Notably, both the Rachlin and the Green and Myerson models accounted for more than 99% of the variance at the group level and for 96% of the variance in the median individual. Because both models provide such good descriptions of the data, model selection will need to be based on other grounds.