Exponential versus hyperbolic discounting of delayed outcomes: Risk and waiting time

Frequently, animals must choose between more immediate, smaller rewards and more delayed, but larger rewards. For example, they often must decide between accepting a smaller prey item versus continuing to search for a larger one, or between entering a leaner patch versus travelling to a richer patch that is further away. In both situations, choice of the more immediate, but smaller reward may be interpreted as implying that the value of the later reward is discounted; that is, the value of the later reward decreases as the delay to its receipt increases. This decrease in value may occur because of the increased risk involved in waiting for rewards, or because of the decreased rate of reward associated with increased waiting time. The present research attempts to determine the form of the relation between value and delay, and examines implications of this relation for mechanisms underlying risk-sensitive foraging. Two accounts of the relation between value and delay have been proposed to describe the decrease in value resulting from increases in delay: an exponential model and a hyperbolic model. Our research demonstrates that, of the two, a hyperbola-like discounting model consistently explains more of the variance in temporal discounting data at the group level and, importantly, at the individual level as well. We show mathematically that the hyperbolic model shares fundamental features with models of prey and patch choice. In addition, the present review highlights the implications of a psychological perspective for the behavioral biology of risk-sensitive foraging, as well as the implications of an ecological perspective for the behavioral psychology of risk-sensitive choice and decision-making.