Delay-amount tradeoffs in choices by pigeons and rats: hyperbolic versus exponential discounting

Abstract

An adjusting-delay procedure was used to study the choices of pigeons and rats when both delay and amount of reinforcement were varied. In different conditions, the choice alternatives included one versus two reinforcers, one versus three reinforcers, and three versus two reinforcers. The delay to one alternative (the standard alternative) was kept constant in a condition, and the delay to the other (the adjusting alternative) was increased or decreased many times a session so as to estimate an indifference point--a delay at which the two alternatives were chosen about equally often. Indifference functions were constructed by plotting the adjusting delay as a function of the standard delay for each pair of reinforcer amounts. The experiments were designed to test the prediction of a hyperbolic decay equation that the slopes of the indifference functions should increase as the ratio of the two reinforcer amounts increased. Consistent with the hyperbolic equation, the slopes of the indifference functions depended on the ratios of the two reinforcer amounts for both pigeons and rats. These results were not compatible with an exponential decay equation, which predicts slopes of 1 regardless of the reinforcer amounts. Combined with other data, these findings provide further evidence that delay discounting is well described by a hyperbolic equation for both species, but not by an exponential equation. Quantitative differences in the y-intercepts of the indifference functions from the two species suggested that the rate at which reinforcer strength decreases with increasing delay may be four or five times slower for rats than for pigeons.

Keywords: exponential discounting; hyperbolic discounting; keypeck; lever press; pigeons; rats; reinforcer amount; reinforcer delay; self-control.

Fig. 1

Predictions of the hyperbolic equation (left panel) and the exponential equation (right panel) are shown for three different ratios of reinforcer amounts.

Fig. 2

Mean adjusting delays at each indifference point are shown for each pigeon in Experiment 1. The lines are regression lines fitted to the data points. The slopes and y-intercepts of the lines and percent variance accounted for are given in Table 3.

Fig. 3

Group means of the indifference points are shown for the pigeons in Experiment 1. The lines are the best-fitting predictions of Equation 4, which were obtained with the following parameter values: K  =  0.64, A₂  =  1.88, and A₃  =  2.43. See text for details.

Fig. 4

Mean adjusting delays at each indifference point are shown for each rat in Experiment 2. The lines are regression lines fitted to the data points. The slopes and y-intercepts of the lines and percent variance accounted for are given in Table 5.

Fig. 5

Group means of the indifference points are shown for the rats in Experiment 2. The lines are the best-fitting predictions of Equation 4, which were obtained with the following parameter values: K  =  0.14, A₂  =  2.00, and A₃  =  2.82. See text for details.