Thinking and Deciding

484 DECISIONS ABOUT THE FUTURE
Figure 19.2: Subjective utility of two different rewards as a function of the time at
which a decision is made. (The utility scale has no units. Rewards are available at
times T1 and T2, respectively.)
of animals who are faced with repeated choices varying in magnitude and delay over
thousands of trials (Chung and Herrnstein, 1967; Herrnstein and Prelec, 1991).
Note that for this effect to occur, the curve on the graph has to have a certain
shape. It has to be very steep just as you approach the time of the reward. The
economists’ idea of a constant discount rate must be incorrect. That idea implies that
the utility of a reward increases by a constant percentage for every unit of time; that
is, the utility at the end of the time unit is some multiple (greater than 1) of the utility
at the beginning of the unit. If this were true, any reward with a higher utility at the
beginning of each time unit would be higher at the end as well, and there would be
no crossover of the sort shown on the graph. Therefore, Ainslie’s theory requires a
particular shape for the curve relating utility to time. 8
This theory predicts that preferences will reverse as a function of the time at
which they are made (relative to the time the rewards are available). Therefore, delay
independence will be violated. Ainslie and Haendel (1983) tested this prediction by
asking subjects to choose between two prizes to be paid by a reliable company. The
larger prize always had a greater delay. For example, the choice could be between
$50 immediately and $100 in six months, or between $50 in three months and $100
8 Ainslie recommends replacing the usual exponential function with a hyperbola. The hyperbola is
consistent with a number of experiments discussed under the category of the “matching law” (see Rachlin,
Logue, Gibbon, and Frankel, 1986, who also provide an interesting account of the certainty effect by
interpreting probability as delay). However, other functions would work.