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Green and Myerson Page 12 of 48
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emphasizes the use of parallel experimental procedures and mathematical modeling techniques
to rigorously evaluate the case for similarities between temporal and probability discounting.
We would emphasize that the discounting framework, through its use of parallel approaches,
also is well suited to reveal theoretically meaningful differences between the two types of
discounting. When different procedures and analytical techniques are used, as is typically the
case, any apparent differences observed between temporal and probability discounting could
be due to the procedural and analytic approaches rather than to true differences in the underlying
processes.
The conceptual similarity of discounting involving delayed and probabilistic rewards may be
seen by considering reversals in preference. Recall that it was argued previously that preference
reversals with delayed rewards occur because the subjective value of the smaller, sooner reward
increases more than the subjective value of the larger, later reward as the delays to both are
decreased equally. Similarly, preference reversals with probabilistic rewards might be
explained by assuming that the subjective value of the smaller, less risky reward decreases
more than the subjective value of the larger, more risky reward if the probabilities of winning
decrease. As was true for temporal discounting, however, preference reversals with risky
rewards do not greatly constrain the mathematical form of the probability discounting function
(although they do preclude a simple expected value model of risky choice).
Given what is now known about the form of the temporal discounting function, the question
becomes whether a similar hyperbola-like mathematical function also describes the
discounting of probabilistic rewards. Recent evidence suggests that, indeed, this is the case.
Moreover, the fact that the same mathematical function describes both temporal and probability
discounting has generated suggestions that the same (or similar) underlying processes might
account for both probability and temporal discounting (e.g., Green & Myerson, 1996; Prelec
& Loewenstein, 1991; Rachlin et al., 1986, 1994; Stevenson, 1986). After presenting two
proposed probability discounting functions that have the same mathematical forms as the
temporal discounting functions discussed previously, we evaluate such single-process
accounts.
Mathematical Descriptions of Probability Discounting
Rachlin et al. (1991) proposed that the value of probabilistic rewards may be described by a
discounting function of the same form (i.e., a hyperbola) as that which they used to describe
delayed rewards:
V = A / (1 + hΘ), (4)
where V represents the subjective value of a probabilistic reward of amount A, h is a parameter
(analogous to k in Equation 2) that reflects the rate of decrease in subjective value, and Θ
represents the odds against receipt of a probabilistic reward (i.e., Θ = [1 − p]/p, where p is the
probability of receipt). When h is greater than 1.0, choice is always risk averse; when h is less
than 1.0, choice is always risk seeking; and when h equals 1.0, the subjective value predicted
by Equation 4 is equivalent to the expected value.
Alternatively, Ostaszewski et al. (1998) suggested that the discounting of probabilistic rewards
may be better described by a hyperbola-like form analogous to that for delayed rewards (i.e.,
Equation 3):
V = A / ( 1 + hΘ) s . (5)
The parameter s may represent the nonlinear scaling of amount and/or odds against and is
usually less than 1.0 (Green et al., 1999a). This is analogous to the role of the exponent in
Psychol Bull. Author manuscript; available in PMC 2006 February 24.