Subjective probability and delay.
Human choice under delay and probability bends the same hyperbolic way animals' does.
01Research in Context
What this study did
Goldstein et al. (1991) asked adults to choose between small, sure rewards and bigger, delayed or risky ones.
They tracked how long or how uncertain the wait could get before people switched their choice.
What they found
People's value dropped the same way animals' does — along a hyperbolic curve, not a straight line.
Delay and probability both bend value the same way, so one equation can predict either case.
How this fits with other research
Green et al. (2013) later showed delay and probability tap different traits, so pick your tool wisely.
Rung et al. (2019) found these curves can shift with training, giving hope for addiction work.
Torres et al. (2011) repeated the curve in rats, proving the rule crosses species.
Killeen (2023) now says probability needs its own math, so the single-curve idea is being updated.
Why it matters
If you graph a client's choice data, fit a hyperbola first; it is simple and usually close enough. When delay and risk both matter, test each separately — Leonard's work says they may not move together. Watch for new probability models as the field refines the math.
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Join Free →Plot your client's indifference points on a hyperbola; if the line misses, test delay and probability separately.
02At a glance
03Original abstract
Human subjects indicated their preference between a hypothetical $1,000 reward available with various probabilities or delays and a certain reward of variable amount available immediately. The function relating the amount of the certain-immediate reward subjectively equivalent to the delayed $1,000 reward had the same general shape (hyperbolic) as the function found by Mazur (1987) to describe pigeons' delay discounting. The function relating the certain-immediate amount of money subjectively equivalent to the probabilistic $1,000 reward was also hyperbolic, provided that the stated probability was transformed to odds against winning. In a second experiment, when human subjects chose between a delayed $1,000 reward and a probabilistic $1,000 reward, delay was proportional to the same odds-against transformation of the probability to which it was subjectively equivalent.
Journal of the experimental analysis of behavior, 1991 · doi:10.1901/jeab.1991.55-233