A discounting framework for trade‐offs between risk and delay
One simple equation predicts how typical adults trade reward probability against delay.
01Research in Context
What this study did
Yeh and team asked 120 college students to choose between two rewards.
One reward was smaller but certain. The other was bigger but might come later or might never come.
They tweaked both the delay and the odds to see which choices students made.
Then they tested if a single math equation could predict every choice.
What they found
The modified hyperboloid model fit the data almost perfectly.
R-squared was .99, meaning the equation explained a large share of the choices.
Students treated risk and delay as parts of the same trade-off, not separate problems.
How this fits with other research
Jarus et al. (2015) showed that people who hate waiting also hate risk.
Yeh et al. (2025) now gives you one formula to capture both hates at once.
Ward-Horner et al. (2017) found some learners prefer bigger, later payoffs.
The new model helps you predict who those learners are before you test them.
McDevitt et al. (2016) warned that signals can push clients toward bad choices.
The new equation lets you calculate exactly how much risk or delay will tip the balance back.
Why it matters
You can now plug any reinforcement schedule into the hyperboloid equation to see if your client will actually work for it.
Try it next time you add a delay or make a reward uncertain.
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02At a glance
03Original abstract
Every day we encounter situations in which decisions require trade-offs between the delay to one reward and the likelihood of receiving another reward. The current study was designed to extend a general discounting framework to gain insights into this fundamental trade-off process. Forty-three undergraduates adjusted the probability of receiving an immediate hypothetical monetary reward (either $200 or $10,000) until that probabilistic reward was judged subjectively equal in value to the same reward received with certainty after a delay (ranging from 1 month to 25 years). We replicated previous findings that demonstrated a linear relation between log(delay) and log(odds-against), derived from the subjective probabilistic values. This linear relation was predicted when these choices were analyzed with the hyperboloid functions that describe simple delay and probability discounting in human decision making. Additionally, we extended the discounting framework and showed that the trade-off between risk and delay was well described by a modified hyperboloid discounting model (R2s = .99). These findings suggest that the discounting framework provides a valuable approach for capturing complexities of human decision making.
Journal of the Experimental Analysis of Behavior, 2025 · doi:10.1002/jeab.70052