A tribute to Howard Rachlin and his two‐parameter discounting model: Reliable and flexible model fitting
The Rachlin two-parameter model gives steady, trustworthy numbers when you fit delay-discounting data, even with messy small samples.
01Research in Context
What this study did
Franck and colleagues ran thousands of fake data sets through the Rachlin two-parameter delay-discounting model. They wanted to see if the model would crash or give wild numbers when data were messy or sample sizes were small.
They also compared the Rachlin model to other two-parameter rivals. The test was purely computational—no human or animal subjects—just laptops grinding numbers.
What they found
The Rachlin model rarely failed. Even with tiny, noisy data sets it returned stable, believable parameter estimates. Other two-parameter models often blew up or gave impossible values.
In short, the Rachlin formula is both flexible and safe. You can trust it to describe how steeply someone devalues delayed rewards without hand-checking every run.
How this fits with other research
Reyes-Huerta et al. (2016) showed that removing numbers from delay-discounting tasks wipes out the magnitude effect. Their finding matters because the Rachlin model must still fit those flattened curves; Franck’s work says it can.
Ainslie et al. (2003) bundled rewards to make rats choose larger-later options. Those data sets are exactly the kind the Rachlin model handles well—steep curves that bend fast.
Kincaid (2023) urged researchers to label procedures clearly. Franck answers by giving us a labeled, reliable model to describe delay curves instead of guessing.
Why it matters
If you study impulsivity or self-control, you can now fit delay-discounting data with one line of code and sleep well at night. No more reruns because the solver crashed. Use the Rachlin model, report the two parameters, and move on to interpreting what the numbers say about your client’s choices.
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02At a glance
03Original abstract
Delay discounting reflects the rate at which a reward loses its subjective value as a function of delay to that reward. Many models have been proposed to measure delay discounting, and many comparisons have been made among these models. We highlight the two‐parameter delay discounting model popularized by Howard Rachlin by demonstrating two key practical features of the Rachlin model. The first feature is flexibility; the Rachlin model fits empirical discounting data closely. Second, when compared with other available two‐parameter discounting models, the Rachlin model has the advantage that unique best estimates for parameters are easy to obtain across a wide variety of potential discounting patterns. We focus this work on this second feature in the context of maximum likelihood, showing the relative ease with which the Rachlin model can be utilized compared with the extreme care that must be used with other models for discounting data, focusing on two illustrative cases that pass checks for data validity. Both of these features are demonstrated via a reanalysis of discounting data the authors have previously used for model selection purposes.
Journal of the Experimental Analysis of Behavior, 2023 · doi:10.1002/jeab.820