On Herrnstein's equation and related forms.
The matching equation can be built from two different ideas, so watch both learning history and response thresholds when data drift.
01Research in Context
What this study did
Kazdin (1977) wrote a math paper. No kids, no rats, no data.
The author asked: where does Herrnstein’s matching equation come from? He showed two roads lead to the same formula. One road says learning builds the curve. The other road says animals have a built-in threshold. Both roads end at the same equation.
What they found
The paper found that the same equation drops out of two very different stories. If the stories are that different, the equation’s numbers should still track each other. That is the testable hint left behind.
How this fits with other research
VERHAVMOLLIVER (1963) came first. That paper showed how to measure “reinforcement value” in the lab with lever presses. Kazdin (1977) later gave the math bones that explain why the value index works.
Baum (1989) took the equation further. M said stop looking at single pecks or presses. Look at the big-picture feedback the schedule gives the animal. M extends E by turning the equation into a tool for daily-life prediction.
Charles Mace (2018) shows the end of the arc. A basic scientist can start with Herrnstein-type math and still end up helping real people. The chain runs: T measures, E explains, M broadens, Charles shows it can matter outside the cage.
Why it matters
You now know the matching law is not tied to one story. If a client’s choice curve looks off, you can think two ways: check the learning history or check the response threshold. Either path keeps the same equation on your graph. Next time you write a token or FR schedule, remember the big feedback loop Baum (1989) talks about—one glance at the overall pay-off may tell you more than counting every single response.
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02At a glance
03Original abstract
In 1970, Herrnstein proposed a simple equation to describe the relation between response and reinforcement rates on interval schedules. Its empirical basis is firm, but its theoretical foundation is still uncertain. Two approaches to the derivation of Herrnstein's equation are discussed. It can be derived as the equilibrium solution to a process model equivalent to familiar linear-operator learning models. Modifications of this approach yield competing power-function formulations. The equation can also be derived from the assumption that response strength is proportional to reinforcement rate, given that there is a ceiling on response rate. The proportional relation can, in turn, be derived from a threshold assumption equivalent to Shimp's "momentary maximizing". This derivation implies that the two parameters of Herrnstein's equation should be correlated, and may explain its special utility in application to internal schedules.
Journal of the experimental analysis of behavior, 1977 · doi:10.1901/jeab.1977.28-163