On Herrnstein's equation and related forms

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.