An analytic comparison of Herrnstein's equations and a multivariate rate equation

Herrnstein's equations are approximations of the multivariate rate equation at ordinary rates of reinforcement and responding. The rate equation is the result of a linear system analysis of variable-interval performance. Rate equation matching is more comprehensive than ordinary matching because it predicts and specifies the nature of concurrent bias, and predicts a tendency toward undermatching, which is sometimes observed in concurrent situations. The rate equation contradicts one feature of Herrnstein's hyperbola, viz., the theoretically required constancy of k. According to the rate equation, Herrnstein's k should vary directly with parameters of reinforcement such as amount or immediacy. Because of this prediction, the rate equation asserts that the conceptual framework of matching does not apply to single alternative responding. The issue of the constancy of k provides empirical grounds for distinguishing between Herrnstein's account and a linear system analysis of single alternative variable-interval responding.