Matching theory, linear system theory, and the constancy of Herrnstein's k

Linear system theory (McDowell & Kessel, 1979) and matching theory (Herrnstein, 1970) are mathematical accounts describing behavior as a function of environmental events. Both accounts generate a hyperbolic equation to describe response rate as a function of reinforcement rate. The two theories make opposing predictions about k, one of the parameters of the hyperbolic equations. Matching theory requires that k remain invariant when reinforcer properties are varied. Linear system theory, on the other hand, states that k should vary with changes in reinforcer properties. Despite the clear distinction between the two theories' predictions, empirical discrepancies exist. In a study to directly test matching theory and linear system theory, McDowell and Wood (1984, 1985) presented data from humans lever-pressing for monetary reinforcement. They demonstrated that k varied as a function of reinforcer magnitude. This result seemed to confirm linear system theory and falsify matching theory. Heyman and Monaghan (1987), however, have recently presented data from rats lever-pressing for water under various water deprivation regimes. Their data seemed to indicate that k is invariant with changes in reinforcer properties. These discrepancies can be resolved by an a priori prediction from linear system theory. Linear system theory states that there are conditions under which k can appear to remain constant despite changes in reinforcer properties. Over a restricted range, the resulting k values can appear invariant. Large variations in k can best be seen when the range of the reinforcer manipulation is large and includes low values. This paper demonstrates that Heyman and Monaghan made erroneous conclusions because their study failed to include an adequate range in their reinforcer manipulation. These results demonstrate that when appropriate conditions are included, k does in fact vary. These results prove that the constant-k requirement of matching theory is not tenable, and matching theory is falsified. Linear system theory predicted the conditions under which k should remain constant. It also specified the form of the variation in k. Thus, the present study provided support for linear system theory. These results are then used to argue that the empirical discrepancies concerning the constancy of k are due to procedural variables, and that these discrepancies can be explained within linear system theory.