# Matching law

In operant conditioning, the **matching law** is a quantitative relationship that holds between the relative rates of response and the relative rates of reinforcement in concurrent schedules of reinforcement. It applies reliably when non-human subjects are exposed to concurrent variable interval schedules; its applicability in other situations is less clear, depending on the assumptions made and the details of the experimental situation.

Stated simply, the matching law suggests that an animal's response rate to a scenario will be proportionate to the amount/duration of positive reinforcement delivered.

The matching law was first formulated by R. J. Herrnstein (1961) following an experiment with pigeons on concurrent variable interval schedules. Pigeons were presented with two buttons in a Skinner box, each which led to varying rates of food reward. The pigeons tended to peck the button that yielded the greater food reward more often than the other button; however, they did so at a rate that was similar to the rate of reward.

If *R*_{1} and *R*_{2} are the rate of responses on two schedules that yield obtained (as distinct from programmed) rates of reinforcement *Rf*_{1} and *Rf*_{2}, the strict matching law holds that the relative response rate *R*_{1}/(*R*_{1}+*R*_{2}) **matches**, that is, equals, the relative reinforcement rate *Rf*_{1}/(*Rf*_{1}+*Rf*_{2}). That is,

<math>\frac{R_1}{R_1+R_2}=\frac{Rf_1}{Rf_1+Rf_2}</math>

This relationship can also be stated in terms of response and reinforcement ratios:

<math>\frac{R_1}{R_2}=\frac{Rf_1}{Rf_2}</math>

Subsequent research has shown that data normally depart from strict matching, but are fitted to a very good approximation by a power function generalization of the strict matching (Baum, 1974),

<math>\frac{R_1}{R_2}=b \left(\frac{Rf_1}{Rf_2}\right)^s</math>

This is more conveniently expressed in logarithmic form

<math>log\left(\frac{R_1}{R_2}\right)=b+s \cdot log\left(\frac{Rf_1}{Rf_2}\right)</math>

The constants *b* and *s* are referred to as **bias** and **sensitivity** respectively. This **generalized matching law** accounts for high proportions of the variance in most experiments on concurrent variable interval schedules in non-humans. Values of *b* depend on details of the experiment set up, but values of *s* are consistently found to be around 0.8, whereas the value required for strict matching would be 1.0 (Baum, 1974; Davison & McCarthy, 1988).

The matching law is theoretically important for two reasons. First, it offers a simple quantification of behaviour which is capable of extension to a number of other situations. Secondly, it appears to offer a lawful, predictive account of choice; as Herrnstein (1970) expressed it, under an operant analysis, choice is nothing but behavior set into the context of other behavior. It thus challenges any idea of free will, in exactly the way B. F. Skinner had argued that the experimental analysis of behavior should, in his book *Beyond freedom and dignity*. However this challenge is only serious if the scope of the matching law can be extended from pigeons to humans. When human participants perform under concurrent schedules of reinforcement, matching has been observed in some experiment (e.g. Bradshaw et al, 1976), but wide deviations from matching have been found in others (e.g. Horne & Lowe, 1993). The matching law has generated a great deal of research, much of it presented to the Society for Quantitative Analysis of Behavior.

## References

- Baum, W. M. (1974). On two types of deviation from the matching law: Bias and undermatching.
*Journal of the Experimental Analysis of Behavior, 22*, 231-242. - Bradshaw, C. M., Szabadi, E., & Bevan, P. (1976). Behavior of humans in variable-interval schedules of reinforcement
*Journal of the Experimental Analysis of Behavior, 26*, 135–141. - Davison, M., & McCarthy, D. (1988).
*The matching law: A research review*. Hillsdale, NJ: Erlbaum. - Herrnstein, R. J. (1961). Relative and absolute strength of responses as a function of frequency of reinforcement.
*Journal of the Experimental Analysis of Behaviour, 4*, 267-272. - Herrnstein, R. J. (1970). On the law of effect.
*Journal of the Experimental Analysis of Behavior, 13*, 243-266. - Horne, P. J., & Lowe, C. F. (1993). Determinants of human performance on concurrent schedules.
*Journal of the Experimental Analysis of Behavior, 59*, 29-60. (doi: 10.1901/jeab.1993.59-29).