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. The matching law simply states that there is a correlation between behavior and the environment. 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. This law has significantly helped behaviour analysts lawfully relate behaviour to environment and write equations that clearly show how these two covary.

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.

There are three ideas on how humans and animals maximize reinforcement, molecular maximizing, molar maximizing and melioration.

• molecular maximizing: organisms always choose whichever response alternative is most likely to be reinforced at the time.
• molar maximizing: organisms distribute their responses among various alternatives so as to maximize the amount of reinforcement they earn over the long run.
• melioration: organisms respond so as to improve the local rates of reinforcement for response alternatives.

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 of 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 R₁ and R₂ are the rate of responses on two schedules that yield obtained (as distinct from programmed) rates of reinforcement Rf₁ and Rf₂, the strict matching law holds that the relative response rate R₁ / (R₁ + R₂) matches, that is, equals, the relative reinforcement rate Rf₁ / (Rf₁ + Rf₂). That is,

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

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),

This is more conveniently expressed in logarithmic form

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.

The matching law is theoretically important for two reasons. First, it offers a simple quantification of behavior 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 experiments, but wide deviations from matching have been found in others. The matching law has generated a great deal of research, much of it presented to the Society for Quantitative Analysis of Behavior.

Although Herrstein was the pioneer in this area, a recent review by McDowell reveals that unlike the generalized matching equation, Herrnstein's original equation fails to accurately describe concurrent-schedule data under a substantial range of conditions. Therefore, the generalized matching equation is a much powerful descriptive tool widely used by behaviour analysts.