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.

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,
 * $$\frac{R_1}{R_1+R_2}=\frac{Rf_1}{Rf_1+Rf_2}$$

This relationship can also be stated in terms of response and reinforcement ratios:
 * $$\frac{R_1}{R_2}=\frac{Rf_1}{Rf_2}$$

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),
 * $$\frac{R_1}{R_2}=b \left(\frac{Rf_1}{Rf_2}\right)^s$$

This is more conveniently expressed in logarithmic form
 * $$log\left(\frac{R_1}{R_2}\right)=b+s \cdot log\left(\frac{Rf_1}{Rf_2}\right)$$

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 experiments (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.

Role in the development of psychopathology
The matching law greatly expands psychologists and behavior analysts understanding of natural phenomena such as development and developmental psychopathology. . Response matching has implications for developmental psychopathology research. . James Snyder and colleauge has found that response matching predicts the use of conflict tacts by children and parents during conflict bouts. . This matching rate predicts future arrests. Even children's use of deviant talk appears to follow a matching pattern. .