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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 behaviour 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.
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 describes 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.
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 colleague has found that response matching predicts the use of conflict tactics 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.
As revealed in a recent and comprehensive review by McDowell (2005), it is clear that whereas the generalized matching equation accurately describes concurrent-schedule data under a substantial range of conditions, Herrnstein's original equation fails to do so.
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