Matching law
In operant conditioning
Operant conditioning
Operant conditioning is a form of psychological learning during which an individual modifies the occurrence and form of its own behavior due to the association of the behavior with a stimulus...

, 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
Reinforcement is a term in operant conditioning and behavior analysis for the process of increasing the rate or probability of a behavior in the form of a "response" by the delivery or emergence of a stimulus Reinforcement is a term in operant conditioning and behavior analysis for the process of...

 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
Skinner box
An operant conditioning chamber is a laboratory apparatus used in the experimental analysis of behavior to study animal behavior. The operant conditioning chamber was created by B. F. Skinner while he was a graduate student at Harvard University...

, 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 R and R are the rate of response
Rate of response
Rate of response is a ratio between two measurements with different units. Rate of responding is the number of responses per minute, or some other time unit. It is usually written as R. Its first major exponent was B.F. Skinner . It is used in the Matching Law.R = # of Responses/Unit of time =...

s on two schedules that yield obtained (as distinct from programmed) rates of reinforcement Rf and Rf, the strict matching law holds that the relative response rate R / (R + R) matches, that is, equals, the relative reinforcement rate Rf / (Rf + Rf). 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
Free will
"To make my own decisions whether I am successful or not due to uncontrollable forces" -Troy MorrisonA pragmatic definition of free willFree will is the ability of agents to make choices free from certain kinds of constraints. The existence of free will and its exact nature and definition have long...

, in exactly the way B.F. Skinner had argued that the experimental analysis of behavior
Experimental analysis of behavior
The experimental analysis of behavior is the name given to the school of psychology founded by B.F. Skinner, and based on his philosophy of radical behaviorism. A central principle was the inductive, data-driven examination of functional relations, as opposed to the kinds of hypothetico-deductive...

 should, in his book Beyond Freedom and Dignity
Beyond Freedom and Dignity
Beyond Freedom and Dignity is a book written by American psychologist B. F. Skinner and first published in 1971. The book argues that entrenched belief in free will and the moral autonomy of the individual hinders the prospect of using scientific methods to modify behavior for the purpose of...

. 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
Society for Quantitative Analysis of Behavior
The Society for Quantitative Analysis of Behavior was founded in 1978 by Michael Lamport Commons and John Anthony Nevin. The first president was Richard J. Herrnstein. In the beginning it was called the Harvard Symposium on 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.

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 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.
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