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. For example, if two response alternatives A and B are offered to an organism, the ratio of response rates to A and B equals the ratio of reinforcements yielded by each response. This law applies fairly well when non-human subjects are exposed to concurrent variable interval schedules (but see below); its applicability in other situations is less clear, depending on the assumptions made and the details of the experimental situation. The generality of applicability of the matching law is subject of current debate. The matching law can be applied to situations involving a single response maintained by a single schedule of reinforcement if one assumes that alternative responses are always available to an organism, maintained by uncontrolled "extraneous" reinforcers. For example, an animal pressing a lever for food might pause for a drink of water. 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, and the ratio of their rates to the two buttons matched the ratio of their rates of reward on the two buttons.
Mathematical statement 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,
R 1 R 1 + R 2 = R f 1 R f 1 + R f 2 {\displaystyle {\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:
R 1 R 2 = R f 1 R f 2 {\displaystyle {\frac {R_{1}}{R_{2}}}={\frac {Rf_{1}}{Rf_{2}}}}
Alternatively stated, it states that there exists a constant C {\displaystyle C} for an individual animal, such that R i R f i = C {\displaystyle {\frac {R_{i}}{Rf_{i}}}=C} for any i {\displaystyle i} . That is, for an individual animal, the rate of response is proportional to rate of reinforcement for any task.
Deviations from matching, and the generalized matching law A recent review by McDowell reveals that Herrnstein's original equation fails to accurately describe concurrent-schedule data under a substantial range of conditions. Three deviations from matching have been observed: undermatching, overmatching, and bias. Undermatching means that the response proportions are less extreme than the law predicts. Undermatching can happen if subjects too often switch between the two response options, a tendency that may be strengthened by reinforcers that happen to occur just after a subject switches. A changeover delay may be used to reduce the effectiveness of such post-switch reinforcers; typically, this is a 1.5 second interval after a switch when no reinforcer is presented. Overmatching is the opposite of undermatching, and is less common. Here the subjects response proportions are more extreme than reinforcement proportions. Overmatching may occur if there is a penalty for switching. A final deviation is bias, which occurs when subjects spend more time on one alternative than the matching equation predicts. This may happen if a subject prefers a certain environment, area in a laboratory, or method of responding. These failures of the matching law have led to the development of the "generalized matching law", which has parameters that reflect the deviations just described. This law is a power function generalization of the strict matching (Baum, 1974), and it has been found to fit a wide variety of matching data.
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