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. 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.[1] 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 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.[2] 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.

Equations for the matching law

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}

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. These failures can be generally characterized as "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.

\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)=\log\left(b\right)+s\cdot\log\left(\frac{Rf_1}{Rf_2}\right)

The constants b and s are referred to as "bias" and "sensitivity" respectively. "Bias" reflects any tendency the subject may have to prefer one response over the other. "Sensitivity" reflects the degree to which the reinforcement ratio actually impacts the choice ratio. When this equation is plotted, the result is straight line; sensitivity changes the slope and bias changes the intercept of this line.

The generalized matching law accounts for high proportions of the variance in most experiments on concurrent variable interval schedules in non-humans. Values of b often 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.[3][4]

Processes underlying the distribution of responses

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

Theoretical importance

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.[5] 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,[6] but wide deviations from matching have been found in others.[7] The matching law has generated a great deal of research, much of it presented to the Society for Quantitative Analysis of Behavior.

Relevance to psychopathology

The matching law, and the generalized matching law, greatly expand psychologists' and behavior analysts' understanding of natural phenomena such as development and developmental psychopathology.[8] Response matching has implications for developmental psychopathology research.[9] James Snyder and colleague have found that response matching predicts the use of conflict tactics by children and parents during conflict bouts.[10] This matching rate predicts future arrests. Even children's use of deviant talk appears to follow a matching pattern.[9]

Notes

  1. ↑ Poling, A., Edwards, T. L., Weeden, M., & Foster, T. (2011). The matching law. Psychological Record, 61(2), 313-322.
  2. ↑ Herrnstein, R.J. (1961). Relative and absolute strength of responses as a function of frequency of reinforcement. Journal of the Experimental Analysis of Behaviour, 4, 267–72.
  3. ↑ Baum, W.M. (1974). On two types of deviation from the matching law: Bias and undermatching. Journal of the Experimental Analysis of Behavior, 22, 231–42.
  4. ↑ Davison, M. & McCarthy, D. (1988). The matching law: A research review. Hillsdale, NJ: Erlbaum.
  5. ↑ Herrnstein, R.J. (1970). On the law of effect. Journal of the Experimental Analysis of Behavior, 13, 243–66.
  6. ↑ Bradshaw, C.M.; Szabadi, E. & Bevan, P. (1976). Behavior of humans in variable-interval schedules of reinforcement Journal of the Experimental Analysis of Behavior, 26, 135–41.
  7. ↑ Horne, P.J. & Lowe, C.F. (1993). Determinants of human performance on concurrent schedules. Journal of the Experimental Analysis of Behavior, 59, 29–60. doi:10.1901/jeab.1993.59-29
  8. ↑ Strand, P.S. (2001) Momentum, Matching, and Meaning: Toward a Fuller Exploitation of Operant Principles. The Behavior Analyst Today, 2(3), 170–84
  9. 1 2 James Snyder, Mike Stoolmiller, Gerald R. Patterson, Lynn Schrepferman, Jessica Oeser, Kassy Johnson, and Dana Soetaert (2003): The Application of Response Allocation Matching to Understanding Risk Mechanisms in Development: The Case of Young Children's Deviant Talk and Play, and Risk for Early-Onset Antisocial Behavior. The Behavior Analyst Today, 4(4), 435–45
  10. ↑ Snyder, J.J. & Patterson, G.R. (1995). Individualized differences in social aggression: A test of the reinforcement hypothesis in the natural environment. Behavior Therapy, 26, 371–91.

References

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