Introduction to the Matching Law
The Matching Law is one of the most robust quantitative relationships in the experimental analysis of behavior. First described by Richard Herrnstein in 1961 based on his research with pigeons on concurrent variable-interval (VI) schedules, the Matching Law states that the relative rate of responding on one alternative matches the relative rate of reinforcement obtained from that alternative. In simpler terms, organisms distribute their behavior in proportion to the reinforcement available.
This principle has profound implications for understanding why individuals choose one behavior over another, including why they may choose problem behavior over appropriate alternatives. For BCBA exam candidates, the Matching Law provides a quantitative framework for understanding choice behavior and designing effective interventions.
The Basic Matching Equation
The simplest form of the Matching Law can be expressed as:
B1 / (B1 + B2) = R1 / (R1 + R2)
Where B1 and B2 represent the rates of responding on two alternatives, and R1 and R2 represent the rates of reinforcement obtained from those alternatives. If 75% of the available reinforcement comes from Alternative 1, then approximately 75% of the individual's responding will be allocated to Alternative 1.
Strict Matching vs. Observed Patterns
In laboratory settings, strict matching (where the response ratio exactly equals the reinforcement ratio) is relatively rare. Instead, researchers observe several common deviations:
- Undermatching: The organism distributes responses more evenly than the reinforcement ratio predicts. This is the most common deviation and occurs when the individual does not fully discriminate between the two schedules.
- Overmatching: The organism allocates even more responding to the richer schedule than predicted. This can occur when there is a cost associated with switching between alternatives.
- Bias: A consistent preference for one alternative that is not explained by the reinforcement rates alone. Bias may result from qualitative differences in the reinforcers, effort differences, or other factors.
The Generalized Matching Equation
To account for these deviations, Baum (1974) proposed the generalized matching equation, which adds parameters for sensitivity and bias. The generalized equation uses a logarithmic transformation and includes a slope parameter (sensitivity, denoted a) and an intercept parameter (bias, denoted b). When sensitivity equals 1.0 and bias equals 0, the equation reduces to strict matching. Sensitivity values below 1.0 indicate undermatching, while values above 1.0 indicate overmatching.
For the BCBA exam, you do not need to compute the generalized matching equation, but you should understand what undermatching, overmatching, and bias mean conceptually and how they relate to clinical scenarios.
Clinical Applications of the Matching Law
Understanding Problem Behavior as Choice
One of the most important clinical applications of the Matching Law is understanding problem behavior as a choice between concurrent schedules of reinforcement. When a child engages in problem behavior (e.g., hitting to escape tasks) rather than appropriate behavior (e.g., requesting a break), the Matching Law suggests that the reinforcement parameters for problem behavior are more favorable. Problem behavior may produce reinforcement on a richer schedule (every time), with shorter delays (immediately), or of higher quality than the appropriate alternative.
This framework leads directly to intervention strategies. If problem behavior contacts reinforcement on a denser schedule than appropriate behavior, the clinician can adjust the schedules to make appropriate behavior more competitive. This is the fundamental logic behind differential reinforcement procedures and functional communication training (FCT).
Designing Effective DRA Procedures
When implementing differential reinforcement of alternative behavior (DRA), the Matching Law tells us that simply providing reinforcement for an alternative is not enough. The alternative must produce reinforcement that matches or exceeds what problem behavior produces in terms of rate, delay, quality, and magnitude. If a child can get immediate escape by hitting but must wait 30 seconds for a response to a break card, the matching law predicts continued hitting. Effective DRA requires making the appropriate alternative more competitive across all reinforcement dimensions.
Understanding Treatment Integrity Failures
The Matching Law also explains why intermittent reinforcement of problem behavior during treatment can be so damaging. If an extinction procedure for problem behavior breaks down even occasionally (say, 20% of the time), the Matching Law predicts that responding will persist at levels proportional to that intermittent reinforcement schedule. This is why treatment integrity is so critical: even small amounts of reinforcement for problem behavior can maintain it at significant levels relative to the alternative.
Factors That Influence Matching
Several reinforcement dimensions beyond rate affect how organisms allocate their behavior:
- Reinforcer quality: Higher-quality reinforcers bias responding toward that alternative
- Reinforcer delay: Shorter delays bias responding toward the immediate alternative
- Reinforcer magnitude: Larger reinforcers bias responding toward that alternative
- Response effort: Lower-effort responses are preferred when reinforcement parameters are otherwise equal
These dimensions interact in complex ways. A higher-quality reinforcer delivered after a delay may be less preferred than a lower-quality reinforcer delivered immediately. Understanding these interactions helps clinicians design reinforcement systems that effectively compete with problem behavior.
BCBA Exam Tips
Matching Law questions on the BCBA exam typically ask you to apply the principle to clinical scenarios rather than perform calculations. Be prepared to:
- Identify why an intervention is failing based on reinforcement schedule analysis
- Recommend changes to reinforcement parameters to make appropriate behavior more competitive
- Explain undermatching in the context of a clinical scenario
- Connect the Matching Law to differential reinforcement and FCT procedures
Remember that the Matching Law is descriptive, not prescriptive. It describes how organisms naturally distribute behavior, and clinicians can use this knowledge to arrange environments that promote adaptive behavior over maladaptive alternatives.