The matching law in applied behavior analysis describes allocation among concurrently available alternatives. Relative responding tends to approximate relative obtained reinforcement. If one response produces twice as many reinforcers, more responding may shift to it.
Matching is quantitative, not conscious calculation. Choice can reflect reinforcement rate, value, immediacy, effort, delay, instructions, history, and bias.
What Is the Matching Law in ABA?
In simple form, relative response rates approximate relative reinforcement rates. Let B1/B2 be response rates and R1/R2 obtained reinforcement rates:
B1 / B2 ≈ R1 / R2
Examine simultaneous alternatives and obtained, not promised, reinforcement. Delayed or missed tokens change the relation.
| Symbol or term | Meaning | Example |
|---|---|---|
| B1 | Behavior allocated to alternative 1. | Requests made through a picture card. |
| B2 | Behavior allocated to alternative 2. | Requests made through a speech device. |
| R1 | Reinforcement obtained from alternative 1. | Six completed requests contact assistance. |
| R2 | Reinforcement obtained from alternative 2. | Three completed requests contact assistance. |
| Response allocation | The relative share of behavior directed to each option. | About two times as many requests use card 1 as device 2. |
The BACB BCBA Test Content Outline, 6th edition includes matching law in B.23. It describes allocation; it does not replace direct measurement or functional assessment.
How to Calculate Matching and Response Allocation
Use the same unit and observation period for both alternatives. Count responses, time, and reinforcers, then calculate each option’s share. For two options, behavior share is:
B1 / (B1 + B2)
The corresponding reinforcement share is:
R1 / (R1 + R2)
With 60 responses on A and 40 on B, response allocation is 60%. If obtained reinforcers are 6 and 4, reinforcement allocation is also 60%, a descriptive match. Use repeated data.
| Step | Calculation | Illustrative result |
|---|---|---|
| 1. Count behavior | Record responses or time for each alternative. | A = 60 responses; B = 40 responses. |
| 2. Count obtained reinforcement | Record reinforcers actually delivered from each alternative. | A = 6; B = 4. |
| 3. Calculate response share | A behavior divided by total behavior. | 60 / 100 = 0.60. |
| 4. Calculate reinforcement share | A reinforcement divided by total reinforcement. | 6 / 10 = 0.60. |
| 5. Compare shares | Inspect whether the two proportions are similar. | Response allocation matches obtained reinforcement allocation. |
Matching is not pass/fail. A 70% response share versus 60% reinforcement share signals a difference to examine for effort, delay, quality, and bias.
Concurrent Schedules and Choice Alternatives
Concurrent schedules make two or more contingencies available at once; learners can switch. Matching is most directly examined in this arrangement.
If Staff A responds more often than Staff B, requests may favor A. Measure actual reinforcement, switching, effort, latency, and interaction quality.
| Concurrent-schedule feature | Question | Why it matters |
|---|---|---|
| Alternatives | What responses or activities are simultaneously available? | Matching requires defined choice options. |
| Contingency | What consequence follows responding on each option? | Different outcomes can change allocation. |
| Schedule | How often is reinforcement available for each alternative? | Relative reinforcement affects predicted allocation. |
| Switching cost | What effort or delay is required to move between options? | Switching effort can produce bias unrelated to reinforcement rate. |
| Obtained reinforcement | What did the learner actually receive? | Actual contact may differ from programmed schedules. |
See the concurrent schedules guide. A choice alone is insufficient; identify contingencies and measure responding.
Applied Examples of the Matching Law
Applied examples include help from a peer or teacher, an appropriate request versus problem behavior, or tasks with different feedback and effort.
If a speech device is slow but a picture card produces a faster break, allocation may favor the card. Changing effort or delay can shift behavior.
| Context | Alternatives | Matching-law question |
|---|---|---|
| Communication | Picture card versus speech device. | Does request allocation track the rate and quality of successful help? |
| Education | Independent work versus help-seeking. | Which option produces faster and more valuable feedback? |
| Staff performance | Two available tasks with different feedback schedules. | Does work allocation correspond to obtained reinforcement? |
| Behavior support | Replacement response versus challenging behavior. | Is the alternative response contacting reinforcement often enough? |
| Daily routine | Two leisure activities available at once. | Does time allocation track access, quality, and effort? |
| Supervision | Ask a supervisor now versus continue independently. | What history makes one route more efficient or reinforcing? |
Use the law to make a safe alternative more valuable and accessible; it never justifies allowing harmful behavior.
Undermatching, Overmatching, and Bias
Undermatching is less sensitivity to reinforcement differences; overmatching is more sensitivity; bias is disproportionate responding not explained by relative reinforcement.
Inspect effort, delay, quality, magnitude, preference, instructions, history, switching cost, and measurement. A rich option may be hard to access.
| Pattern | Definition | Possible variables to inspect |
|---|---|---|
| Matching | Response allocation approximately tracks reinforcement allocation. | Stable concurrent schedules and comparable measurement. |
| Undermatching | Behavior changes less than the difference in reinforcement rates. | Switching cost, effort, delayed reinforcers, or weak discrimination. |
| Overmatching | Behavior changes more than the difference in reinforcement rates. | Quality differences, instructions, or strong stimulus bias. |
| Bias | Consistent preference for one alternative beyond relative reinforcement. | Position, history, social variables, response topography, or accessibility. |
| Indifference | Similar allocation despite meaningful reinforcement differences. | Reinforcer value, measurement, satiation, or unclear contingencies. |
A mismatch signals investigation, not manipulation. Ask what makes the richer option difficult, delayed, inaccessible, or less preferred.
Using the Matching Law to Design Interventions
Make a socially valid alternative easier, faster, clearer, and reliably reinforced when appropriate. Do not create unsafe delay or remove essential care.
If screaming gets immediate attention but help requests do not, allocation follows the environment. Reinforce help quickly, adjust staff response, and measure the shift.
| Intervention lever | Possible change | Measurement |
|---|---|---|
| Response effort | Simplify the alternative response or make materials accessible. | Latency, completion, and switching frequency. |
| Delay | Shorten the time between the alternative response and reinforcement. | Time to reinforcement and response allocation. |
| Quality | Improve the social or activity outcome produced by the alternative. | Engagement and preference across options. |
| Magnitude | Adjust the amount or duration of reinforcement when justified. | Relative reinforcement and response share. |
| Schedule | Increase reliability or density of reinforcement for the alternative. | Obtained reinforcement per opportunity. |
| Competing response | Reduce accidental reinforcement for the problem behavior while protecting safety. | Problem behavior, replacement behavior, and side effects. |
Keep intervention individualized and ethical. Include preference, assent, communication rights, accessibility, and context.
Measuring Response Allocation in Practice
Match measurement to the choice: rate for discrete responses, duration for activities, and obtained reinforcer counts for each option. Define the window and switching rules.
Analyze stable sessions, graph relative response and reinforcement shares, and explain limits when options are not independently available.
| Data component | Operational question | Quality check |
|---|---|---|
| Response count | How many responses occurred on each alternative? | Observers use the same response definitions. |
| Time allocation | How much time did the learner spend with each option? | Timing rules handle switching and pauses consistently. |
| Obtained reinforcement | How many reinforcers were actually delivered from each option? | Delivery, consumption, and delay are recorded. |
| Opportunity count | How many choice opportunities were available? | Unavailable or interrupted trials are marked separately. |
| Relative shares | Do response and reinforcement proportions track? | Use the same time window and denominator logic. |
| Context variables | What effort, quality, delay, instruction, or bias was present? | Interpretation includes relevant environmental variables. |
Measurement shows a relation, not that reinforcement rates alone caused behavior. Improve definitions and procedures before strong claims.
Matching Law, Behavioral Contrast, and Related Concepts
Matching concerns concurrent alternatives; behavioral contrast concerns responding across components of a multiple schedule. They are not interchangeable.
Preference assessment identifies selections under a procedure; matching analyzes allocation relative to obtained reinforcement. Effort, delay, and history can change the relation.
| Concept | Core question | Distinction |
|---|---|---|
| Matching law | How does response allocation relate to relative reinforcement? | Describes choice across concurrently available alternatives. |
| Concurrent schedule | What contingencies operate at the same time? | Provides the arrangement in which matching is often analyzed. |
| Multiple schedule | How does responding change across distinct stimulus components? | Components alternate rather than remain concurrently available. |
| Behavioral contrast | Does a change in one component alter responding in another? | Focuses on cross-component response changes. |
| Preference assessment | Which options are selected under the assessment procedure? | Selection does not by itself establish reinforcement or matching. |
See the behavioral contrast guide. Matching questions usually show simultaneous options, relative reinforcement, and response allocation.
Ethical and Clinical Limits of the Matching Law
Use matching to improve environments, not to arrange deprivation or control choices without consent. A person is not a ratio calculation.
Check whether alternatives are accessible, safe, culturally responsive, and meaningful. Improve context when a replacement requires more effort; do not blame the learner.
| Boundary | Risk | Safer practice |
|---|---|---|
| Deprivation | Withholding basic needs to increase responding. | Use preference, assent, natural opportunities, and least restrictive arrangements. |
| Communication | Delaying help or refusing a request until a target allocation occurs. | Protect functional communication and safety access. |
| Problem behavior | Allowing dangerous behavior to contact reinforcement for the sake of observation. | Use risk-informed assessment and reinforce a safe competing response. |
| Choice restriction | Calling a forced selection “preference.” | Offer meaningful choices and record refusal or nonselection. |
| Overinterpretation | Treating one data point as proof of a law or cause. | Use repeated measurement, procedural checks, and cautious language. |
The practical question is what makes a safe alternative easier and worthwhile. Matching can guide that question with assessment, consent, and data.
BCBA Exam Traps, Checklist, and FAQ
Matching predicts a distribution across available alternatives, not always the richest single choice. Use obtained reinforcement, and distinguish matching from contrast, preference, and matching-to-sample.
| Exam clue | Likely concept | Reasoning move |
|---|---|---|
| Two response options are available at the same time. | Concurrent schedules and matching law. | Compare response allocation with obtained reinforcement allocation. |
| One option produces twice as many reinforcers. | Relative reinforcement predicts greater response allocation. | Calculate proportions rather than predict a single choice. |
| Response allocation changes less than reinforcement allocation. | Undermatching. | Inspect effort, delay, quality, and measurement. |
| Response allocation changes more than reinforcement allocation. | Overmatching or bias. | Look for instructions, stimulus bias, or quality differences. |
| A learner selects a sample-matching comparison. | Matching-to-sample and conditional discrimination. | Do not confuse “matching” language with the matching law. |
| Responding changes in one alternating component after another changes. | Behavioral contrast. | Analyze multiple-schedule components, not concurrent allocation. |
See the free BCBA mock exam. Checklist:
- Identify concurrent alternatives.
- Define each response or time measure.
- Count obtained reinforcement.
- Calculate response shares.
- Calculate reinforcement shares.
- Check effort, delay, quality, and magnitude.
- Check instructions, switching cost, and bias.
- Distinguish matching law from matching-to-sample.
- Separate preference and behavioral contrast.
- Improve safe alternatives without restricting communication.
What is the matching law? Response allocation across concurrent alternatives tends to approximate relative obtained reinforcement.
What is the formula? B1/B2 ≈ R1/R2; proportions compare shares.
Does matching mean the learner always chooses the richest option? No. Effort, delay, quality, bias, history, and access affect allocation.
What is undermatching? Allocation less sensitive to reinforcement differences than predicted.
How can a BCBA use it clinically? Measure alternatives and obtained reinforcement, then make a safe functional alternative easier and more reliable.
Matching connects relative behavior with relative obtained reinforcement; applied analysis adds effort, delay, quality, history, and ethics.
