The matching law describes mathematical equations used to understand the relationship of one person’s behavior based on reinforcers, which may be internal or external. The individual’s behavior is the rate of response, or how often they choose a specific behavior based on the presence of certain reinforcers.

Reinforcers may be something positive, like choosing to hang out with a specific friend because they make you happy. However, when applied behavior analysis (ABA) therapists use the term, they almost always examine negative reinforcers, or behaviors that creep into the environment and accidentally reinforce a maladaptive behavior that the therapist is attempting to change.

ABA Therapy

Applied behavior analysis (ABA) is a field of behavioral therapy that focuses on improving specific behaviors in areas like communication, learning, socializing, and academics. Approaches to behavioral change and management also work for tasks in daily life, from personal hygiene to job skills.

ABA therapy is the most used evidence-based approach for treating symptoms of autism. Therapists certified in this practice can work in specialized clinics, schools, community centers, and clients’ homes.

The Matching Law: Math Describes Behavior to Support Evidence-Based Practices

ABA therapy is an evidence-based therapy practice, meaning therapists are trained to make observations of their clients’ behaviors and determine whether the treatment plan is effective at leading to positive (or adaptive) behavioral change or if the client is not benefitting from the therapist’s approach. One way that ABA therapists may determine that is through a mathematical formula called the matching law.

The matching law describes the rate of responses to the rate of reinforcement. The equation typically shows that when someone is offered a choice between two behaviors that are equally likely to occur, the individual will allocate their responses to the behavior that has resulted in more positive reinforcement or “feeling better” historically for them.

For example, someone may associate feeling good with eating ice cream; however, they are gaining weight and need to stop eating ice cream. Despite this, when offered a choice between a piece of fruit or ice cream, they will prefer the ice cream because of their history of feeling better after consuming this food.

Another example often cited is the behavior of children on a playground. Observing their behavior allows a researcher to gather distributed information of where the children go with playmates, which equipment they choose, and if they choose more or less physically active games. Aggregating responses can show the children’s preferences based on their patterns, which suggests that they have a specific rate of response to the reinforcement of the playground environment.

The one-to-one connection between the rate of response and the rate of reinforcement is called strict matching. While it is a useful concept, it is no longer considered realistic. There are systematic departures from strict matching in the matching equation, so a generalized version is more often used in modern behavioral analysis to describe collected data.

What Is the Matching Law?

The matching law lets behavioral therapists or researchers use a formula for behavior, make observations, gather data, and use a systematic approach to hypothesize what a person will do. Equations from the matching law end up with two types of graphs.

  1. Generalized matching: This is a linear function between two or more behavior-reinforcement alternatives, often when the reinforcer is internal.
  2. Single-alternative matching: This looks like a hyperbolic function describing one target behavior in relation to outside reinforcement.

The basic formula of the matching law is:

B1/B1 + B2 = R1/R1 + R2

Behavior 1 and 2 are the response rates for two behaviors. R1 and R2 are the reinforcement rates for the two behaviors.

These methods of graphing behavior provide important visual information to ABA therapists about their clients, so they can improve treatment outcomes. The correlations seen in the data help therapists assess behaviors that their clients have had reinforced, either through positive or negative reinforcement, and determine how to increase adaptive behaviors by changing negative reinforcement and adjusting the application of positive reinforcement. Depending on where the behavior analyst works, these changes may be implemented throughout the child’s home, in their classroom, or in the wider school environment.

Research on the Matching Law

Two articles on the matching law are often cited as part of ABA therapy, justifying the use of the equation to understand the relationships between a behavior from a client, like a child with autism, and a reinforcer, like the negative reaction of the parent.

A paper published in 1981, for example, cited a case study of a boy with autism who had self-injuring behaviors, which are maladaptive. The boy’s family verbally reprimanded him when he injured himself, but using the matching equation, his injurious behavior was correlated with the verbal reprimands as reinforcing that behavior. When there were more reprimands, the boy was more likely, rather than less likely, to continue injuring himself. This study showed that the matching law functions as a theory in ABA therapy.

Positive reinforcement is most often used in ABA therapy. It’s important for both the therapist and the child with autism to understand how negative reinforcers creep in and cause stagnation or regression in efforts to create adaptive behaviors.

A study published in 2002 used the matching law to understand how four people with developmental disabilities had specific behaviors reinforced, in a clinical setting with their therapist and in their homes with their caregivers. Researchers recorded potential reinforcers, appropriate behaviors, and problematic behaviors. Using functional analysis, the researchers identified the reinforcers in these settings that maintained the problem behaviors. After the data was analyzed, the researchers reported that the proportional rate of appropriate to problem behaviors learned by the participants appeared to correlate with the rate of reinforcement of problem behaviors.

Another study examining interactions between 10 dyads of a mother and her son, who was labeled socially aggressive, gathered data on the parent-child interactions. The study found that in many instances, the mother’s reaction to her son’s behavior was unintentionally the reinforcer of the child’s behavior. The researchers examined both positive and negative verbal interactions from the sons after they interacted with their mothers.

The Matching Law: Useful but Not Foolproof

In most instances, there is not a one-to-one correlation between the response rate and the reinforcement rate. The matching law does state that, everything being equal between two choices, an organism will choose the option that is associated with immediate reinforcement. When described this cleanly, the relationship between the organism’s association and the organism’s choice seems clear.

Both humans and animals do not make choices with such clear delineation very often, however. A study on the alleged tautology of the matching law when applied to consumer behavior, tracking amount spent with amount purchased, was artificial in its simplification. The independent and dependent variables listed in the consumer behavior analysis were defined and measured in a way that was manipulated, rather than representative of the consumers’ “natural” environment.


Applied Behavior Analysis. Psychology Today.

The Matching Law: A Tutorial for Practitioners. (2011). Association for Behavioral Analysis International (ABAI).

The Matching Law. (2011). The Psychological Record.

Matching Law Visualization Tool. (May 2015). KU ScholarWorks.

Quick Summary of the Matching Law: A Mathematical Equation for Why People Make Choices. (July 2015). Psych Central.

An Application of the Matching Law to Severe Problem Behavior. (2002). Journal of Applied Behavior Analysis.

An Application of the Matching Law to Social Dynamics. (2007). Journal of Applied Behavior Analysis.

On the Tautology of the Matching Law in Consumer Behavior Analysis. (May 2010). Behavioral Processes.