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Matching Law: Practical Applications in ABA

The field of Applied Behavior Analysis (ABA) offers a variety of insights into human behavior. Applying this information in real-life situations is the challenge most professionals face on a daily basis. Professionals study the research to find effective interventions for their clients and become discouraged when their interventions don’t align with the researchers. Often what works in controlled, experimental settings is problematic or even unacceptable in an applied setting. This leads professionals to alter the implementation of the intervention, leading to varying results.

The Matching Law is a behavioral principle that states that behavior occurs in direct proportion to reinforcement available for each behavior. Essentially, when 2 or more concurrent schedules exist, preference is shown to the behavior that achieves the highest amount of reinforcement. This relationship can be calculated mathematically, but how does this apply to your everyday programming?

The intentional use of the Matching Law allows you to manipulate concurrent schedules to influence behavior. This technique is especially useful when one schedule of reinforcement is outside of your control or when you want to avoid the negative effects of extinction. By increasing the magnitude of reinforcement for a desired behavior, you increase the likelihood of that behavior occurring over other behaviors.

To utilize the Matching Law effectively, you must first understand concurrent schedules and reinforcer magnitude.

Concurrent Schedules of Reinforcement

Concurrent schedules occur when multiple simple schedules of reinforcement (i.e. FR, FI, VR, VI) are available at the same time for 2 or more different behaviors. These schedules work independently of one another. The individual can receive reinforcement for both behaviors, but the reinforcement is different along some element such as magnitude or quality. This is the essence of choice. Learn more about specific schedules of reinforcement in our post: Understanding Consequence Interventions: Punishment vs Reinforcement

Example of concurrent schedules of reinforcement

An apple and a cookie sitting next to each other on the counter present a concurrent schedule of reinforcement.

  • Selecting the apple results in reinforcement depending on your preference. Maybe you feel the satisfaction of knowing you ate something healthy that also tastes good.
  • Selecting the cookie also results in reinforcement. Maybe you remember eating the cookies your grandmother made when you were young and you really like this particular cookie.
Concurrent schedule of reinforcement example

These concurrent schedules influence the behavior of the individual depending on the individual’s preference for the reinforcer (Herrnstein, 1961). As you can see both intrinsic and extrinsic motivation may play a role in the preference toward a particular reinforcer.

Sometimes the use of concurrent schedules is intentional while other times it’s not. In highly controlled settings, it’s possible to minimize unintentional reinforcement, but as the environment becomes more natural and less contrived, the opportunities for inadvertent reinforcement increases. This creation of concurrent schedules of reinforcement impacts the rate at which behavior occurs.

Take a look at this example:

The reality of natural environments and human behavior demonstrates repeatedly that concurrent schedules occur, often without intent. As in the example above, a parent trying to implement DRA at home to reduce access-maintained tantrums may require the use of PECS or functional communication under most circumstances, but when visitors come to the house she may give access during challenging behavior to reduce the burden on the guests or avoid feeling embarrassed. These concurrent schedules compete against each other, both applying some effect on behavior.

Reinforcer Magnitude

Reinforcer magnitude refers to the size or amount of the reinforcer available each time reinforcement is provided. You can increase or decrease magnitude by changing:

  • Time spent with a specified reinforcer
  • Amount of a particular tangible reinforcer

You can also alter the quality of the specified reinforcer to impact preference for that reinforcer..

Remember that satiation and deprivation also impact preference for a particular reinforcer. As time or quantity increases, satiation may begin to impact motivation for the reinforcer.

Reinforcer magnitude impacting satiation and deprivation GIF

Matching Law

When behaviors receive concurrent schedules of reinforcement, the matching law states that behavior occurs in proportion to the reinforcement available. The individual tends to choose the behavior receiving the highest magnitude of reinforcement. Visually, the Matching Law looks like:

Matching law and the impact of reinforcer magnitude on behavior change

Mathematically, with all things being equal, the matching law can be calculated using the following equation:

Mathematical calculation of the matching law

In reality, humans are complex and their decisions are never quite so clearly understood. As in the example above comparing the concurrent schedules of the apple and the cookie, there are additional variables that an observer may not be aware of that impacts the individual’s choices (a desire to eat healthy and a memory of grandmother baking cookies).

Another way to look at it is if an individual receives twice as much reinforcement for behavior 1 as he does for behavior 2 that individual is likely to engage in behavior 1 twice as much as behavior 2. The behavior receiving the greatest magnitude of reinforcement will occur most frequently. This is true, of course, only if all other factors are the same.

Matching law example

You value money. Last month, you started a business mowing lawns for people living in your neighborhood. You receive $25 for mowing the lawn for your neighbor on the left and $50 for mowing the lawn for your neighbor on the right. Whose lawn are you more likely to mow, all other factors being equal?

Matching law example

The key to that example is that other factors must remain equal. Once other factors start to change, the preference for the reinforcer changes with it. If it takes you 3 times as long to mow the lawn on your left, then which reinforcement schedule would you prefer? If you really like the neighbor on your right, how does that impact your choice?

In the example with Steffy and Diana above, functional communication received reinforcement across conditions (when Steffy was home alone with Diana and when they had guests) and tantrums received reinforcement during only one of the conditions (when guests were at the house). If tantrums received a lower magnitude (i.e. less frequent, lower quality, or shorter duration) of reinforcement than functional communication, Diana would continue to be more likely to choose functional communication over tantrums to gain access to what she wants.

Example of the matching law, reinforcer magnitude and behavior change

Collateral Effects of Interventions

Many situations exist in the natural environment where the collateral effects of interventions make those interventions either inappropriate, unethical, or impractical for implementation. Myerson and Hale (1984) discussed the use of the Matching Law when extinction or punishment were not advisable. Intentionally creating concurrent schedules where the maladaptive behavior receives a lower magnitude of reinforcement than the adaptive replacement behavior allows interventionists to shape behavior without the potential risks of extinction or punishment.

For the Matching Law to be effective, it’s critical to accurately identify the function of the maladaptive behavior. This allows you to identify:

  • A functionally-equivalent replacement behavior
  • All existing variables that maintain the behavior

When a behavior is maintained by multiple variables, you must account for these variables. Borrero and Vollmer (2002) noted the importance of identifying all of the potential sources of reinforcement when attempting to predict responding using the Matching Law. For example, if challenging behavior is maintained both by access to a tangible and attention, you must consider both of these factors when selecting reinforcement for the alternative behavior and for altering the magnitude of the current reinforcement received for the target behavior. Failure to include both sources of reinforcement will result in a heavier lean of reinforcement toward the challenging behavior than intended.

The Matching Law as an Alternative to Extinction

Much of the early research demonstrates that differential reinforcement procedures are generally one of the most effective means of changing behavior. These procedures typically involve reinforcing one behavior or response class while withholding reinforcement for another (extinction). Extinction is known to have potentially dangerous side effects including extinction bursts and aggression.

The Matching Law can become an alternative to extinction by adjusting the reinforcer used following each behavior. Apply a higher magnitude of reinforcement to adaptive alternative behavior and a lower magnitude of reinforcement for the behavior targeted for reduction. Planning for this occurrence minimizes the chances of unintentionally applying an effective reinforcer following the targeted behavior.

Make sure you are adjusting the magnitude for all variables maintaining the behavior (i.e. attention and access to a tangible).

Example of the Matching Law Instead of Extinction:

You work with a mother, Sami, and her 5-year-old son, Mac. You work with Sami to identify aggression toward siblings as the behavior targeted for reduction. Through direct and indirect assessment, you identify attention from both Sami and Mac’s siblings as the function of the aggression. Although you know that withholding attention following aggression may lead to the quickest reduction in the behavior, you recognize that Mac’s mom and siblings probably will not consistently ignore this behavior and the extinction burst is likely to be unsafe. Instead, you decide to set up concurrent schedules of reinforcement:

Schedule 1: Occurs when Mac engages in aggression. When he hits or kicks his siblings, Sami and/or his siblings will tell him in a neutral voice “that hurts, don’t hit/kick.” They will then leave the room (i.e. go into their bedroom or kitchen if they are in the living room).

Schedule 2: Occurs when Mac uses words to ask for attention from his siblings (FCT/DRA). When Mac appropriately attempts to interact with his siblings in any way, Sami and his siblings will provide lavish attention in the form of tickles, high fives and hugs (all identified as reinforcing for Mac). They will maintain this attention for at least 2 minutes even if it is disruptive to their current activity.

Providing Sami and Mac’s siblings with a specific way to respond to Mac’s aggression overcomes several hurdles:

  • Parents and siblings react to challenging behavior without thinking
  • You mitigate concerns about a potential extinction burst
  • Sami and the siblings are empowered with clear direction no matter the behavior

Application of the Matching Law

The definition of the matching law should describe it as a tendency for individuals to choose behavior that receive a higher magnitude or better quality of reinforcer than alternative behaviors. It’s impossible to fully understand the preferences of someone else and the different factors that alter those preferences. When applying the matching law in practice, use data to determine if the learner is actually choosing the behavior you expect.

If behavior does not change in the way you expect, re-evaluate the different variables that might be at play. Could there be factors that you can’t see (i.e. thoughts, feelings, memories, et.c) impacting the behavior?

The matching law can be a practical, effective alternative to the use of extinction that is generally preferred by both learners and interventionists.

References and Further Reading

Borrero, J. C., & Vollmer, T. R. (2002). An application of the matching law to severe problem behaviorJournal of Applied Behavior Analysis35(1), 13-27.

Herrnstein R. J. Relative and absolute strength of response as a function of frequency of reinforcementJournal of the Experimental Analysis of Behavior. 1961;4:267–272. 

Hoch, H., McComas, J. J., Thompson, A. L., & Paone, D. (2002). Concurrent reinforcement schedules: Behavior change and maintenance without extinctionJournal of Applied Behavior Analysis35(2), 155-169.

MacNaul, H. L., & Neely, L. C. (2018). Systematic review of differential reinforcement of alternative behavior without extinction for individuals with autism. Behavior Modification42(3), 398-421.

Myerson, J., & Hale, S. (1984). Practical implications of the matching lawJournal of Applied Behavior Analysis17(3), 367-380.

Poling, A., Edwards, T. L., Weeden, M., & Foster, T. M. (2011). The matching law. The Psychological Record61(2), 313-322.

Reed, D. D., & Kaplan, B. A. (2011). The matching law: A tutorial for practitioners. Behavior analysis in practice4(2), 15-24.

Trump, C. E., Ayres, K. M., Quinland, K. K., & Zabala, K. A. (2020). Differential reinforcement without extinction: A review of the literature. Behavior Analysis: Research and Practice20(2), 94.

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