Learners almost never face one response option at a time. At any moment they could work, play, ask for help, wander off, or engage in problem behavior. The matching law describes how behavior tends to be divided among options like these.
Why it matters
When a client has two ways to get the same thing, such as screaming or signing to get attention, the question is not only "Is screaming reinforced?" but "How much is screaming reinforced compared with signing?" The matching law gives you that comparison. It explains why a replacement behavior can fail even when you reinforce it, and why problem behavior can drop without full extinction when the alternative pays off better.
In practice, that means predicting how responding will split between two options, computing a simple proportion, and telling undermatching, overmatching, and bias apart.
Key ideas
- Choice is studied with concurrent schedules. On a concurrent schedule, two or more schedules run at the same time, each tied to a different response. The learner is free to switch between them.
- Responding matches relative reinforcement. Richard Herrnstein found that the proportion of responses on an alternative roughly equals the proportion of reinforcers obtained from it. In simple form: B1 / (B1 + B2) = R1 / (R1 + R2), where B is responding and R is reinforcement on each alternative.
- It is relative, not absolute. The same amount of reinforcement for a behavior can support a lot of responding when nothing else pays off, and much less when a richer alternative is available.
- Time allocation follows the same rule. Matching applies to how long a learner spends on each activity, not only to how many responses occur.
- Other dimensions matter too. Allocation also shifts with reinforcer magnitude, quality, immediacy, and the response effort each option requires. A lower-effort option that delivers a bigger, better, quicker reinforcer tends to draw more behavior.
- The generalized matching law adds two parameters. Written as B1 / B2 = b(R1 / R2)^a, it includes sensitivity (a) and bias (b). When a = 1 and b = 1, you have strict matching.
Definitions
- Matching law: the relative rate of responding on an alternative approximately equals the relative rate of reinforcement obtained from that alternative.
- Concurrent schedules: two or more schedules of reinforcement in effect at the same time, each for a different response.
- Undermatching: responding is less extreme than reinforcement predicts; the preference for the richer option is weaker than strict matching (a below 1). This is the most common deviation.
- Overmatching: responding is more extreme than reinforcement predicts; the richer option gets an even larger share (a above 1).
- Bias: a consistent preference for one alternative that the measured reinforcement does not explain, such as a favored location, an easier response, or an unmeasured reinforcer.
- Changeover delay: a brief delay after the learner switches to an alternative before a response on it can be reinforced, used in laboratory work to discourage rapid back-and-forth switching.
Examples from a session
Matching in everyday settings
- Classroom: Teacher attention follows calling out about 10 times per hour and hand raising about 30 times per hour. Matching predicts that about 75% of Malik's bids for attention (30 of 40 reinforcers) will be hand raises.
- Clinic: Rosa, age 5, can sign "break" or push materials away. When pushing materials away produced a break almost every time and signing rarely did, nearly all her escape responses were pushes. After the team made signing produce quick, longer breaks while pushing produced shorter, less frequent ones, most of her responding moved to signing.
- Home: Ken, a 15-year-old, divides his evening between video games and homework. His parents praise homework, but the game delivers frequent, immediate, high-quality reinforcement. Unless homework earns something richer, matching predicts most of his time goes to the game.
- Day program: Two vocational tasks earn Ines, an adult client, reinforcement at the same rate, yet she spends about 80% of her time on the one at the table by the window. That consistent tilt, beyond what the scheduled reinforcement predicts, is bias.
The treatment lesson: when extinction is hard to carry out with full integrity, you can still shift behavior by making the replacement response the richer option. Differential reinforcement of an alternative behavior works best when the alternative earns more frequent, larger, better, quicker reinforcement for less effort than the problem behavior does.
Common mistakes / exam traps
- Predicting all-or-none choice. Matching predicts proportional responding, not exclusive responding on the richer option. A 3:1 reinforcement ratio predicts roughly 75% of responses on the richer side, not 100%.
- Using absolute numbers. Divide each option's reinforcers by the total; that proportion is the matching prediction for how responding will split.
- Ignoring other dimensions. Two options with equal reinforcement rates can still draw unequal responding if they differ in effort, delay, magnitude, or quality.
- Forgetting the alternative. Problem behavior can decrease with no change to its own consequences if reinforcement for other behavior increases.
Exam trap
Undermatching vs. bias. Undermatching is a weaker-than-predicted preference for whichever option is richer, so it changes sides when the richer option changes. Bias is a steady tilt toward one particular option (a location, an easier response) that persists even when reinforcement is equal or favors the other side.
Exam trap
Matching vs. resurgence or behavioral contrast. If a response that was never on extinction gains ground because its concurrent alternative now pays less, that is matching. Resurgence is the return of a previously extinguished response; behavioral contrast is an opposite change across alternating components of a multiple schedule.
Quick check
Ari, age 8, can request help with a worksheet or rip the page. Requests produce help about 8 times per hour and ripping produces help about 2 times per hour. What share of his help-getting responses does the matching law predict will be requests?