The application of the matching law to simple ratio schedules.
The matching law predicts response stoppage on ratio schedules that real data do not show—treat the equation with caution.
01Research in Context
What this study did
Atnip (1977) looked at a math rule called the matching law. The rule says animals split their time between two choices in the same ratio as the rewards they get.
The paper focused on simple ratio schedules. These schedules give a reward only after a set number of responses. The author warned the rule predicts the animal will stop working, but data do not show that.
What they found
The math equation said animals should quit responding on lean ratio schedules. Real data showed they keep working.
The paper told researchers to be careful when using the matching law with ratio schedules.
How this fits with other research
Glenn (1988) later tested rats on concurrent VR schedules. Matching held only when both counters went up together. If only the active counter advanced, rats picked the rich schedule every time. This shows the equation needs extra rules.
Davison et al. (1995) found pigeons under-match when reward ratios get extreme. Their data matched the warning that simple equations miss real behavior.
Pear et al. (1984) still told clinicians to use VI schedules to out-compete problem behavior. They trusted the law for applied work, but only with VI, not ratio, schedules. The warning and the advice line up.
Why it matters
When you write a concurrent schedule, check the math assumptions. If you use ratio schedules, do not expect perfect matching. Watch for exclusive preference or under-matching. Test your data before you trust the equation. Adjust reinforcer size or add brief VI elements if you need smoother choice.
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Join Free →Plot your learner’s response counts across two ratio schedules and check if choice matches the reinforcer ratio—if not, tweak the schedule or add a VI component.
02At a glance
03Original abstract
To account for performance under simple ratio sched- ules, Pear (1975) derived the following equation from the matching law (Herrnstein, 1970):where P is number of responses, k is a constant, n is the average number of responses required for one reinforcement, and R. is the reinforcement for other alternatives.2This paper considers briefly the empirical adequacy of this equation in predicting responding un- der ratio schedules.Equation 1 predicts two effects of questionable generality: (a) responding should decrease as the average value of the ratio increases until (b) at some value of the ratio, responding goes to zero.The prediction of an inverse relation between number of responses and ratio size appears contradicted by considerable data.
Journal of the experimental analysis of behavior, 1977 · doi:10.1901/jeab.1977.27-215