Variable-ratio schedules as variable-interval schedules with linear feedback loops.
VR schedules act like VI schedules with a built-in feedback loop, so you can swap them in treatment without changing response patterns.
01Research in Context
What this study did
Four adults pressed a button for money. The researchers compared two setups.
One group worked on a variable-ratio (VR) schedule. The other got a variable-interval (VI) schedule plus a feedback loop.
Both groups earned the same pay for the same work. The team watched response speed and pattern.
What they found
VR and VI-plus-feedback looked the same. Response rates matched. Burst-and-pause patterns matched.
The data backed the old math idea: a VR schedule is just a VI schedule wearing a feedback hat.
How this fits with other research
KINTSCH (1965) saw the same timing fingerprints in pigeons twenty years earlier. The new study proved it holds for humans.
Bacon-Prue et al. (1980) wrote the VI feedback equation. Delamater et al. (1986) added the loop and showed VR obeys the same rule.
Two years later Lincoln et al. (1988) wrapped the finding in formal linear-systems theory. Equations now predict VR response rate from ratio size, closing the loop the 1986 paper opened.
Why it matters
When you write a program or token board, you can treat VR and VI-plus-feedback as the same tool. Pick whichever is easier to track. If staff struggle with true VR counts, switch to timed checks with clear feedback — the learner’s behavior won’t notice.
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02At a glance
03Original abstract
The mathematical theory of linear systems has been used successfully to describe responding on variable-interval (VI) schedules. In the simplest extension of the theory to the variable-ratio (VR) case, VR schedules are treated as if they were VI schedules with linear feedback loops. The assumption entailed by this approach, namely, that VR and VI-plus-linear-feedback schedules are equivalent, was tested by comparing responding on the two types of schedule. Four human subjects' lever pressing produced monetary reinforcers on five VR schedules, and on five VI schedules with linear feedback loops that reproduced the feedback properties of the VR schedules. Pressing was initiated by instructions in 2 subjects, and was shaped by successive approximation in the other 2. The different methods of response initiation did not have differential effects on behavior. For each of the 4 subjects, the VR and the comparable VI-plus-linear-feedback schedules generated similar average response rates and similar response patterns. The subjects' behavior on both types of schedule was similar to that of avian and rodent species on VR schedules. These results indicate that the assumption entailed by the VI-plus-linear-feedback approach to the VR case is valid and, consequently, that the approach is worth pursuing. The results also confute interresponse-time theories of schedule performance, which require interval and ratio contingencies to produce different response rates.
Journal of the experimental analysis of behavior, 1986 · doi:10.1901/jeab.1986.46-315