The linear system theory's account of behavior maintained by variable-ratio schedules.
Linear-systems math now gives BCBAs a single formula that predicts response rate on both VR and VI schedules.
01Research in Context
What this study did
Lincoln et al. (1988) built math equations that describe how fast animals respond on variable-ratio (VR) schedules. The team used linear-systems theory, a tool from engineering, to turn VR schedules into VI schedules plus a feedback loop.
They tested whether the same equations could fit old VR data and also predict new schedule differences. The work is purely theoretical; no new animals were run.
What they found
The equations matched existing VR response-rate data and cleanly separated ratio size from reinforcement rate. One set of numbers now predicts both VR and VI performance.
The model shows that raising the ratio requirement slows responding in a predictable, straight-line way.
How this fits with other research
Delamater et al. (1986) had already shown that VR and VI-plus-feedback schedules produce the same response patterns in humans. The 1988 paper turns that idea into ready-to-use formulas.
Herrnstein et al. (1979) wrote a multivariate rate equation for VI schedules. The 1988 work extends the same math to VR, so one equation family now covers both schedule types.
Davison et al. (1984) warned that matching on concurrent VI VR might be a statistical artifact. The new equations do not assume matching; they derive response rates from feedback alone, sidestepping the critique.
Why it matters
If you write token boards, piece-rate pay, or any VR-like program, you can now estimate response rate before you start. Plug the ratio and planned reinforcer rate into the linear equation; the answer tells you roughly how fast the client will work. Use it to set realistic production targets or to explain to teachers why a thinner ratio feels harder even though payoff looks the same.
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02At a glance
03Original abstract
The mathematical theory of linear systems, which has been used successfully to describe behavior maintained by variable-interval schedules, is extended to describe behavior maintained by variable-ratio schedules. The result of the analysis is a pair of equations, one of which expresses response rate on a variable-ratio schedule as a function of the mean ratio requirement (n) that the schedule arranges. The other equation expresses response rate on a variable-ratio schedule as a function of reinforcement rate. Both equations accurately describe existing data from variable-ratio schedules. The theory accounts for two additional characteristics of behavior maintained by variable-ratio schedules; namely, the appearance of strained, two-valued (i.e., zero or very rapid) responding at large ns, and the abrupt cessation of responding at a boundary n. The theory also accounts for differences between behavior on variable-interval and variable-ratio schedules, including (a) the occurrence of strained responding on variable-ratio but not on variable-interval schedules, (b) the abrupt cessation of responding on occurrence of higher response rates on variable-ratio than on variable-interval schedules. Furthermore, given data from a series of variable-interval schedules and from a series of concurrent variable-ratio variable-interval schedules, the theory permits quantitative prediction of many properties of behavior on single-alternative variable-ratio schedules. The linear system theory's combined account of behavior on variable-interval and variable-ratio schedules is superior to existing versions of six other mathematical theories of variable-interval and variable-ratio responding.
Journal of the experimental analysis of behavior, 1988 · doi:10.1901/jeab.1988.49-143