A multivariate rate equation for variable-interval performance.

The authors built a new math formula for variable-interval (VI) schedules. The old Herrnstein hyperbola only used reinforcer rate. This new equation also added reinforcer "power" — a weight for how strong each reward feels.

They tested the formula against published pigeon data. The goal was to predict response rate and the matching pattern seen on concurrent VI VI schedules.

The rate-plus-power equation fit the data better than Herrnstein’s single-rate version. It tracked both how fast the birds pecked and how they split time between the two keys.

In plain words, counting both how often and how "potent" rewards are gives sharper forecasts of VI performance.

Alba et al. (1972) had already shown pigeons match on concurrent VI VI as long as change-over delays stay short. Herrnstein et al. (1979) now supply the math that describes those very curves.

Santi (1978) pushed the idea further, showing rats match under VI negative reinforcement (timeout from shock). The new equation should, in theory, cover these escape data too.

Davison et al. (1984) throw a wrench in the works. Their simulation and pigeon data say matching on concurrent VI VR can be an artifact of the schedule feedback loops, not a true mental process. If they are right, the equation’s success on VI VR mixes might be luck, not law.

If you write protocols that use VI schedules — like thinning reinforcement in a classroom — remember that both rate and "power" (quality, size, novelty) drive behavior. Try boosting power instead of density when thinning risks drop-offs. Also, stay cautious with VI VR mixes; the pretty matching curve might fool you into thinking the learner is "following" the math when the schedule itself is doing the work.