Undermatching on concurrent variable-interval schedules and the power law.

The author built a math model. The model shows why animals "undermatch" on two-choice VI schedules. Undermatching means the animal does not quite follow the richer schedule as much as the reinforcer ratio predicts.

The model uses a power function. It turns reinforcer time into a transformed value. The response ratio then matches this transformed value, not the raw reinforcer ratio.

The power law with a fractional exponent reproduced the flat, undermatched curves seen in earlier pigeon data. No new birds were run; the paper is pure theory.

Jarrold et al. (1994) took the same undermatching idea into a classroom token economy. Middle-school students split their time between two math worksheets. Reinforcer ratios controlled time, but kids still undermatched unless the teacher added change-over delays or timers. The lab model became a teaching tool.

Hunter et al. (1985) asked how fast pigeons adjust when the reinforcer ratio changes daily. They found the response ratio needs about five sessions to settle. Wearden (1980) gave the static picture; I et al. gave the speed limit. Use both facts when you shuffle token rates across days.

McLean et al. (2018) later showed that rapid daily changes wear down sensitivity. Their rats matched within a session but got worse across sessions. This does not break the power-law idea; it just warns that history matters.

If you run concurrent reinforcement in the classroom, clinic, or token board, expect undermatching. You can lean on the power-law insight: learners will not fully follow the richer side. Add brief change-over delays, timers, or clear signals to tighten the fit. Check for at least five sessions before you call a new ratio "stable." These small moves turn a 1980 pigeon equation into better student engagement tomorrow.