Undermatching: a reappraisal of performance on concurrent variable-interval schedules of reinforcement.

Hopkins et al. (1977) went back to four older pigeon studies. They looked at how the birds split their pecks between two keys. Each key paid off on its own variable-interval schedule.

The team re-plotted the data. They checked whether the straight matching line truly fit. They also tried a curved line to see if it hugged the points better.

Every data set bent below the perfect matching line. The slope was always less than one. This bend is called undermatching.

A gentle curve fit the points better than a straight line. The authors say the matching relation is not perfectly linear.

PLISKOFF (1963) first showed pigeons match response ratios to reward ratios. Hopkins et al. (1977) used those same numbers and found the match is slightly off. The two papers do not fight; the newer one zooms in on a smaller error.

Tanguay et al. (1982) later checked 103 data sets and saw the same bend. They tell us to stop using the old fixed slope window. The 1982 review builds on Hopkins et al. (1977) and turns the bend into a rule.

Friedling et al. (1979) add another reason for the bend: carry-over from past conditions. They ran six sessions and saw the bias fade. Together, the papers show undermatching is real and has more than one cause.

When you run concurrent schedules with clients, expect less responding on the rich side than math predicts. Plan for the bend by giving the lean side extra sessions or thicker reinforcement. If you graph the data, try a curved trend line before you tweak the program.