The generalized matching law as a description of multiple-schedule responding.
The matching law survives multiple schedules but under-matching grows with longer or more complex setups.
01Research in Context
What this study did
Nakamura et al. (1986) looked at 30 years of pigeon choice data. They asked one simple question: does the generalized matching law still fit when birds face three, four, or more schedules in a row?
They pooled every published multiple-schedule experiment they could find. Then they ran the same math on each set: slope, bias, and R-squared.
What they found
The law held up. Across 128 data sets it explained a median a large share of the variance. That is almost perfect prediction.
But the fit got worse as setups grew. More schedules per session, longer time inside each component, or extra training days all pushed the slope below 1.0. Birds "under-matched" more under heavier procedures.
How this fits with other research
Villarreal et al. (2019) extends the same numbers game. They swapped 1980s regression for modern Bayesian graphics. You now get pretty pictures of the same slope and bias plus credible intervals.
Baum (2021) sounds like a contradiction. He says stop counting lever presses and treat behavior as a continuous stream. Yet K’s review counted discrete pecks to get a large share R-squared. The two views meet in practice: the matching math still works even if the raw data are artificial slices of a richer process.
Byrne (2025) shows the multiple-schedule spirit lives on. His one-hour planarian lab uses the same across-behaviors baseline logic, just with worms instead of pigeons and light instead of grain.
Why it matters
If you run concurrent-schedule reinforcer assessments, expect under-matching. Add extra components or stretch the session and the client’s choices may look flatter than the true reinforcer rates. Track slope in your Excel sheet; if it drops below 0.8, shorten components or reduce the number of options before you call the data stable.
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Join Free →Graph your last concurrent-schedule data, add a trend line, and check if the slope is under 0.8—if so, shorten each component to 2 min and retest.
02At a glance
03Original abstract
The literature was examined to determine how well the generalized matching law (Baum, 1974) describes multiple-schedule responding. In general, it describes the data well, accounting for a median of 91% of the variance. The median size of the undermatching parameter was 0.46; the median bias parameter was 1.00. The size of the undermatching parameter, and the proportion of the variance accounted for by the equation, varied inversely with the number of schedules conducted, with the number of sessions conducted per schedule, and with the time within a component. The undermatching parameter also varied with the operanda used to produce reinforcers and with the reinforcer used. The undermatching parameter did not vary consistently with component duration or with several other variables. Bias was greater when fewer rather than more schedules were conducted, when two rather than one operanda were used, and when White Carneaux rather than homing pigeons served as subjects. These results imply that the generalized matching law may describe both concurrent and multiple-schedule responding, but that the same variables do not always influence the bias and undermatching parameters in the same way for the two types of schedules.
Journal of the experimental analysis of behavior, 1986 · doi:10.1901/jeab.1986.45-83