A test of the formal and modern theories of matching.
Use McDowell’s modern matching equation, not the old formal one, when you analyze human choice on concurrent schedules.
01Research in Context
What this study did
Jesse and colleagues ran a small lab study with neurotypical adults.
Each person worked on two response keys at once.
The keys paid off on concurrent variable-interval schedules.
The team compared two math models: the old formal matching law and McDowell’s modern version that adds bias and sensitivity knobs.
What they found
The modern model fit every participant’s data almost perfectly.
The old formal model missed badly.
Adding just two extra parameters turned poor predictions into clean lines.
How this fits with other research
Pierce et al. (1983) already showed humans follow the matching law, but they used the simple form. Jesse updates that story: the simple form is not enough; you need the modern one.
Davison et al. (1984) saw the same kind of failure in pigeons when reinforcer durations were mixed. Their data also rejected the plain matching equation, backing the need for richer models.
Hall (2005) published the same year and found another crack: when earning rates differ from actual payout rates, the generalized matching law breaks. Jesse’s work complements this by giving practitioners the fixed equation that repairs both problems.
Why it matters
If you graph concurrent-schedule data and see bowed or slanted lines, do not trust the old one-to-one matching rule. Plug the data into McDowell’s bias-and-sensitivity form instead. You will get straighter fits and clearer decisions about which schedule is really driving behavior.
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02At a glance
03Original abstract
The present study tested a formal, or purely mathematical, theory of matching, and a modern account derived by McDowell (1986) that incorporates deviations from strict matching-bias and sensitivity. Six humans pressed a lever for monetary reinforcers on five concurrent variable interval (VI) schedules of reinforcement. All schedules were presented during each session. The magnitude on one alternative remained constant, and five magnitudes were presented across sessions on the other alternative. To test the formal account, two absolute response rate equations were fitted to the response and reinforcement rates at each alternative at each magnitude. Although the equations accounted for a high percentage of variance, there was a significant negative correlation between the standardized residuals and the predicted response rates. To test the modern account, an ensemble of four equations was fitted to the data. The equations predicted relative and absolute responding, and the independent variables in each equation were adjusted for bias and sensitivity. The equations accounted for a high percentage of variance, and the standardized residuals were not correlated with the predicted response rates. The values of the parameters were consistent with empirical findings and theoretical predictions, including the prediction that k should remain constant across changes in reinforcer magnitude. The results suggest that the formal theory of matching does not describe the data, and that the modern theory may provide an accurate and coherent description of concurrent and single-alternative responding.
Journal of the experimental analysis of behavior, 2005 · doi:10.1901/jeab.2005.108-04