A test of the formal and modern theories of matching

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.