Mathematical principles of reinforcement and resistance to change

Although Killeen's mathematical principles of reinforcement (MPR) apply to the asymptotic rate of a free operant after extended exposure to a single schedule of reinforcement, they can be extended to resistance to change in multiple schedules via alterations in the parameter representing the activating effects of reinforcers. MPR's predictions of resistance to change in relation to reinforcer rate on variable-interval (VI) schedules are empirically correct and agree with behavioral momentum theory (BMT). However, both MPR and BMT encounter problems in accounting for the effects of delayed reinforcement on resistance to change, relative to immediate reinforcement at the same rate. Further problems are raised by differences in resistance to change between variable-ratio (VR) and variable-interval performances maintained by the same reinforcer rate. With both delayed versus immediate reinforcement and variable-ratio versus variable-interval reinforcement, differential resistance to change is negatively correlated with the log ratios of baseline response rates when reinforcer rates are equated. Cases where resistance to change varies despite equated reinforcer rates, and the correlations among behavioral measures, provide challenges and opportunities for both MPR and BMT.