The mathematical principles of reinforcement (MPR) constitute of a set of mathematical equations set forth by Peter Killeen and his colleagues attempting to describe and predict the most fundamental aspects of behavior (Killeen & Sitomer, 2003).
The three key principles of MPR, arousal, constraint, and coupling, describe how incentives motivate responding, how time constrains it, and how reinforcers become associated with specific responses, respectively.
Mathematical models are provided for these basic principles in order to articulate the necessary detail of actual data.
An increase in activity level following repeated presentations of incentives is a fundamental aspect of conditioning.
Killeen, Hanson, and Osborne (1978) proposed that adjunctive (or schedule induced) behaviors are normally occurring parts of an organism's repertoire.
Delivery of incentives increases the rate of adjunctive behaviors by generating a heightened level of general activity, or arousal, in organisms.
Killeen & Hanson (1978) exposed pigeons to a single daily presentation of food in the experimental chamber and measured general activity for 15 minutes after a feeding.
They showed that activity level increased slightly directly following a feeding and then decreased slowly over time.
Competing behaviors such as goal tracking or hopper inspection are at a minimum directly after food presentation.
These behaviors increase as the interval elapses, so the measure of general activity would slowly decrease.
The increase in activity level with repeated presentation of incentives is called cumulation of arousal.
The first principle of MPR states that arousal level is proportional to rate of reinforcement,
, where: A= arousal level a= specific activation r= rate of reinforcement (Killeen & Sitomer, 2003).
This limiting factor must be taken into account in order to correctly characterize what responding could be theoretically, and what it will be empirically.
According to Killeen & Sitomer (2003), the IRT consists of two subintervals, the time required to emit a response,
may be the best measure to use, as the nature of the operandum may change arbitrarily within an experiment (Killeen & Sitomer, 2003).
Coupling is the final concept of MPR that ties all of the processes together and allows for specific predictions of behavior with different schedules of reinforcement.
The third principle of MPR states that the degree of coupling between a response and reinforcer decreases with the distance between them (Killeen & Sitomer, 2003).
The sum of this series is the coupling coefficient for fixed-ratio schedules: The continuous approximation of this is: where
According to Killeen & Sitomer (2003), the duration of a response can affect the rate of memory decay.
The sum of this process up to infinity is (Killeen 2001, Appendix): The coupling coefficient for VR schedules ends up being:
MN= lò e-lndn 0 This is the degree of saturation in memory of all responses, both target and non-target, elicited in the context (Killeen, 1994).
Expanding into a power series gives the following approximation: c» rlbt This equation predicts serious instability for non-contingent schedules of reinforcement.
To derive the coupling coefficient, the probability of the schedule not having ended, weighted by the contents of memory, must be integrated.
Simply adding b to the VT equation gives: M= b+ lò e-n’t/te-ln’ dn’ Solving the integral and multiplying by r gives the coupling coefficient for VI schedules: c= b+(1-b) rlbt The coupling coefficients for all of the schedules are inserted into the activation-constraint model to yield the predicted, overall response rate.
The third principle of MPR states that the coupling between a response and a reinforcer decreases with increased time between them (Killeen & Sitomer, 2003).
Mathematical principles of reinforcement describe how incentives fuel behavior, how time constrains it, and how contingencies direct it.
It is a general theory of reinforcement that combines both contiguity and correlation as explanatory processes of behavior.
Specific models are provided for the three basic principles to articulate predicted response patterns in many different situations and under different schedules of reinforcement.
Coupling coefficients for each reinforcement schedule are derived and inserted into the fundamental equation to yield overall predicted response rates.