Matching theory and induction explain operant performance

Matching theory is a general framework for understanding allocation of behavior among activities. It applies to choice in concurrent schedules and was extended to single schedules by assuming that other unrecorded behavior competes with operant behavior. Baum and Davison (2014) found that the competing activities apparently are induced by the “reinforcers” (phylogenetically important events, e.g., food) according to power functions. Combined with power-function induction, matching theory provides new equations with greater explanatory power. Four pigeons were exposed to conditions in which 7 different schedules of food delivery were presented within each experimental session. We replicated earlier results with variable-interval schedules: (a) a negatively accelerated increase of peck rate as food rate increased in the low range of food rates; (b) an upturn in pecking at higher rates; and (c) a downturn in pecking at extremely high food rates. When the contingency between pecking and food was removed, the food continued to induce pecking, even after 20 sessions with no contingency. A ratio schedule inserted in place of 1 variable-interval schedule maintained peck rates comparable to peck rates maintained by short interval schedules. We explained the results by fitting equations that combined matching theory, competition, and induction.     

Some parameters of behavioral contrast and allocation of interim behavior in rats

Two experiments examined the effects of baseline reinforcement rate and component duration on behavioral contrast and on re-allocation of interim behavior in rats. Positive behavioral contrast occurred during multiple variable-interval 10-second extinction (VI 10 EXT) after a multiple VI 10 VI 10 baseline condition, but not during multiple VI 60 EXT following multiple VI 60 VI 60 baseline. Component duration had no significant effect on contrast. These results differed from those found in studies of pigeons' key pecking. Contrast was accompanied by an increased rate of drinking in the changed component, but drinking in the constant component did not decrease. These results are not consistent with the competition theory of contrast, but are consistent with the predictions based on the matching law. However, no current theory seems to account for all instances of behavioral contrast.     

Rethinking reinforcement: allocation, induction, and contingency

The concept of reinforcement is at least incomplete and almost certainly incorrect. An alternative way of organizing our understanding of behavior may be built around three concepts: allocation, induction, and correlation. Allocation is the measure of behavior and captures the centrality of choice: All behavior entails choice and consists of choice. Allocation changes as a result of induction and correlation. The term induction covers phenomena such as adjunctive, interim, and terminal behavior-behavior induced in a situation by occurrence of food or another Phylogenetically Important Event (PIE) in that situation. Induction resembles stimulus control in that no one-to-one relation exists between induced behavior and the inducing event. If one allowed that some stimulus control were the result of phylogeny, then induction and stimulus control would be identical, and a PIE would resemble a discriminative stimulus. Much evidence supports the idea that a PIE induces all PIE-related activities. Research also supports the idea that stimuli correlated with PIEs become PIE-related conditional inducers. Contingencies create correlations between "operant" activity (e.g., lever pressing) and PIEs (e.g., food). Once an activity has become PIE-related, the PIE induces it along with other PIE-related activities. Contingencies also constrain possible performances. These constraints specify feedback functions, which explain phenomena such as the higher response rates on ratio schedules in comparison with interval schedules. Allocations that include a lot of operant activity are "selected" only in the sense that they generate more frequent occurrence of the PIE within the constraints of the situation; contingency and induction do the "selecting."     

Response and time allocation in concurrent second-order schedules

Six pigeons were trained on two-key concurrent variable-interval schedules in which the required response was the completion of a fixed number of key pecks. When the required number of pecks was equal on the two keys, response- and time-allocation ratios under-matched obtained reinforcement rate ratios. A similar result was found when the required number of pecks was unequal, except that performance, measured in response terms, was biased to the shorter required number of pecks and was less sensitive to reinforcement-rate changes. No such differences were found in the data on time spent responding. When the variable-interval schedules were kept constant and the required numbers of pecks were systematically varied, response ratios changed inversely with the ratio of the required number of pecks, but time-allocation ratios varied directly with the same independent variable. Thus, on response measures, pigeons “prefer” the schedule with the smaller peck requirement, but on time measures they “prefer” the schedule with the larger peck requirement. This finding is inconsistent with a commonsense notion of choice, which sees response and time-allocation measures as equivalent.     

Matching, contrast, and equalizing in the concurrent lever-press responding of rats

Five rats pressed levers for food reinforces delivered by several concurrent variable-interval variable-interval schedules. The rate of reinforcement available for responding on one component schedule was held constant at 60 reinforcers per hour. The rate of reinforcement available for responding on the other schedule varied from 30 to 240 reinforcers per hour. The behavior of the rats resembled the behavior of pigeons pecking keys for food reinforcers. The ratio of the overall rates of responding emitted under, and the ratio of the time spent responding under, the two components of each concurrent schedule were approximately equal to the ratio of the overall rates of reinforcement obtained from the components. The overall rate of responding emitted under, and the time spent responding under, the variable component schedule varied directly with the overall rate of reinforcement from that schedule. The overall rate of responding emitted under, and the time spent responding under, the constant component schedule varied inversely with the overall rate of reinforcement obtained from the variable component. The local rates of responding emitted under, and the local rates of reinforcement obtained from, the two components did not differ consistently across subjects. But they were not exactly equal either.     

The identities hidden in the matching laws, and their uses

Various theoretical equations have been proposed to predict response rate as a function of the rate of reinforcement. If both the rate and probability of reinforcement are considered, a simple identity, defining equation, or "law" holds. This identity places algebraic constraints on the allowable forms of our mathematical models and can help identify the referents for certain empirical or theoretical coefficients. This identity can be applied to both single and compound schedules of reinforcement, absolute and relative measures, and to local, global and overall rates and probabilities. The rate matching equations of Hernstein and Catania appear to have been approximations to, and to have been evolving toward, one form of this algebraic identity. Estimates of the bias and sensitivity terms in the generalized ratio and logarithmic matching models are here held to be averaging artifacts arising from fitting procedures applied to models that violate or conceal the underlying identities.     

Run length, visit duration, and reinforcers per visit in concurrent performance

The contingencies in each alternative of concurrent procedures consist of reinforcement for staying and reinforcement for switching. For the stay contingency, behavior directed at one alternative earns and obtains reinforcers. For the switch contingency, behavior directed at one alternative earns reinforcers but behavior directed at the other alternative obtains them. In Experiment 1, responses on the main lever, in S1, incremented stay and switch schedules and obtained a stay reinforcer when it became available. Responses on the switch lever changed S1 to S2 and obtained switch reinforcers when available. In S2, neither responses on the main lever nor on the switch lever were reinforced, but a switch response changed S2 to S1. Run lengths and visit durations were a function of the ratio of the scheduled probabilities of reinforcement (staying/switching). From run lengths and visit durations, traditional concurrent performance was synthesized, and that synthesized performance was consistent with the generalized matching law. Experiment 2 replicated and extended this analysis to concurrent variable-interval schedules. The synthesized results challenge any theory of matching that requires a comparison among the alternatives.