Undermatching and overmatching: The fixed-ratio changeover requirement

Concurrent variable-interval two-minute and six-minute schedules were arranged while the fixed-ratio changeover requirement was varied among one, two, and four responses. A four-response requirement produced overmatching in the response and time data. A one-response requirement produced consistent undermatching in the time data but mixed results in the response data. The two-response requirement showed undermatching in the time data and overmatching in the response data. The results are discussed in relation to previous research using changeover requirements of five and ten responses, which produced clear tendencies toward overmatching, especially with response data. Taken together, these findings suggest that matching is not a unique result, and that undermatching or overmatching can be produced by continuous variation of the changeover requirements.     

Free-operant choice behavior: A molecular analysis

Pigeons' pecks to two concurrent initial-link stimuli occasionally produced one of two mutually exclusive terminal links. Further responding to the terminal-link stimulus produced food under fixed-interval or fixed-ratio schedules. In such concurrent chained schedules, investigators rarely use a changeover delay to control superstitious switching, although it is customary to use a changeover delay in simple concurrent schedules in which choice responses produce food directly. When terminal-link fixed-interval schedules were equal or similar in duration and no changeover delay was employed, conditional probabilities of choice for a schedule were found to be lower if the last choice was for that schedule than if the last choice was for the other schedule (“switching dependency”). Imposition of a changeover delay with equal or unequal terminal links produced the opposite pattern: conditional probabilities of choice for a schedule were higher if the last choice was for that schedule than if the last choice was for the other schedule. Turning off all chamber lights during the changeover delay interval attenuated these “repetition dependencies.” The results indicate that excessive switching can complicate the interpretation of data from concurrent chains much as from simple concurrent schedules, and that using blackouts to control switching may be preferable to using a changeover delay.     

Undermatching and overmatching as deviations from the matching law

A model of performance under concurrent variable-interval reinforcement schedules that takes as its starting point the hypothetical “burst” structure of operant responding is presented. Undermatching and overmatching are derived from two separate, and opposing, tendencies. The first is a tendency to allocate a certain proportion of response bursts randomly to a response alternative without regard for the rate of reinforcement it provides, others being allocated according to the simple matching law. This produces undermatching. The second is a tendency to prolong response bursts that have a high probability of initiation relative to those for which initiation probability is lower. This process produces overmatching. A model embodying both tendencies predicts (1) that undermatching will be more common than overmatching, (2) that overmatching, when it occurs, will tend to be of limited extent. Both predictions are consistent with available data. The model thus accounts for undermatching and overmatching deviations from the matching law in terms of additional processes added on to behavior allocation obeying the simple matching relation. Such a model thus enables processes that have been hypothesized to underlie matching, such as some type of reinforcement rate or probability optimization, to remain as explanatory mechanisms even though the simple matching law may not generally be obeyed.     

Choice, rate of reinforcement, and the changeover delay

Pigeons distribute their responses on concurrently available variable-interval schedules in the same proportion as reinforcements are distributed on the two schedules only when a changeover delay is used. The present study shows that this equality between proportions of responses and proportions of reinforcements (“matching”) is obtained when the value of the changeover delay is varied. When responses are partitioned into the set of rapid response bursts occurring during the delay interval and the set of responses occurring subsequently, the proportion of neither set of responses matches the proportion of reinforcements. Instead, each set deviates from matching but in opposite directions. Matching on the gross level results from the interaction of two patterns evident in the local response rates: (I) the lengthening of the changeover delay response burst is accompanied by a commensurate decrease in the number of changeovers; (2) the changeover delay response burst is longer than the scheduled delay duration. When delay responses are eliminated by introducing a blackout during the delay interval, response matching is eliminated; the pigeon, however, continues to match the proportion of time spent responding on a key to the proportion of reinforcements obtained on that key.     

Chained concurrent schedules: reinforcement as situation transition

Pigeons' pecks at two white response keys (initial-link situation) occasionally turned both keys red (terminal-link situation). When the two keys were red, pecks occasionally produced food, after which the keys were again white. In both situations, a changeover delay prevented the response-produced outcome from immediately following a change of responding from either key to the other. In the initial-link situation, the ratio of pecks at the keys closely paralleled the ratio of transitions into the terminal-link situation produced by the pecks, conforming to the well-known matching relation. In the terminal-link situation, the peck ratios deviated from the matching relation toward indifference. Overall response rate and rate of changeover were generally higher in the terminal-link situation than in the initial-link situation. The finding of matching in the initial-link situation supports a definition of reinforcement as situation transition. The differences in performance between the two situations, viewed in the light of other recent findings, suggest that the effects of a changeover delay depend on the overall reinforcing value of the choice alternatives.     

Matching, maximizing, and hill-climbing

In simple situations, animals consistently choose the better of two alternatives. On concurrent variable-interval variable-interval and variable-interval variable-ratio schedules, they approximately match aggregate choice and reinforcement ratios. The matching law attempts to explain the latter result but does not address the former. Hill-climbing rules such as momentary maximizing can account for both. We show that momentary maximizing constrains molar choice to approximate matching; that molar choice covaries with pigeons' momentary-maximizing estimate; and that the “generalized matching law” follows from almost any hill-climbing rule.     

Residence time and choice in concurrent foraging schedules

Five pigeons were trained on a concurrent-schedule analogue of the “some patches are empty” procedure. Two concurrently available alternatives were arranged on a single response key and were signaled by red and green keylights. A subject could travel between these alternatives by responding on a second yellow “switching” key. Following a changeover to a patch, there was a probability (p) that a single reinforcer would be available on that alternative for a response after a time determined by the value of λ, a probability of reinforcement per second. The overall scheduling of reinforcers on the two alternatives was arranged nonindependently, and the available alternative was switched after each reinforcer. In Part 1 of the experiment, the probabilities of reinforcement, ρ_red and ρ_green, were equal on the two alternatives, and the arranged arrival rates of reinforcers, λ_red and λ_green, were varied across conditions. In Part 2, the reinforcer arrival times were arranged to be equal, and the reinforcer probabilities were varied across conditions. In Part 3, both parameters were varied. The results replicated those seen in studies that have investigated time allocation in a single patch: Both response and time allocation to an alternative increased with decreasing values of λ and with increasing values of ρ, and residence times were consistently greater than those that would maximize obtained reinforcer rates. Furthermore, both response- and time-allocation ratios undermatched mean reinforcer-arrival time and reinforcer-frequency ratios.     

Choice, contingency discrimination, and foraging theory

Four pigeons were trained on eight or nine pairs of independent concurrent variable-interval schedules. The range of reinforcement ratios included extreme ratios (up to 532 to 1). Large samples of stable performance were gathered. Contrary to the findings of Davison and Jones (1995), the generalized matching law described choice more accurately than a contingency-discriminability model. Taking small samples (5 to 10 sessions) and applying a more liberal stability criterion used by Davison and Jones only increased the unsystematic variance in the data and in estimates of generalized-matching-law sensitivity. Because changing to dependent scheduling and inserting a changeover delay had no systematic effect, the deviations from generalized matching reported by Davison and Jones probably arose from imperfectly discriminated stimuli. Analysis of visits revealed that visits to the nonpreferred alternative were brief and approximately constant. When choice between the preferred (rich) and nonpreferred (lean) alternatives, regardless of position, was analyzed according to the generalized matching law, sensitivities approximated 1.0, with bias in favor of the lean alternative. This bias, which arose from an excessive frequency of visits to the lean alternative, explains undermatching as the result of fitting one line to a choice relation that consists of two displaced lines, both with a slope of 1.0. The pattern of deviation from the generalized matching line confirmed this account. The findings suggest an alternative analysis of choice that focuses on probability of visiting the lean alternative as the dependent variable. This probability was directly proportional to ratio of reinforcement. Matching, undermatching, and overmatching may all be explained by a view of concurrent performance based on foraging theory, in which responding occurs primarily at the rich alternative and is occasionally interrupted by brief visits to the lean alternative.     

Local response-rate constancy on concurrent variable-interval schedules of reinforcement

Concurrent variable-interval schedules were arranged with a main key that alternated in color and schedule assignment, along with a changeover key on which a small fixed ratio was required to changeover. Acceptable matching was observed with pigeons in two replications, but there was a tendency toward overmatching. Local response rates were found to differ for unequal schedules of a concurrent pair: local response rate was greater for the variable-interval schedule with the smaller average interreinforcement interval, but qualifications based on an interresponse-time analysis were discussed. In a second experiment, two 3-minute variable-interval schedules were arranged concurrently, and the experimental variable was the changeover procedure: either a changeover delay was incurred by each changeover or a small fixed ratio on a changeover key was required to complete a changeover. Changeover delays of 2 and 5 seconds were compared with a fixed-ratio changeover of five responses. The response output on the main key (associated with the variable-interval schedules) was greater when a changeover delay was arranged than when a fixed ratio was required to changeover. A detailed analysis of stripchart records showed that a 2-second delay generated an increased response rate for 3 seconds after a changeover, while the fixed-ratio requirement generated an increased rate during the first second only, followed by a depressed response rate for 2 seconds.     

Matching under concurrent fixed-ratio variable-interval schedules of food presentation

Four pigeons were exposed to concurrent fixed-ratio, variable-interval schedules of food presentation. The fixed-ratio requirement was either 25, 50, 75, or 100 responses, with the variable-interval schedule parameter held constant at 4 minutes. A delay time was imposed between a changeover from one schedule to the other and subsequent food availability. The delay time was varied at each ratio requirement over four values; no delay, 0-second delay, 1.5-second delay, and 5.0-second delay. As the fixed-ratio requirement or the delay time increased, a greater proportion of the total responses and time spent responding occurred under the variable-interval schedule relative to the proportion of food deliveries under that schedule. Neither relative overall response rate nor relative time spent responding equalled the relative frequency of food presentation, as would be predicted by a linear “matching” model. Rather, these data were described by power functions with slopes of approximately 1.0 and intercepts greater than 1.0. In the terms of Baum's (1974) analysis, these deviations from linear matching represent bias in favor of responding under the interval schedule. Bias, as reflected in the intercept of the power function, was greater for the ratio of time than the ratio of responses.     

IS THERE A DECISIVE TEST BETWEEN MATCHING AND MAXIMIZING?

Reinforcers under typical concurrent variable-interval, variable-ratio schedules may be (a) earned and obtained during the variable-interval component, (b) earned and obtained during the variable-ratio component, or (c) earned during the variable-ratio component and obtained during the variable-interval component. Categories a and b, which have no bearing on matching versus maximizing accounts of choice, were set at zero. The rate of Category c reinforcers and the duration of a changeover delay were varied. Simple matching, which predicts exclusive choice of the variable-interval component, and strict maximizing of overall reinforcement rate, which predicts a bias towards the variable-ratio component, were both disconfirmed: Subjects spent approximately 25% of their time in the variable-ratio component, contrary to the matching prediction, but earned only about one third of the reinforcers predicted by strict maximizing. However, maximizing describes the findings functionally in terms of discounting of delayed reinforcers; matching may describe the data in terms of a restructuring of the alternatives. Matching and maximizing are not competing theories about the fundamental nature of choice, but compatible points of view that may reveal environmental function and behavioral structure.     

A model for residence time in concurrent variable interval performance

A component-functions model of choice behavior is proposed for performance on interdependent concurrent variable-interval (VI) variable-interval schedules based on the product of two component functions, one that enhances behavior and one that reduces behavior. The model is the solution to the symmetrical pair of differential equations describing behavioral changes with respect to two categories of reinforcers: enhancing and reducing, or excitatory and inhibitory. The model describes residence time in interdependent concurrent VI VI schedules constructed from arithmetic and exponential distributions. The model describes the data reported by Alsop and Elliffe (1988) and Elliffe and Alsop (1996) with a variance accounted for of 87% compared to 64% accounted for by the Davison and Hunter (1976) model and 42% by Herrnstein's (1970) hyperbola. The model can explain matching, undermatching, and overmatching in the same subject under different procedures and has the potential to be extended to performance on concurrent schedules with more than two alternatives, multiple schedules, and single schedules. Thus, it can be considered as an alternative to Herrnstein's quantitative law of effect.     

Preference after training with differential changeover delays

Pigeons were trained on a multiple schedule in which each component consisted of concurrent variable-interval (VI) 30-s VI 60-s schedules. The two components of the multiple schedule differed only in terms of the changeover delays (COD): For one component short CODs were employed, and in the second component long CODs were used. After approximate matching was obtained in each component, probe tests involving new combinations of stimuli were presented (e.g., the VI 30-s schedule from each component) to determine how the different CODs affected preference. Despite shorter CODs producing higher changeover rates, the COD value had no systematic effect on preference on the probe trials. However, differences in reinforcement rate always produced preference for the schedule with the higher reinforcement rate. The results thus show that the the pattern of changeover behavior per se is not a critical determinant of choice in the probe-trial procedure.     

Time allocation in human vigilance

Three human subjects detected unpredictable signals by pressing either of two telegraph keys. The relative frequencies with which detections occurred for the two alternatives were varied. The procedure included a changeover delay and response cost for letting go of a key. All subjects matched the relative time spent holding each key to the relative number of detections for that key, in conformity with the matching law. One subject's performance, which at first deviated from the relation, came into conformity with it when response cost was increased. Another subject's performance approximated matching more closely when the changeover delay was increased. The results confirm and extend the notions that choice consists in time allocation and that all behavior can be measured on the common scale of time.     

Preference for signaled over unsignaled shock schedules: Ruling out asymmetry and response fixation as factors

Experiment 1 tested whether a “symmetrical” choice procedure yields results different from those previously reported using the “unidirectional” standard changeover procedure (e.g., Badia & Culbertson, 1972). Subjects could change at any time from unsignaled to signaled shock by pressing a lever and from signaled to unsignaled shock by pressing a second lever. Results were identical to those of the standard procedure and showed that the standard procedure is fully adequate. Experiment 2 tested whether choice of high density signaled shock over low-density unsignaled shock (Badia, Coker, & Harsh, 1973) resulted from initial training with equal-density schedules. Subjects were trained and tested with signaled shock twice as dense as unsignaled shock. Three of four subjects strongly preferred the signaled condition, thus ruling out carry-over and “response fixation” as alternative explanations.     

Effects of the discriminability of alternatives in three-alternative concurrent-schedule performance

Six pigeons were trained on two- and three-alternative concurrent schedules in which the alternatives were signaled by different wavelengths of light on the main pecking key. The schedules were arranged according to a switching-key procedure in which pecks on a white side key produced a 3-s blackout and, intermittently, a change in the variable-interval schedule of food programmed on the main (center) key after the blackout. In Part 1, a two-alternative concurrent variable-interval schedule was arranged in which the alternatives were signaled by 560 nm and 630 nm. Parts 2 and 3 arranged three-alternative concurrent variable-interval schedules with the alternatives signaled by 560 nm, 600 nm, and 630 nm (Part 2) and 560 nm, 623 nm, and 630 nm (Part 3). Within each part, the relative rate of food reinforcers available on the alternatives was varied across a wide range. In all parts of the experiment, the ratios of responses emitted between pairs of alternatives were more extreme than the ratios of reinforcers obtained on the pairs of alternatives, a result termed overmatching. In Parts 2 and 3, generalized matching sensitivities between pairs of alternatives were found to be higher when the reinforcer rate on the third alternative was low than when it was high—an apparent failure of the constant-ratio rule. The data were well described by an extension of the Davison and Jenkins (1985) model, which assumes differing discriminabilities between concurrent-schedule alternatives in combination with a punishing effect of blackout following changeovers.     

On the falsifiability of matching theory.

Herrnstein's matching theory requires the parameter, k, which appears in the single-alternative form of the matching equation, to remain invariant with respect to changes in reinforcement parameters like magnitude or immediacy. Recent experiments have disconfirmed matching theory by showing that the invariant-k requirement does not hold. However, the theory can be asserted in a purely algebraic form that does not require an invariant k and that is not disconfirmed by the recent findings. In addition, both the original and the purely algebraic versions of matching theory can be asserted in forms that allow for commonly observed deviations from matching (bias, undermatching, and overmatching). The recent finding of a variable k does not disconfirm these versions of matching theory either. As a consequence, matching remains a viable theory of behavior, the strength of which lies in its general conceptualization of all behavior as choice, and in its unified mathematical treatment of single- and multialternative environments.     

Leaving patches: Effects of travel requirements

Five pigeons were trained in an analogue foraging procedure in which, by completing a travel requirement, they entered a “patch” in which a reinforcer might be available after an unpredictable time. They also had the opportunity, by emitting a defined response, to exit the patch and travel to another patch. Prey availability in a patch was not signaled. Data were collected on the length of time that subjects stayed in patches before exiting (residence times) as a function of various travel requirements: travel for a fixed time in blackout, fixed-interval schedule traveling, fixed-time traveling with an added response required to terminate traveling, and fixed-ratio traveling. For each of these conditions, the required amount of travel (time or responses) was varied over a wide range. As previously reported, residence times increased with increases in fixed-time traveling, as they did with increasing fixed-interval or fixed-ratio traveling. There was no evidence that adding response or work requirements systematically affected residence time except via increased travel time, although 3 of the 5 birds stayed longer in a patch under higher fixed-ratio values. A “threshold-maximization” model described the data well with a single parameter that was consistent across subjects, procedures, and experiments.     

Choice: Effects of changeover schedules on concurrent performance

The components of concurrent schedules were separated temporally by placing interval schedules on the changeover key. The rates of responding on both the main and changeover keys were examined as a function of the reinforcement rates. In the first experiment, the sensitivity of main-key performance to changing reinforcement rates was inversely related to the temporal separation of components, and changeover performance was monotonically related to the ratio of the reinforcement rates. In the second experiment, when the ratio of the reinforcement rates was scheduled to remain constant while the frequency of reinforcement was varied, changeover performance did not remain constant. A “sampling” interpretation of changeover responding was proposed and subsequently tested in a third experiment where extinction was always scheduled in one component and the frequency of reinforcement was varied in the second component. It was concluded that changeover performance can be interpreted using molar measures of reinforcement and that animals sample activities available to them at rates which are controlled by relative reinforcement rates.     

The matching law

The matching law may be viewed either as an empirical generalization, and therby subject to disproof, or as part of a system of equations used to define the utility (“value”) of a reinforcer. In the latter case it is tautologous, and not subject to disproof within the defining context. A failure to obtain matching will most often be a signal that the independent variables have not been properly scaled. If, however, the proper transformations have been made on the independent variables, and matching is not obtained, the experimental paradigm may be outside the purview of the matching law. At that point, reinterpretations or revisions of the law are called for. The theoretical matching law is but one of many possible ways to define utility, and it may eventually be rejected in favor of a more useful definition.