Choice behavior in a discrete-trial concurrent VI-VR: A test of maximizing theories of matching

Rats were trained on a discrete-trial procedure in which one alternative (VR) was correlated with a constant probability of reinforcement while the other was correlated with a VI schedule which ran during the intertrial intervals and held the scheduled reinforcer until they were obtained by the next VI response. Relative reinforcement rate was varied in series of conditions in which the VR schedule was varied and in series in which the VI was varied. Choice behavior was described well by the generalized matching law, although moderate undermatching occurred for all subjects. Contrary to the predictions of molar maximizing (optimality) theories, there was no consistent bias in favor of the ratio alternative, and the sensitivity to reinforcement allocation was not systematically affected by whether the ratio or interval schedule was varied. The results were also contrary to momentary maximizing accounts, as there was no correspondence between the probability of a changeover to the VI behavior and the time since the last response to the VI alternative. Neither variety of maximizing theory appears to provide a general explanation of matching in concurrent schedules.     

WITHIN‐SESSION TRANSITIONS IN CHOICE: A STRUCTURAL AND QUANTITATIVE ANALYSIS

The present study used within‐session transitions between two concurrent schedules to evaluate choice in transition. Eight female Long‐Evans rats were trained to respond under concurrent schedules of reinforcement during experimental sessions that lasted 22 hr. The generalized matching equation was used to model steady‐state behavior at the end of each session, while transitional behavior that emerged following the change in reinforcement schedules was modeled using a logistic equation. The generalized matching and logistic equations were appropriate models for behavior generated during single‐session transitions. A local analysis of behavior on the two response alternatives during acquisition was used to determine the source of preference as revealed in response ratios. The number of “low‐response” visits, those containing three to five responses, remained stable. Preference ratios largely reflected a sharp increase in the number of visits with long response bouts on the rich alternative and a decrease in the number of such visits to the leaner alternative.     

Concurrent-schedule performance in dairy cows: persistent undermatching.

Performance of dairy cows responding under concurrent variable-interval variable-interval schedules of food delivery was examined, with results analyzed in terms of the generalized matching equation. In Experiment 1, bias measures indicated that crushed barley was preferred over meatmeal when these foods were available under the alternative schedules. For whole-session data, substantial undermatching of response and time-allocation ratios to obtained reinforcement ratios was evident. Postreinforcement pause time ratios approximately matched obtained reinforcement rates. Subtracting these times from total time-allocation values yielded net time-allocation ratios that undermatched obtained reinforcement ratios to a greater degree than did whole-session time-allocation ratios. In Experiment 2, substantial undermatching was evident when the same foods (hay for 2 cows, crushed barley for 2 others) were available under the alternative schedules. Food-related activities and other defined behavior not related to food were quantified by direct observation, and were found to occupy a substantial proportion (roughly 40% to 80%) of experimental sessions. Subtracting the time spent in these activities from the time allocated to each component schedule did not reduce the degree of undermatching obtained. Across all conditions in both experiments, slopes of regression lines relating behavioral outputs to environmental inputs characteristically were below 0.6, which agrees with prior findings and suggests that, contrary to suggestions in the literature, undermatching in dairy cows is not the result of using different foods under alternative schedules or differential pausing under those schedules.     

Concurrent schedule assessment of food preference in cows

Six dairy cows (Bos taurus) were trained on several pairs of concurrent variable-interval schedules with different types of food available on each alternative. The required response was a plate press made by the animal's muzzle. Performance generally replicated that found with other species. The generalized matching law accounted for the preference data, showing that food preference could be quantitatively analyzed as a special case of response bias. The preference functions showed that the response- and time-allocation ratios were not as extreme as obtained reinforcement rate ratios (undermatching).     

Matching, undermatching, and overmatching in studies of choice

Almost all of 103 sets of data from 23 different studies of choice conformed closely to the equation: log (B(1)/B(2)) = a log (r(1)/r(2)) + log b, where B(1) and B(2) are either numbers of responses or times spent at Alternatives 1 and 2, r(1) and r(2) are the rates of reinforcement obtained from Alternatives 1 and 2, and a and b are empirical constants. Although the matching relation requires the slope a to equal 1.0, the best-fitting values of a frequently deviated from this. For B(1) and B(2) measured as numbers of responses, a tended to fall short of 1.0 (undermatching). For B(1) and B(2) measured as times, a fell to both sides of 1.0, with the largest mode at about 1.0. Those experiments that produced values of a for both responses and time revealed only a rough correspondence between the two values; a was often noticeably larger for time. Statistical techniques for assessing significance of a deviation of a from 1.0 suggested that values of a between .90 and 1.11 can be considered good approximations to matching. Of the two experimenters who contributed the most data, one generally found undermatching, while the other generally found matching. The difference in results probably arises from differences in procedure. The procedural variations that lead to undermatching appear to be those that produce (a) asymmetrical pausing that favors the poorer alternative; (b) systematic temporal variation in preference that favors the poorer alternative; and (c) patterns of responding that involve changing over between alternatives or brief bouts at the alternatives.     

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.     

Concurrent choice: Effects of overall reinforcer rate and the temporal distribution of reinforcers

Six pigeons responded on a series of concurrent exponential variable-interval schedules, offering a within-subject comparison with previously published data from concurrent arithmetic variable-interval schedules. Both relative and overall reinforcer rates were varied between conditions. The generalized matching law described the data well, with undermatching much more frequent than strict matching. Time-allocation sensitivity consistently exceeded response-allocation sensitivity for both schedule types, and exponential-schedule sensitivity exceeded arithmetic-schedule sensitivity for both measures of choice. A further set of conditions using variable-interval schedules whose shortest interval was correlated with the mean interval, like arithmetic schedules, but that provided a constant conditional probability of reinforcement, like exponential schedules, produced sensitivities between those produced by conventional arithmetic and exponential schedules. Unlike previous arithmetic-schedule results, exponential sensitivity changed nonmonotonically with changes in overall reinforcer rate. The results clarify our knowledge of the effects of arithmetic and exponential schedules but confuse our understanding of the effects of overall reinforcer rate on concurrent choice.     

Human group choice: discrete-trial and free-operant tests of the ideal free distribution

Ideal free distribution theory predicts that foragers will form groups proportional in number to the resources available in alternative resource sites or patches, a phenomenon termed habitat matching. Three experiments tested this prediction with college students in discrete-trial simulations and a free-operant simulation. Sensitivity to differences in programmed reinforcement rates was quantified by using the sensitivity parameter of the generalized matching law (s). The first experiment, replicating prior published experiments, produced a greater degree of undermatching for the initial choice (s = 0.59) compared to final choices (s = 0.86). The second experiment, which extended prior findings by allowing only one choice per trial, produced comparable undermatching (s = 0.82). The third experiment used free-operant procedures more typical of laboratory studies of habitat matching with other species and produced the most undermatching (s = 0.71). The results of these experiments replicated previous results with human groups, supported predictions of the ideal free distribution, and suggested that undermatching represents a systematic deviation from the ideal free distribution. These results are consistent with a melioration account of individual behavior as the basis for group choice.     

An analytic comparison of Herrnstein's equations and a multivariate rate equation

Herrnstein's equations are approximations of the multivariate rate equation at ordinary rates of reinforcement and responding. The rate equation is the result of a linear system analysis of variable-interval performance. Rate equation matching is more comprehensive than ordinary matching because it predicts and specifies the nature of concurrent bias, and predicts a tendency toward undermatching, which is sometimes observed in concurrent situations. The rate equation contradicts one feature of Herrnstein's hyperbola, viz., the theoretically required constancy of k. According to the rate equation, Herrnstein's k should vary directly with parameters of reinforcement such as amount or immediacy. Because of this prediction, the rate equation asserts that the conceptual framework of matching does not apply to single alternative responding. The issue of the constancy of k provides empirical grounds for distinguishing between Herrnstein's account and a linear system analysis of single alternative variable-interval responding.     

Maximizing versus matching on concurrent variable-interval schedules

Maximization and matching predictions were examined for a time-based analogue of the concurrent variable-interval variable-ratio schedule. One alternative was a variable interval whose time base operated relatively independent of the schedule chosen, and the other was a discontinuous variable interval for which timing progressed only when selected. Pigeons switched between schedules by pecking a changeover key. The maximization hypothesis predicts that subjects will show a bias toward the discontinuous variable interval and undermatching; however the obtained results conformed closely to the predictions of the matching law. Finally, a quantitative comparison was made of the bias and sensitivity estimates obtained in published concurrent variable-interval variable-ratio analogue studies. Results indicated that only the ratio-based analogue of the concurrent variable interval variable ratio studied by Green, Rachlin, and Hanson (1983) produced significant bias toward the variable-ratio alternative and undermatching, as predicted by reinforcement maximization.     

The generalized matching law as a description of multiple-schedule responding

The literature was examined to determine how well the generalized matching law (Baum, 1974) describes multiple-schedule responding. In general, it describes the data well, accounting for a median of 91% of the variance. The median size of the undermatching parameter was 0.46; the median bias parameter was 1.00. The size of the undermatching parameter, and the proportion of the variance accounted for by the equation, varied inversely with the number of schedules conducted, with the number of sessions conducted per schedule, and with the time within a component. The undermatching parameter also varied with the operanda used to produce reinforcers and with the reinforcer used. The undermatching parameter did not vary consistently with component duration or with several other variables. Bias was greater when fewer rather than more schedules were conducted, when two rather than one operanda were used, and when White Carneaux rather than homing pigeons served as subjects. These results imply that the generalized matching law may describe both concurrent and multiple-schedule responding, but that the same variables do not always influence the bias and undermatching parameters in the same way for the two types of schedules.     

Reinforcement uncertainty enhances preference for choice in humans

Under concurrent‐chains schedules of reinforcement, participants often prefer situations that allow selection among alternatives (free choice) to situations that do not (forced choice). The present experiment examined the effects of reinforcement probability on choice preferences. Preferences for free versus forced choice were measured under a condition in which participants' choices were always reinforced (reinforcement probability of 1.0) and a condition in which outcomes were uncertain (reinforcement probability of 0.5). Forty‐four college students participated and preferences were examined under a concurrent‐chains schedule of reinforcement. Participants preferred free choice under uncertain reinforcement, but a bias toward free choice was not observed when reinforcement was certain. These results align with previous findings of preference for free choice under conditions of uncertainty, but suggest that preference may be dependent upon probabilistic reinforcement contingencies in the terminal links of the concurrent‐chains arrangement. Thus, reinforcement probability is an important variable to consider when conducting similar studies on the value of choice.     

Limited matching on concurrent-schedule reinforcement of academic behavior

Three adolescent students with special educational needs were given a choice between completing one of two available sets of math problems. Reinforcers (nickels) across these alternatives were arranged systematically in separate experimental phases according to three different concurrent variable-interval schedules (reinforcement ratios of 2:1, 6:1, and 12:1). Time allocated to the two stacks of math problems stood in linear relationship to the reinforcement rate obtained from each stack, although substantial undermatching and bias were observed for all subjects. However, changes in the schedules were not followed by changes in allocation patterns until adjunct procedures (e.g., changeover delays, limited holds, timers, and demonstrations) were introduced. The necessity of adjunct procedures in establishing matching in applied situations is discussed as a limitation to quantitative applications of the matching law in applied behavior analysis.     

QUANTIFICATION OF ETHANOL'S ANTIPUNISHMENT EFFECT IN HUMANS USING THE GENERALIZED MATCHING EQUATION

Increases in rates of punished behavior by the administration of anxiolytic drugs (called antipunishment effects) are well established in animals but not humans. The present study examined antipunishment effects of ethanol in humans using a choice procedure. The behavior of 5 participants was placed under six concurrent variable‐interval schedules of monetary reinforcement. In three of the six concurrent schedules, punishment, in the form of monetary loss, was superimposed on one alternative. Data were analyzed according to the generalized matching equation which distinguishes between bias (allocation of behavior beyond what matching to relative reinforcer densities would predict) and sensitivity to reinforcement (how well behavior tracks relative reinforcer densities). In addition, participants completed a pencil‐tapping test. Under placebo punishment conditions, all participants demonstrated low response rates and a bias against the alternative associated with punishment, despite a resultant loss of available reinforcers. Bias against the punished alternative was dose‐dependently reduced in participants shown to be most sensitive to ethanol (0.6, 1.2, and 1.8 g/kg) in measures of overall responding and on the pencil‐tapping test. No ethanol‐induced change in bias was noted when punishment was not imposed. Sensitivity to reinforcement also decreased for participants shown to be sensitive to ethanol. In addition to extending antipunishment effects to humans, these results also show that antipunishment effects can be quantified via the matching equation.     

Response Form, Force, And Number: Effects On Concurrent-schedule Performance

Six hens responded on concurrent variable-interval (key-peck) variable-interval (door-push) schedules of reinforcement in which the second-order (fixed-ratio) requirements on the alternatives (Experiment 1) or the required door forces (Experiment 2) were varied. The key-peck and door-push response (measured as fixed-ratio completion) and time data were well described by the generalized matching law. However, the manipulations of fixed-ratio requirement and required response force differed in their effects. The manipulations of fixed-ratio size affected the response and time measures differently, producing fairly constant, multiplicative biases only in terms of response allocation. It was argued that variations in fixed-ratio size necessarily change the time allocated to that response unit, and thus changes in time bias were not necessarily a fundamental effect of changing the ratio. In contrast, the changes in response bias were a fundamental result of changes in ratio size. The response-force manipulations produced similar bias shifts in terms of response and time allocation, but they appeared to combine with relative reinforcement rate to affect choice interactively. Specifically, behavior appeared to be biased towards the least effortful (i.e., key-peck) response, but the increases in door force had a larger effect on bias when the hens were making this response infrequently (on a lean schedule). The different effects of the fixed-ratio and response-force manipulations on concurrent performance were partially accounted for by the differing times required to complete each response unit under those manipulations, but this would not account for the interaction. The interaction would be consonant with increased response effort decreasing the effective value of the associated reinforcement schedule.     

Choice, changeover, and travel

Since foraging in nature can be viewed as instrumental behavior, choice between sources of food, known as “patches,” can be viewed as choice between instrumental response alternatives. Whereas the travel required to change alternatives deters changeover in nature, the changeover delay (COD) usually deters changeover in the laboratory. In this experiment, pigeons were exposed to laboratory choice situations, concurrent variable-interval schedules, that were standard except for the introduction of a travel requirement for changeover. As the travel requirement increased, rate of changeover decreased and preference for a favored alternative strengthened. When the travel requirement was small, the relations between choice and relative reinforcement revealed the usual tendencies toward matching and undermatching. When the travel requirement was large, strong overmatching occurred. These results, together with those from experiments in which changeover was deterred by punishment or a fixed-ratio requirement, deviate from the matching law, even when a correction is made for cost of changeover. If one accepted an argument that the COD is analogous to travel, the results suggest that the norm in choice relations would be overmatching. This overmatching, however, might only be the sign of an underlying strategy approximating optimization.     

The generalized matching law in elite sport competition: football play calling as operant choice

A mathematical model of operant choice, the generalized matching law was used to analyze play-calling data from the 2004 National Football League season. In all analyses, the relative ratio of passing to rushing plays was examined as a function of the relative ratio of reinforcement, defined as yards gained, from passing versus rushing. Different analyses focused on season-aggregate data for the league as a whole, game-by-game data for the league as a whole, and game-by-game data for individual teams. In all analyses except those for a few individual teams, the generalized matching law accounted for a majority of variance in play calling. The typical play-calling pattern reflected undermatching (suggesting imperfect sensitivity of play calling to yardage-gained reinforcers) and a bias for calling rushing plays. Bias was found to be a function of both the relative risk of turnovers and the relative variability in yards gained associated with passing versus rushing plays. The external validity of the matching analyses was supported by significant correlations between parameters of the generalized matching law and team success on offense and season winning percentage. These results illustrate the broad applicability of the generalized matching law to problems outside of the laboratory.