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.     

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.     

Computational behavior dynamics: an alternative description of Nevin (1969).

A computational processing behavior-dynamic model was instantiated in the form of a computer program that "behaved" on the task developed by Nevin (1969). In this classic discrete-trials experiment, the relative frequency of choosing a response alternative matched the relative frequency of reinforcement for that alternative, the local structure of responding was opposite that predicted by momentary maximizing (i.e., the probability of a changeover decreased with run length), and absolute and relative response rates varied independently. The behavior-dynamic model developed here qualitatively reproduced these three results (but not in quantitative and specific detail) and also generated some interesting, as-yet-untested predictions about performance in Nevin's task. The model was discussed as an example of a stochastic behavior-dynamic alternative to algebraic behavior theory.     

Matching in a network

Pigeons were given practice choosing between pairs of alternatives yielding different frequencies of reinforcement. Four individual alternatives were set into four pairwise choices. Averaged over subjects, the distribution of responses in each choice approximated matching. The four individual alternatives were then presented, two by two, in two pairwise choices for which there had been no initial practice. No further reinforcement was given during the tests with the new pairs. Transfer to the two test pairs deviated systematically from matching in most cases by exaggerating the preference for the alternative that had had the higher frequency of reinforcement.     

GROUP FORAGING SENSITIVITY TO PREDICTABLE AND UNPREDICTABLE CHANGES IN FOOD DISTRIBUTION: PAST EXPERIENCE OR PRESENT CIRCUMSTANCES?

The ideal free distribution theory (Fretwell & Lucas, 1970) predicts that the ratio of foragers at two patches will equal the ratio of food resources obtained at the two patches. The theory assumes that foragers have "perfect knowledge" of patch profitability and that patch choice maximizes fitness. How foragers assess patch profitability has been debated extensively. One assessment strategy may be the use of past experience with a patch. Under stable environmental conditions, this strategy enhances fitness. However, in a highly unpredictable environment, past experience may provide inaccurate information about current conditions. Thus, in a nonstable environment, a strategy that allows rapid adjustment to present circumstances may be more beneficial. Evidence for this type of strategy has been found in individual choice. In the present experiments, a flock of pigeons foraged at two patches for food items and demonstrated results similar to those found in individual choice. Experiment 1 utilized predictable and unpredictable sequences of resource ratios presented across days or within a single session. Current foraging decisions depended on past experience, but that dependence diminished when the current foraging environment became more unpredictable. Experiment 2 repeated Experiment I with a different flock of pigeons under more controlled circumstances in an indoor coop and produced similar results.     

Undermatching: a reappraisal of performance on concurrent variable-interval schedules of reinforcement

The extant data for pigeons' performance on concurrent variable-interval schedules were examined in detail. Least-squares lines relating relative pecks and time to the corresponding relative reinforcements were obtained for four studies. The between-study group slopes for time and pecks and five of seven within-study group slopes from individual studies were less than 1.00. This suggested the generality that pigeons respond less to the richer reinforcement schedule than predicted by matching. For pecks, a nonparametric test for distribution of points also supported this concept of undermatching (to the richer reinforcement schedule). In addition, using mean squared error as the criterion, a cubic curve fit the peck proportion data better than any line or other polynomial. This indicates that the relation between peck and reinforcement proportions may be nonlinear.     

Matching and conditioned reinforcement rate

Attempts to examine the effects of variations in relative conditioned reinforcement rate on choice have been confounded by changes in rates of primary reinforcement or changes in the value of the conditioned reinforcer. To avoid these problems, this experiment used concurrent observing responses to examine sensitivity of choice to relative conditioned reinforcement rate. In the absence of observing responses, unsignaled periods of food delivery on a variable-interval 90-s schedule alternated with extinction on a center key (i.e., a mixed schedule was in effect). Two concurrently available observing responses produced 15-s access to a stimulus differentially associated with the schedule of food delivery (S+). The relative rate of S+ deliveries arranged by independent variable-interval schedules for the two observing responses varied across conditions. The relation between the ratio of observing responses and the ratio of S+ deliveries was well described by the generalized matching law, despite the absence of changes in the rate of food delivery. In addition, the value of the S+ deliveries likely remained constant across conditions because the ratio of S+ to mixed schedule food deliveries remained constant. Assuming that S+ deliveries serve as conditioned reinforcers, these findings are consistent with the functional similarity between primary and conditioned reinforcers suggested by general choice theories based on the concatenated matching law (e.g., contextual choice and hyperbolic value-added models). These findings are inconsistent with delay reduction theory, which has no terms for the effects of rate of conditioned reinforcement in the absence of changes in rate of primary reinforcement.     

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.     

Matching, delay-reduction, and maximizing models for choice in concurrent-chains schedules.

Models of choice in concurrent-chains schedules are derived from melioration, generalized matching, and optimization. The resulting models are compared with those based on Fantino's (1969, 1981) delay-reduction hypothesis. It is found that all models involve the delay reduction factors (T - t2L) and (T - t2R), where T is the expected time to primary reinforcement and t2L, t2R are the durations of the terminal links. In particular, in the case of equal initial links, the model derived from melioration coincides with Fantino's original model for full (reliable) reinforcement and with the model proposed by Spetch and Dunn (1987) for percentage (unreliable) reinforcement. In the general case of unequal initial links, the model derived from melioration differs from the revised model advanced by Squires and Fantino (1971) only in the factors affecting the delay-reduction terms (T - t2L) and (T - t2R). The models of choice obtained by minimizing the expected time to reinforcement depend on the type of feedback functions used. In particular, if power feedback functions are used, the optimization model coincides with that obtained from melioration.     

Toward a quantitative theory of punishment

In two experiments, pigeons' key pecking for food on concurrent variable-interval schedules was punished with electric shock according to concurrent variable-interval punishment schedules. With unequal frequencies of food but equal rates of punishment associated with the two keys and at several intensities of shock, the response and time allocation of all six pigeons overmatched the obtained relative frequency of food. The overmatching was predicted by a subtractive model of the interaction between punishment and positive reinforcement but not by two alternative models. Increases in the k and r_e parameters of the generalized matching law could not account for the observed shifts in preference.     

Separating The Effects Of Trial-specific And Average Sample-stimulus Duration In Delayed Matching To Sample In Pigeons

Pigeons were studied in two experiments employing delayed matching-to-sample (DMTS) tasks in which the reduction in delay to reinforcement signaled by the onset of the sample stimulus was manipulated by varying sample-stimulus duration. In Experiment 1, the duration of the sample stimulus was either 5 s or 10 s for one sample stimulus and 10 s or 20 s for the other. Subjects matched more frequently when the sample duration was 10 s following the sample associated with the shorter average duration. This finding is analogous to the memory distribution effect found by Honig (1987) in a successive DMTS task that varied retention interval. In Experiment 2, sample duration was either 5 s or 15 s. In Phases 1 and 3 each sample duration was correlated with a particular sample color, and in Phase 2 sample duration and color were uncorrelated. When sample duration was 5 s, subjects matched more frequently when sample duration and color were correlated than when they were uncorrelated. Overall, subjects matched more frequently when sample duration and color were correlated. The data from both experiments support Wixted's (1989) model, which states that one determinant of choice in a DMTS task is the delay-reduction value of the sample stimulus.     

The Kent Face Matching Test

This study presents the Kent Face Matching Test (KFMT), which comprises 200 same-identity and 20 different-identity pairs of unfamiliar faces. Each face pair consists of a photograph from a student ID card and a high-quality portrait that was taken at least three months later. The test is designed to complement existing resources for face-matching research, by providing a more ecologically valid stimulus set that captures the natural variability that can arise in a person's appearance over time. Two experiments are presented to demonstrate that the KFMT provides a challenging measure of face matching but correlates with established tests. Experiment 1 compares a short version of this test with the optimized Glasgow Face Matching Test (GFMT). In Experiment 2, a longer version of the KFMT, with infrequent identity mismatches, is correlated with performance on the Cambridge Face Memory Test (CFMT) and the Cambridge Face Perception Test (CFPT). The KFMT is freely available for use in face-matching research.     

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.     

On the limits of the matching concept in monkeys (Cebus apella).

Two cebus monkeys, with many years of experience matching a variety of static visual stimuli (forms and colors) within a standard matching-to-sample paradigm, were trained to press a left lever when a pair of displayed static stimuli were the same and to press a right lever when they were different. After learning the same/different task, the monkeys were tested for transfer to dynamic visual stimuli (flashing versus steady green disks), with which they had no previous experience. Both failed to transfer to the dynamic stimuli. A third monkey, also with massive past experience matching static visual stimuli, was tested for transfer to the dynamic stimuli within our standard matching paradigm, and it, too, failed. All 3 subjects were unable to reach a moderate acquisition criterion despite as many as 52 sessions of training with the dynamic stimuli. These results provide further evidence that, in monkeys, the matching (or identity) concept has a very limited reach; they consequently do not support the view held by some theorists that an abstract matching concept based on physical similarity is a general endowment of animals.     

On the analysis of studies of choice

In a review of 103 sets of data from 23 different studies of choice, Baum (1979) concluded that whereas undermatching was most commonly observed for responses, the time measure generally conformed to the matching relation. A reexamination of the evidence presented by Baum concludes that undermatching is the most commonly observed finding for both measures. Use of the coefficient of determination by both Baum (1979) and de Villiers (1977) for assessing when matching occurs is criticized on statistical grounds. An alternative to the loss-in-predictability criterion used by Baum (1979) is proposed. This alternative statistic has a simple operational meaning and is related to the usual F-ratio test. It can therefore be used as a formal test of the hypothesis that matching occurs. Baum (1979) also suggests that slope values of between .90 and 1.11 can be considered good approximations to matching. It is argued that the establishment of a fixed interval as a criterion for determining when matching occurs, is inappropriate. A confidence interval based on the data from any given experiment is suggested as a more useful method of assessment.     

四卡问题解决中的匹配偏向再探

选取经典的四卡问题作为实验材料,深入探讨了“综合考虑证真证伪作用”的提示以及逻辑分析过程对被试解决四卡问题不能产生促进效应的原因。结果发现:(1)多数被试能对卡片P和-Q进行正确的逻辑推断,但是最后却仍然倾向于选择卡片Q而非-Q,这种错误并非是由于附加的认知任务使得被试的短时记忆容量超载所致。(2)元音偶数组与元音非偶数组的被试对四张卡片作出正确逻辑推断的人数百分比基本一致,但是后者选择P-Q的人数百分比却显著高于前者,这表明多数被试似乎并不依据逻辑分析的结果及其命题检验的规则来作出选择,而是采用匹配策略,错误地选择了Q卡片。     

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.     

Molecular maximizing characterizes choice on Vaughan's (1981) procedure

Pigeons keypecked on a two-key procedure in which their choice ratios during one time period determined the reinforcement rates assigned to each key during the next period (Vaughan, 1981). During each of four phases, which differed in the reinforcement rates they provided for different choice ratios, the duration of these periods was four minutes, duplicating one condition from Vaughan's study. During the other four phases, these periods lasted six seconds. When these periods were long, the results were similar to Vaughan's and appeared compatible with melioration theory. But when these periods were short, the data were consistent with molecular maximizing (see Silberberg & Ziriax, 1982) and were incompatible with melioration, molar maximizing, and matching. In a simulation, stat birds following a molecular-maximizing algorithm responded on the short- and long-period conditions of this experiment. When the time periods lasted four minutes, the results were similar to Vaughan's and to the results of the four-minute conditions of this study; when the time periods lasted six seconds, the choice data were similar to the data from real subjects for the six-second conditions. Thus, a molecular-maximizing response rule generated choice data comparable to those from the short- and long-period conditions of this experiment. These data show that, among extant accounts, choice on the Vaughan procedure is most compatible with molecular maximizing.