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.     

The assumptions underlying the generalized matching law

Allen (1981) derived the power-function generalization of the matching law from a functional equation involving relative response rates on three concurrently available schedules of reinforcement. This paper defines the conditions (relative homogeneity and independence) under which a more general class of behavioral laws reduces to the power law. The proof also removes two deficiencies of Allen's result (discussed by Houston, 1982), which are, first, that his derivation produces a power law without a bias coefficient, and second, that it holds only for experiments with three or more concurrent schedules.     

Concurrent variable-ratio schedules: Implications for the generalized matching law

Rats' responses were reinforced on concurrent variable-ratio variable-ratio schedules in which responses on one lever incremented the ratio counter and responses on a second lever changed the schedule and correlated stimulus. The relative frequency of reinforcement was varied from .10 to .99. In one set of conditions, responding on the main lever incremented both ratio counters, but reinforcement required a response in the presence of the stimulus correlated with the ratio that had been completed. In a second set of conditions, responses on the main lever incremented only the ratio correlated with the stimulus that was currently present. When main-lever responses incremented both ratio counters, subjects distributed responding and time in a manner consistent with the generalized matching law. When responses on the main lever incremented only the schedule currently in effect, the rats responded almost exclusively on the schedule producing the higher frequency of reinforcement. These results extend the applicability of the generalized matching law to dependent ratio schedules.     

Choice with a fixed requirement for food, and the generality of the matching relation

Pigeons were trained on choice procedures in which responses on each of two keys were reinforced probabilistically, but only after a schedule requirement had been met. Under one arrangement, a fixed-interval choice procedure was used in which responses were not reinforced until the interval was over; then a response on one key would be reinforced, with the effective key changing irregularly from interval to interval. Under a second, fixed-ratio choice procedure, responses on either key counted towards completion of the ratio and then, once the ratio had been completed, a response on the probabilistically selected key would produce food. In one experiment, the schedule requirements were varied for both fixed-interval and fixed-ratio schedules. In the second experiment, relative reinforcement rate was varied. And in a third experiment, the duration of an intertrial interval separating choices was varied. The results for 11 pigeons across all three experiments indicate that there were often large deviations between relative response rates and relative reinforcement rates. Overall performance measures were characterized by a great deal of variability across conditions. More detailed measures of choice across the schedule requirement were also quite variable across conditions. In spite of this variability, performance was consistent across conditions in its efficiency of producing food. The absence of matching of behavior allocation to reinforcement rate indicates an important difference between the present procedures and other choice procedures; that difference raises questions about the specific conditions that lead to matching as an outcome.     

The case against Allen's generalization of the matching law

Allen (1981) claims to have established the formal validity of a power-function generalization of the matching law. This paper argues that Allen's proof is not correct when schedules are presented in pairs and that his initial assumptions are too restrictive when all schedules are simultaneously available.     

The applied importance of research on the matching law

In this essay, we evaluate the applied implications of two articles related to the matching law and published in the Journal of the Experimental Analysis of Behavior, May 1994. Building on Mace's (1994) criteria for increasing the applied relevance of basic research, we evaluate the applied implications of basic research studies. Research by Elsmore and McBride (1994) and Savastano and Fantino (1994) involve an extension of the behavioral model of choice. Elsmore and McBride used rats as subjects, but arranged a multioperant environment that resembles some of the complex contingencies of human behavior. Savastino and Fantino used human subjects and extended the matching law to ratio and interval contingencies. These experiments contribute to a growing body of knowledge on the matching law and its relevance for human behavior.     

Formal properties of the matching law

The matching law implies that any form of behavior approaches an asymptotic frequency as its reinforcement approaches 100 per cent of the total reinforcement being obtained at a given time. This asymptote is formally independent of the kind or quantity of drive or reinforcement associated with the response in question or with any competing response.     

Testing the generality of the name letter effect: name initials and everyday attitudes

The name letter effect is the tendency to evaluate alphabetical letters in one's name, especially initials, particularly favorably. Recent evidence suggests that name initials may even predict career choices. The authors investigated whether people possess favorable attitudes toward basic attitude objects beginning with name initials, both between individuals (e.g., does Judy like jam more than does Doug?) and within individuals (e.g., does Judy like jam more than honey?). Ratings of animals, foods, leisure activities (Studies 1-4) and national groups (Studies 2-4) revealed no object preference as a function of matching name initials. However, the name letter effect emerged (Studies 3-4), as did a clear preference for brand names starting with one's name initials (Study 4). Self-esteem, narcissism, and stimuli characteristics did not reliably influence these effects. Implications for extending name letter effects to basic attitude processes are discussed.     

On the tautology of the matching law

The generalized matching law, that organisms divide their time between alternatives in proportion to the value of the reinforcement consequent on the choice, is derivable from the assumption that an organism choosing between alternatives is under no constraints except those the contingencies of reinforcement impose. Hence, the law is not subject to empirical test. Its value lies in its simple codification of assumptions underlying choice experiments.     

Testing the generality of the zoom-lens model: Evidence for visual-pathway specific effects of attended-region size on perception

There are volumes of information available to process in visual scenes. Visual spatial attention is a critically important selection mechanism that prevents these volumes from overwhelming our visual system’s limited-capacity processing resources. We were interested in understanding the effect of the size of the attended area on visual perception. The prevailing model of attended-region size across cognition, perception, and neuroscience is the zoom-lens model. This model stipulates that the magnitude of perceptual processing enhancement is inversely related to the size of the attended region, such that a narrow attended-region facilitates greater perceptual enhancement than a wider region. Yet visual processing is subserved by two major visual pathways (magnocellular and parvocellular) that operate with a degree of independence in early visual processing and encode contrasting visual information. Historically, testing of the zoom-lens has used measures of spatial acuity ideally suited to parvocellular processing. This, therefore, raises questions about the generality of the zoom-lens model to different aspects of visual perception. We found that while a narrow attended-region facilitated spatial acuity and the perception of high spatial frequency targets, it had no impact on either temporal acuity or the perception of low spatial frequency targets. This pattern also held up when targets were not presented centrally. This supports the notion that visual attended-region size has dissociable effects on magnocellular versus parvocellular mediated visual processing.     

The relation between the generalized matching law and signal-detection theory

The generalized matching law can be applied to a signal-detection matrix to give two equations. The first relates responding in the presence of the stimulus to the reinforcements for the responses, and the second relates responding in the absence of the stimulus to the reinforcements for the responses. Evidence for stimulus discrimination is given by biases that are opposite in sign in the two equations. As the logarithmic ratio and z proportion transformations are similar, the combination of the absolute values of the two logarithmic biases gives a measure equivalent to the signal-detection measures d′ and η. The two equations can also be combined to eliminate the biases caused by the signalling stimuli and to produce a generalized matching-law statement relating overall performance to the obtained reinforcements.     

The matching law applies to wagtails' foraging in the wild

Field data concerning the time budgets and foraging success of pied wagtails (Motacilla alba yarrelli, Gould) are reanalyzed. It is found that the data are well described by the generalized matching law, with a marked bias towards spending time on the territory. In this case matching is not the result of maximizing reward rate, but it remains possible that it results from an allocation of time that maximizes survival.     

The matching law in and within groups of rats

In each of the two experiments, a group of five rats lived in a complex maze containing four small single-lever operant chambers. In two of these chambers, food was available on variable-interval schedules of reinforcement. In Experiment I, nine combinations of variable intervals were used, and the aggregate lever-pressing rates (by the five rats together) were studied. The log ratio of the rates in the two chambers was linearly related to the log ratio of the reinforcement rates in them; this is an instance of Herrnstein's matching law, as generalized by Baum. Summing over the two food chambers, food consumption decreased, and response output increased, as the time required to earn each pellet increased. In Experiment II, the behavior of individual rats was observed by time-sampling on selected days, while different variable-interval schedules were arranged in the two chambers where food was available. Individual lever-pressing rates for the rats were obtained, and their median bore the same “matching” relationship to the reinforcement rates as the group aggregate in Experiment I. There were differences between the rats in their distribution of time and responses between the two food chambers; these differences were correlated with differences in the proportions of reinforcements the rats obtained from each chamber.     

Concurrent random-interval schedules and the matching law

In Experiment I, a group of eight pigeons performed on concurrent random-interval schedules constructed by holding probability equal and varying cycle time to produce ratios of reinforcer densities of 1:1, 3:1, and 5:1 for key pecking. Schedules for a second group of seven were constructed with equal cycle times and unequal probabilities. Both groups deviated from simple matching, but the two forms of the schedules appeared to produce no consistent patterns of deviation. The data were found to be consistent with those obtained in concurrent variable-interval situations. The parameters of the matching equation in the form of Y=k X^a were estimated; the value of k was unity and a was 0.84. In Experiment II, six pigeons were exposed to two conc RI RI schedules in which one component increasingly approximated an FI schedule. The value of k was not 1.0. Concurrent RI RI schedules were shown to represent a continuum from conc FI VI to conc VI VI schedules. The use of the exponential equation in testing “matching laws” suggests that a<1 will continue to be observed, and this will set limits on the form of new laws and the assumed or rational values of the component variables in these laws.     

The specificity of the generality problem

In “Why the generality problem is everybody’s problem,” Michael Bishop argues that every theory of justification needs a solution to the generality problem. He contends that a solution is needed in order for any theory to be used in giving an acceptable account of the justificatory status of beliefs in certain examples. In response, first I will describe the generality problem that is specific to process reliabilism and two other sorts of problems that are essentially the same. Then I will argue that the examples that Bishop presents pose no such problem for some theories. I will illustrate the exempt theories by describing how an evidentialist view can account for the justification in the examples without having any similar problem. It will be clear that other views about justification are likewise unaffected by anything like the generality problem.