On the law of effect

Experiments on single, multiple, and concurrent schedules of reinforcement find various correlations between the rate of responding and the rate or magnitude of reinforcement. For concurrent schedules (i.e., simultaneous choice procedures), there is matching between the relative frequencies of responding and reinforcement; for multiple schedules (i.e., successive discrimination procedures), there are contrast effects between responding in each component and reinforcement in the others; and for single schedules, there are a host of increasing monotonic relations between the rate of responding and the rate of reinforcement. All these results, plus several others, can be accounted for by a coherent system of equations, the most general of which states that the absolute rate of any response is proportional to its associated relative reinforcement.     

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.     

Dynamics of time matching: Arousal makes better seem worse

Matching of time allocation across alternatives in proportion to relative reinforcement rates is a ubiquitous finding in the animal-learning literature on choice. The dynamics of the underlying mechanism, however, remain poorly understood. A recent finding by Belke (1992) profoundly challenges scalar expectancy theory (SET; Gibbon et al., 1988) and other accounts of matching in concurrent variable interval (VI) schedules. He studied concurrent probe tests of stimuli associated with equal VIs but trained in alternative concurrent pairs. In training, one was preferred and the other not. Unreinforced probes revealed a strong preference for the alternative preferred in training. An experiment is reported replicating this result and showing that it is not due to generalization of preference levels from training. When the probe is between the two preferred training stimuli, the richer schedule is unpreferred. A SET account of these results is presented which implicates two processes in time allocation: (1) the choice between alternatives based on memory for delays to reinforcement, and (2) the times at which such choices are made. The former process is sensitive to reinforcement scheduling; the latter is sensitive to arousal levels induced by overall reinforcement rates in training.     

More on concurrent interval-ratio schedules: a replication and review.

It has been suggested that the failure to maximize reinforcement on concurrent variable-interval, variable-ratio schedules may be misleading. Inasmuch as response costs are not directly measured, it is possible that subjects are optimally balancing the benefits of reinforcement against the costs of responding. To evaluate this hypothesis, pigeons were tested in a procedure in which interval and ratio schedules had equal response costs. On a concurrent variable time (VT), variable ratio-time (VRT) schedule, the VT schedule runs throughout the session and the VRT schedule is controlled by responses to a changeover key that switches from one schedule to the other. Reinforcement is presented independent of response. This schedule retains the essential features of concurrent VI VR, but eliminates differential response costs for the two alternatives. It therefore also eliminates at least one significant ambiguity about the reinforcement maximizing performance. Pigeons did not maximize rate of reinforcement on this procedure. Instead, their times spent on the alternative schedules matched the relative rates of reinforcement, even when schedule parameters were such that matching earned the lowest possible overall rate of reinforcement. It was further shown that the observed matching was not a procedural artifact arising from the constraints built into the schedule.     

Probabilistically reinforced choice behavior in pigeons

A single principle, "momentary maximizing", may account for much of a pigeon's steady-state behavior in both probability learning and concurrent variable interval experiments. The principle states that a pigeon tends to choose the alternative that momentarily has the higher probability of reinforcement. A successive discrimination procedure, which produced matching in an earlier experiment, produced here a tendency to maximize if training were adequately extended. Maximizing was produced also by other procedures, in which no reinforcing event was presented on some trials: one procedure did and two did not provide a bird with information about the availability of reinforcement on a key after an unreinforced response on the other key. The latter two procedures were analogous to concurrent variable interval schedules in two respects: the reinforcement probability on one key increased while a bird responded on the other key; and they produced matching. But sequential statistics suggested that matching resulted from momentary maximizing. Depending on the procedure, the tendency to maximize produced different relative frequencies of pecking a key for a fixed relative frequency of reinforcement. Computer simulation of maximizing behavior in several concurrent variable interval schedules produced matching and sequential statistics similar to those produced by a real bird.     

Reinforcement of least-frequent sequences of choices

When a pigeon's choices between two keys are probabilistically reinforced, as in discrete trial probability learning procedures and in concurrent variable-interval schedules, the bird tends to maximize, or to choose the alternative with the higher probability of reinforcement. In concurrent variable-interval schedules, steady-state matching, which is an approximate equality between the relative frequency of a response and the relative frequency of reinforcement of that response, has previously been obtained only as a consequence of maximizing. In the present experiment, maximizing was impossible. A choice of one of two keys was reinforced only if it formed, together with the three preceding choices, the sequence of four successive choices that had occurred least often. This sequence was determined by a Bernoulli-trials process with parameter p. Each of three pigeons matched when p was ½ or ¼. Therefore, steady-state matching by individual birds is not always a consequence of maximizing. Choice probability varied between successive reinforcements, and sequential statistics revealed dependencies which were adequately described by a Bernoulli-trials process with p depending on the time since the preceding reinforcement.     

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.     

Stimuli, reinforcers, and behavior: an integration

We propose that a fundamental unit of behavior is the concurrent discriminated operant, and we discuss in detail a quantitative model of the concurrent three-term contingency that is based on the notion that an animal's behavior is controlled to differing extents by both stimulus—behavior and behavior—reinforcer relations. We show how this model can describe performance in a variety of experimental procedures: conditional discrimination and matching to sample, both with and without reinforcement for responses that are traditionally identified as errors; conditional discrimination with more than two stimuli and choice alternatives; delayed matching to sample and delayed reinforcement in matching to sample; second-order and complex conditional discrimination; and multiple and concurrent schedules. Although the model is incomplete in its coverage, and may be incorrect, we believe that this conceptual approach will bear fruit in the development of behavior theory.     

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.     

The effects of different component response requirements in multiple and concurrent schedules

Six pigeons were trained on multiple and concurrent schedules. The reinforcement rates were varied systematically (a) when lever pressing was required in one component and key pecking in the successive component; (b) when lever pressing was required in both multiple components; (c) when key pecking was required in both multiple components; and (d) when key pecking was required on one schedule and lever pressing was required on the concurrently-available schedule. Only the absolute level of responding was changed by different response requirements. Analyzed by the generalized matching law, performance under different response requirements resulted in a bias toward key pecking, and the measured response bias was the same in multiple and concurrent schedule arrangements. The bias in time measures obtained from concurrent schedule performance was reliably smaller than the obtained response biases. The sensitivity to reinforcement-rate changes was ordered: concurrent key-lever; multiple key-key; multiple lever-key; and, the least sensitive, multiple lever-lever. The results confirm that requirements of different topographical responses can be handled by the generalized matching law mainly in the bias parameter, but problems for this type of analysis may be caused by the changing sensitivity to reinforcement in multiple schedule performance as response requirements are changed.     

Concurrent variable-interval variable-ratio schedules can provide only weak evidence for matching

Herrnstein and Heyman (1979) showed that when pigeons' pecking is reinforced on concurrent variable-interval variable-ratio schedules, (1) their behavior ratios match the ratio of the schedules' reinforcer frequencies, and (2) there is more responding on the variable interval. Since maximizing the reinforcement rate would require responding more on the variable ratio, these results were presented as establishing the primacy of matching over maximizing. In the present report, different ratios of behavior were simulated on a computer to see how they would affect reinforcement rates on these concurrent schedules. Over a wide range of experimenter-specified choice ratios, matching obtained — a result suggesting that changes in choice allocation produced changes in reinforcer frequencies that correspond to the matching outcome. Matching also occurred at arbitrarily selected choice ratios when reinforcement rates were algebraically determined by each schedule's reinforcement-feedback function. Additionally, three birds were exposed to concurrent variable-interval variable-ratio schedules contingent on key pecking in which hopper durations were varied in some conditions to produce experimenter-specified choice ratios. Matching generally obtained between choice ratios and reinforcer-frequency ratios at these different choice ratios. By suggesting that reinforcer frequencies track choice on this procedure, instead of vice versa, this outcome questions whether matching-as-outcome was due to matching-as-process in the Herrnstein and Heyman study.     

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: its acquisition and generalization

Choice typically is studied by exposing organisms to concurrent variable-interval schedules in which not only responses controlled by stimuli on the key are acquired but also switching responses and likely other operants as well. In the present research, discriminated key-pecking responses in pigeons were first acquired using a multiple schedule that minimized the reinforcement of switching operants. Then, choice was assessed during concurrent-probe periods in which pairs of discriminative stimuli were presented concurrently. Upon initial exposure to concurrently presented stimuli, choice approximated exclusive preference for the alternative associated with the higher reinforcement frequency. Concurrent schedules were then implemented that gave increasingly greater opportunities for switching operants to be conditioned. As these operants were acquired, the relation of relative response frequency to relative reinforcement frequency converged toward a matching relation. An account of matching with concurrent schedules is proposed in which responding exclusively to the discriminative stimulus associated with the higher reinforcement frequency declines as the concurrent stimuli become more similar and other operants-notably switching-are acquired and generalize to stimuli from both alternatives. The concerted effect of these processes fosters an approximate matching relation in commonly used concurrent procedures.     

Short-term and long-term effects of reinforcers on choice

The relation between molar and molecular aspects of time allocation was studied in pigeons on concurrent variable-time variable-time schedules of reinforcement. Fifteen-minute reinforcer-free periods were inserted in the middle of every third session. Generalized molar matching of time ratios to reinforcer ratios was observed during concurrent reinforcement. Contrary to melioration theory, preference was unchanged during the reinforcer-free periods as well as in extinction. In addition to this long-term effect of reinforcement, short-term effects were observed: Reinforcers increased the duration of the stays during which they were delivered but had little consistent effect either on the immediately following stay in the same schedule or on the immediately following stay in the alternative schedule. Thus, an orderly effect of reinforcer delivery on molecular aspects of time allocation was observed, but because of its short-term nature, this effect cannot account for the matching observed at the molar level.     

Optimal behavior and concurrent variable interval schedules

Behavior maintained with 2-component concurrent variable interval schedules of reinforcement (CONC VIVI) is described well by the matching law. Deviations from matching behavior have been handled by adding free parameters to the matching law equation. With CONC VIVI schedules there are infinitely many solutions to the matching law equation at each value of the procedural parameters. However, at each value of the procedural parameters, only one combination of durations of intervals spent in each VI component (dwell times) yields the combined maximum reinforcement rate. The equations that yield the optimal dwell times solution for CONC VIVI schedules are mathematically incompatible with the matching law. Optimal performance and matching coincide only when the parameter values of the two VI components are equal. It seems reasonable to use optimal behavior to assess performance in these schedules. Researchers have not compared optimal and empirical performances in CONC VIVI possibly because the equations for optimal dwell times (ODT) can be solved only numerically. We present a table of ODT for a wide range of VIs and changeover delays. We also derive a function m that can be used to compare matching data and the matching behavior predictions of optimization. We prove that 0.5<m<1.003502, and we describe some of the more nteresting properties of the function.     

Concurrent performances: inhibition of one response by reinforcement of another

In an analysis of interactions between concurrent performances, variable-interval reinforcement was scheduled, in various sequences, for both keys, for only one key, or for neither key of a two-key pigeon chamber. With changeover delays of 0.5 or 1.0 sec, and with each key's reinforcements discriminated on the basis of key-correlated feeder stimuli, reinforcement of pecks on one key reduced the pecking maintained by reinforcement on the other key. The decrease in pecking early after reinforcement was discontinued on one key was not substantially affected by whether pecks on the other key were reinforced, but after reinforcement was discontinued on both keys, reinstatement of reinforcement for one key sometimes produced transient increases in pecking on the other key. Correlating the availability of right-key reinforcements with a stimulus, which maintained right-key reinforcement while reducing right-key pecking to negligible levels, demonstrated that these interactions depended on concurrent reinforcement, not concurrent responding. Thus, reinforcement of a response, but not necessarily the occurrence of the response, inhibits other reinforced responses. Compared with accounts in terms of excitatory effects of extinction, often invoked in treatments of behavioral contrast, this inhibitory account has the advantage of dealing only with observed dimensions of behavior.     

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).     

Concurrent schedules of positive and negative reinforcement: differential-impact and differential-outcomes hypotheses

Considerable evidence from outside of operant psychology suggests that aversive events exert greater influence over behavior than equal-sized positive-reinforcement events. Operant theory is largely moot on this point, and most operant research is uninformative because of a scaling problem that prevents aversive events and those based on positive reinforcement from being directly compared. In the present investigation, humans' mouse-click responses were maintained on similarly structured, concurrent schedules of positive (money gain) and negative (avoidance of money loss) reinforcement. Because gains and losses were of equal magnitude, according to the analytical conventions of the generalized matching law, bias (log b (double dagger) 0) would indicate differential impact by one type of consequence; however, no systematic bias was observed. Further research is needed to reconcile this outcome with apparently robust findings in other literatures of superior behavior control by aversive events. In an incidental finding, the linear function relating log behavior ratio and log reinforcement ratio was steeper for concurrent negative and positive reinforcement than for control conditions involving concurrent positive reinforcement. This may represent the first empirical confirmation of a free-operant differential-outcomes effect predicted by contingency-discriminability theories of choice.     

A contextual model of concurrent-chains choice

An extension of the generalized matching law incorporating context effects on terminal-link sensitivity is proposed as a quantitative model of behavior under concurrent chains. The contextual choice model makes many of the same qualitative predictions as the delay-reduction hypothesis, and assumes that the crucial contextual variable in concurrent chains is the ratio of average times spent, per reinforcement, in the terminal and initial links; this ratio controls differential effectiveness of terminal-link stimuli as conditioned reinforcers. Ninety-two concurrent-chains data sets from 19 published studies were fitted to the model. Averaged across all studies, the model accounted for 90% of the variance in pigeons' relative initial-link responding. The model therefore demonstrates that a matching law analysis of concurrent chains—the assumption that relative initial-link responding equals relative terminal-link value—remains quantitatively viable. Because the model reduces to the generalized matching law when terminal-link duration is zero, it provides a quantitative integration of concurrent schedules and concurrent chains.     

Choice, matching, and human behavior: A review of the literature

This review concerns human performance on concurrent schedules of reinforcement. Studies indicate that humans match relative behavior to relative rate of reinforcement. Herrnstein's proportional matching equation describes human performance but most studies do not evaluate the equation at the individual level. Baum's generalized matching equation has received strong support with humans as subjects. This equation permits the investigation of sources of deviation from ideal matching and a few studies have suggested variables which control such deviations in humans. While problems with instructional control are raised, the overall findings support the matching law as a principle of human choice.