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.     

Concurrent reinforcement schedules for problem behavior and appropriate behavior: experimental applications of the matching law

This study evaluated how children who exhibited functionally equivalent problem and appropriate behavior allocate responding to experimentally arranged reinforcer rates. Relative reinforcer rates were arranged on concurrent variable-interval schedules and effects on relative response rates were interpreted using the generalized matching equation. Results showed that relative rates of responding approximated relative rates of reinforcement. Finally, interventions for problem behavior were evaluated and differential reinforcement of alternative behavior and extinction procedures were implemented to increase appropriate behavior and decrease problem behavior. Practical considerations for the application of the generalized matching equation specific to severe problem behavior are discussed, including difficulties associated with defining a reinforced response, and obtaining steady state responding in clinical settings.     

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.     

Choice and reinforcement delay

Previous studies of choice between two delayed reinforcers have indicated that the relative immediacy of the reinforcer is a major determinant of the relative frequency of responding. Parallel studies of choice between two interresponse times have found exceptions to this generality. The present study looked at the choice by pigeons between two delays, one of which was always four times longer than the other, but whose absolute durations were varied across conditions. The results indicated that choice is not uniquely determined by the relative immediacy of reinforcement, but that absolute delays are also involved. Models for concurrent chained schedules appear to be more applicable to the present data than the matching relation; however, these too failed to predict choice for long delays.     

Optimization and the matching law as accounts of instrumental behavior

The interaction between instrumental behavior and environment can be conveniently described at a molar level as a feedback system. Two different possible theories, the matching law and optimization, differ primarily in the reference criterion they suggest for the system. Both offer accounts of most of the known phenomena of performance on concurrent and single variable-interval and variable-ratio schedules. The matching law appears stronger in describing concurrent performances, whereas optimization appears stronger in describing performance on single schedules.     

Choice, experience, and the generalized matching law

Five pigeons were exposed to different pairs of concurrent variable-interval, variable-interval schedules on nine experimental conditions of 30 sessions each. For every session, the parameters of the generalized matching equation were computed for the first five, six, seven, eight, and nine experimental conditions. The exponent a, both for response and time distribution, tended to decrease with increases in number of experimental conditions and to increase with number of sessions per condition, but values of k (bias) varied unsystematically. When the subjects were exposed to five new pairs of schedules, with 55 sessions per condition, the findings were confirmed. Data from the literature on the generalized matching law suggest that the variability of exponent values may be explained in part by the use of naive or experienced subjects in different investigations and by the variability in number of experimental conditions and in number of sessions per condition.     

Differences in delay, not ratios, control choice in concurrent chains

In two experiments, pigeons were trained with concurrent-chains schedules, wherein responding to equal initial links measured preference between variable-interval terminal links. Absolute terminal-link duration was varied by keeping constant the difference between the terminal-link delays and forcing their ratio to change. Delay-reduction theory scales value relative to a common temporal context and requires that delay differences control choice. Thus, preference should remain invariant. Most competing accounts, including the matching law and a strong form of Weber's law, require that preference vary with the delay ratio. Experiment 1 employed standard concurrent chains, in which terminal-link position and color were confounded. Although average preference remained constant, individual preferences were highly variable and inconsistent, possibly due to carryover of position biases across conditions. In an attempt to reduce variability, Experiment 2 used a modified concurrent-chains procedure. Preference at different terminal-link durations was assessed simultaneously to prevent order effects, and terminal-link position was alternated randomly across trials to minimize the impact of position biases. In Experiment 2, both individual and mean preferences showed the constant-difference invariance. Overall, choice was controlled by terminal-link differences, not ratios.     

Asymmetry of reinforcement and punishment in human choice

The hypothesis that a penny lost is valued more highly than a penny earned was tested in human choice. Five participants clicked a computer mouse under concurrent variable-interval schedules of monetary reinforcement. In the no-punishment condition, the schedules arranged monetary gain. In the punishment conditions, a schedule of monetary loss was superimposed on one response alternative. Deviations from generalized matching using the free parameters c (sensitivity to reinforcement) and log k (bias) were compared in the no-punishment and punishment conditions. The no-punishment conditions yielded values of log k that approximated zero for all participants, indicating no bias. In the punishment condition, values of log k deviated substantially from zero, revealing a 3-fold bias toward the unpunished alternative. Moreover, the c parameters were substantially smaller in punished conditions. The values for bias and sensitivity under punishment did not change significantly when the measure of net reinforcers (gains minus losses) was applied to the analysis. These results mean that punishment reduced the sensitivity of behavior to reinforcement and biased performance toward the unpunished alternative. We concluded that a single punisher subtracted more value than a single reinforcer added, indicating an asymmetry in the law of effect.     

The relative law of effect: effects of shock intensity on response strength in multiple schedules

Key pecking of four birds was reinforced with food according to a two-component multiple variable-interval 1-minute variable-interval 4-minute schedule. In addition, key pecking was punished by a brief shock according to a variable-interval 30-second schedule during both components of the multiple schedule. The intensity of the shock was varied. For all birds, punishment had a stronger suppressive effect on the responding maintained by the leaner food schedule, and the ratio of responding during the two components of the multiple schedule became closer to the ratio of reinforcement as shock intensity was increased, as the relative law of effect predicts. At the higher shock intensity, there was some evidence that the ratio of responses overmatched the ratio of reinforcements.     

Resistance to change and the law of effect

Three experiments using multiple schedules of reinforcement explored the implications of resistance-to-change findings for the response-reinforcer relation described by the law of effect, using both steady-state responding and responding recorded in the first few sessions of conditions. In Experiment 1, when response-independent reinforcement was increased during a third component, response rate in Components 1 and 2 decreased. This response-rate reduction was proportionately greater in a component in which reinforcer magnitude was small (2-s access to wheat) than in the component in which it was large (6-s access to wheat). However, when reinforcer rates in the two components were varied together in Experiments 2 and 3, response-rate change was the same regardless of the magnitude of reinforcers used in the two components, so that sensitivity of response rates to reinforcer rates (Experiment 2) and of response-rate ratios to reinforcer-rate ratios (Experiment 3) was unaffected by the magnitude of the reinforcers. Therefore, the principles determining resistance to change, described by behavioral momentum theory, seem not to apply when the source of behavior change is the variation of reinforcement contingencies that maintain the behavior. The use of extinction as a manipulation to study resistance to change is questioned.     

The effect of punishment on free-operant choice behavior in humans

During Phase I, three female human subjects pressed a button for monetary reinforcement in five variable-interval schedules specifying different frequencies of reinforcement. On alternate days, responding was also punished (by subtracting money) according to a variable-ratio 34 schedule. In the absence of punishment, response rates conformed to Herrnstein's equation for single variable-interval schedules. Punishment suppressed responding at all frequencies of reinforcement. This was reflected in a change in the values of both constants in Herrnstein's equation: the value of the theoretical maximum response-rate parameter was reduced, and the parameter describing the reinforcement frequency corresponding to the half-maximal response rate was elevated. During Phase II, the same five schedules (A) were in operation (without punishment), but in addition, a concurrent variable-interval schedule (B) of standard reinforcement frequency was introduced. On alternate days, responding in Component B was punished according to a variable-ratio 34 schedule. In the absence of punishment, absolute response rates conformed to equations proposed by Herrnstein to describe performance in concurrent schedules; the ratios of the response rates in the two components and the ratios of the times spent in the two components conformed to the Matching Law. When responding in Component B was punished, response rates in Component B were reduced and those in Component A were elevated, these changes being reflected in distortions of the matching relationship.     

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 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.     

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.     

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.     

Nonstable concurrent choice in pigeons

Six pigeons were trained on concurrent variable-interval schedules in which the arranged reinforcer ratios changed from session to session according to a 31-step pseudorandom binary sequence. This procedure allows a quantitative analysis of the degree to which performance in an experimental session is affected by conditions in previous sessions. Two experiments were carried out. In each, the size of the reinforcer ratios arranged between the two concurrent schedules was varied between 31-step conditions. In Experiment 1, the concurrent schedules were arranged independently, and in Experiment 2 they were arranged nonindependently. An extended form of the generalized matching law described the relative contribution of past and present events to present-session behavior. Total performance in sessions was mostly determined by the reinforcer ratio in that session and partially by reinforcers that had been obtained in previous sessions. However, the initial exposure to the random sequence produced a lower sensitivity to current-session reinforcers but no difference in overall sensitivity to reinforcement. There was no evidence that the size of the reinforcer ratios available on the concurrent schedules affected either overall sensitivity to reinforcement or the sensitivity to reinforcement in the current session. There was also no evidence of any different performance between independent and nonindependent scheduling. Because of these invariances, this experiment validates the use of the pseudorandom sequence for the fast determination of sensitivity to reinforcement.     

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 general matching law describes choice on concurrent variable-interval schedules of wheel-running reinforcement.

Six male Wistar rats were exposed to concurrent variable-interval schedules of wheel-running reinforcement. The reinforcer associated with each alternative was the opportunity to run for 15 s, and the duration of the changeover delay was 1 s. Results suggested that time allocation was more sensitive to relative reinforcement rate than was response allocation. For time allocation, the mean slopes and intercepts were 0.82 and 0.008, respectively. In contrast, for response allocation, mean slopes and intercepts were 0.60 and 0.03, respectively. Correction for low response rates and high rates of changing over, however, increased slopes for response allocation to about equal those for time allocation. The results of the present study suggest that the two-operant form of the matching law can be extended to wheel-running reinforcement. 'I'he effects of a low overall response rate, a short Changeover delay, and long postreinforcement pausing on the assessment of matching in the present study are discussed.     

Choice between rewards differing in amount and delay: Toward a choice model of self control

A concurrent-chain procedure was used to study pigeons' choices between rewards differing in both amount and delay. The shorter delay terminated with a 2-second access to grain whereas the longer delay terminated with a 6-second access to grain. The ratio of the delays was constant within a given condition while their absolute values were varied. Over conditions, ratios of 6:1, 3:1, and 3:2 were studied. As the absolute values of the delays to reinforcement increased, preference for the longer-delayed but larger reward decreased under both the 6:1 and 3:1 ratios, but increased under the 3:2 ratio. These results are inconsistent with choice models predicting no change in preference when the ratios of delays and amounts are held constant. In addition, the change in preference under the 3:1 ratio is inconsistent with a simple multiplicative interaction of the trade off between reinforcer amount and delay, and suggests that delay is a more potent determinant of choice than is amount. These results have implications for models that view choice between small immediate rewards and large but delayed rewards as underlying the behavior commonly called self control.