A General Model of Learned Performance Under Schedule Constraint

P5ychological Review 1983, Vol. 90, No.3.

American

261--;282

Regulation

Copyright 1983 by the Psychological Association, Inc.

During Challenge: A General Model of Learned Performance Under Schedule Constraint Stephen Jose Hanson and William Indiana University

Timberlake

This article developsa general behavior-regulation model of learned performance related to the equilibrium approach of Timberlake ( 1980) and Timberlake and Allison ( 1974).The model is basedon four assumptions: (a) Both the instrumental and contingent responsesare regulated with respect to their own set point~; (b) these set points can be measured in a free baseline when both responses are relatively unconstrained and simultaneously available; (c) a reinforcement schedule can be seen as a constraint function that cross-couples the environmental effects of regulatory systems underlying the instrumental and contingent responses,thereby challenging their set points; and (d) molar behavior change under a schedule representsa compromise between the deviations from set points forced by the constraint function. These assumptions are translated into a set of coupled differential equations describing two regulatory systems related by a schedule. After providing an exact solution for this model, we derive as special casestwo current alternative models of learned performance (Allison, 1976;Staddon, 1979). Finally, we demonstrate that the model is consistent in form with data from a variety of simple schedules.

The study of learned performance in animals has roots in at least two traditions: the study of associations, and the study of the adaptive, regulatory functions of behavior. Until recently, research has focused primarily on the nature and the strength of associations formed through experience. Regulatory effects have been largely ignored or masked by the use of highly deprived subjects working during short sessionsunder stingy schedules. The renewed interest in a regulatory approach can be attributed to severalimportant advantages. In contrast to associative theories, a regulatory approach can identify potential reinforcers in an a priori and noncircular fashion, predict characteristics of asymptotic performance under schedules, and be used to select schedules that produce reinforcement, punishment, or no change in This article resulted from a year-long collaboration between the authors. We thank each other for our patience, enthusiasm, and good humor. Completion of the article was supported in part by National Science Foundation Grants BS79 15117 and BNS 82 10139 and Bell Laboratories. Requests for reprints should be sent to Stephen Jose Hanson, Bell Laboratories, Lincroft, New Jersey 07738 or to William Timberlake, Psychology Department, Indiana University, Bloomington, Indiana 47405.

behavior (Timberlake, 1980; Timberlake & Allison, 1974). Moreover, it can be argued that an accurate regulatory model of learned perfonnance is a necessary prerequisite for evaluating the contribution of associative factors. Certainly in terms of an organism 's evolutionary and individual history, learning ~ost often serves within a regulatory framework. The capacity to learn evolved and is typically expressedas an adaptive mechanism to promote survival and reproduction through the more efficient regulation of an organism 's internal and external environment. In this view, regulation precedes and underlies learning, setting the context for its nature and expression. This article translates the equilibrium approaCh to the study of learned perfonnance (Timberlake, 1980; Timberlake & Allison, 1974) into the concepts of control theory. From the resultant behavior-regulation theory, we develop a set of coupled differential equations that model behavior under schedule constraint. These equations embody the assumption that learned perfonnance results from the simultaneous regulation of behavior under schedule constraint with respect to the behavioral set points of boih instrumental and contingent responding. From the model

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STEPHEN JosE HANSON AND WILLIAM

we are able' to derive as special cases the models of Allison ( 1976) and Staddon ( 1979).. Finally, we show that the coupled-regulation model can account for a variety of data from simple fixed-ratio, fixed-interval, anq variable-interval schedules.

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as well (Atkinson & Birch, 1970; McDowell & Kessel, 1979; Myerson & Miezen, 1980; Rachlin, Kagel, & Battalio, 1980). This trend reflects several advantages that accrue to the modeler when he or she is framing theoretical assumptions in terms of a differential equation. First, a rate-change description often captures intuitions about the mechanisms of The Regulatory Approach behavior more readily than an attempt to Basic Elements describe performance directly. It is easier to The application of a regulatory analysis to mathematically represent processesin terms of discrete "snapshots" than by continuous behavior began early in this century with work on guided orientation in insects (Cro-" specification through time. Second, the difzier & Hoagland, 1934; Loeb, 1918) and .ferential equation approach is powerful in physiological homeostasisin mammals (Can- that simple verbal assumptions can often be non, 1934). In the late 1940s and 1950s a mapped to and from nonintuitive, nonrageneral control theory was developed in a tional functional forms. For example, assuming a proportional relation in a differential form that clarified the nature of regulation and the concept of feedback. Beginning in equation results in an exponential function. Finally, the functional form resulting from the late 1960s and 1970s researcherssuch as a differential equation model is a unique McFarland (1971) and Powers(1978) applied product of the assumptions that define it. regulatory concepts to 'the prediction of Functional equations developed primarily on choice and sequencing of behaviors. Most the basis of intuition or curve fitting are at recently, researchershave used control theory in describing and modeling the time course best consistent with the modeler's verbal asof eating and drinking (Booth, 1978; Toates, sumptions rather than being forced by them (e.g., Allison, 1976; Herrnstein, 1970; Hull, 1980) and behavior under reinforcement schedules (Baum, 1981; Pring- Mill, 1979; 1943). A potential problem with such a ..consistency" derivation is that the functional Staddon, 1980). The basic elements of a regulatory ap- form described by the model is very unlikely to represent a unique mapping of underlying proach are (a) One or more stable set points or reference inputs that the system is orga- assumptions. As a result, fitting tests of a nized around and tends to move toward as particular model are inconclusive because they cannot distinguish among the alternathat system is perturbed, (b) a constraint tive sets of assumptions. For example, the function that represents challenges that drive the system away from its set points, and (c) form of the matching law (Hermstein, 1961) a control function that describes exactly how has been derived from several different and the system responds to the challenge imposed apparently incompatible sets of assumptions (Baum, 1981; Herrnstein, 1970, 1979; by the constraint function. The regulatory Hermstein & Vaughn, 1980; Killeen, 1979, approach is t?xtremely general and can be 1982a; Myerson & Miezen, 1980; Staddon, applied to both physical and biological sys1977; Staddon & Motheral, 1978). Currently, tems under many different forms of challenge. In this article we will focus on its ap- none of these derivations has been shown to be related nor has one been shown to be a plication to asymptotic learned performance unique mapping of any set of basic assumpunder schedule challenge. tions underlying matching. In sum, the use of differential equations Differential Equation Models allows the tracing of important steps from Recent developments in the modeling of basic assumptions to functional forms. The behavior have leaned heavily on the use of fact that the researcher is forced to map asdifferential equations, not only in realizations sumptions directly to differential equations of control theory (e.g., McFarland, 1971; may provide and suggestcritical tests of the Toates, 1980),but in other theoretical scheJnes model. Differential equations have been used

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increasingly to model behavior, although the specific differential equation is often obscured by the absence of a deductive derivation, the particular method of solution (e.g., Laplace transforms-McDowell & Kessel, 1979), or a focus on stationary analysis (e.g., optimization theory-Rachlin, Kage1, & Battalio, 1980). The remainder of this article shows how a general set of differential equations can be developed from a regulatory approach and applied to the prediction of asymptotic learned performance under a contingency. A Behavior Regulation Approach Learned Performance

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ganisms cannot achieve both set points at once. (d) The change in each response under the schedule is determined by simultaneous regulation of both systems with respect to their set points. In the next three sections we develop more completely the control-theory view of equilibrium theory by considering the nature of behavioral set points, constraint functions, and control functions as revealed in these assumptions. The following section then translates the resultant behavior-regulation theory into a set of differential equations suitable for modeling learned performance.

to

Our model is a more precise statement and generalization of the molar-equilibrium approach outlined by Timberlake ( 1980), which is, in turn, a more precise and general statement of the response-deprivation approach of Timberlake and Allison ( 1974). Both of these approaches find their roots in earlier work by Premack (1959, 1965). According to the equilibrium approach, changes in instrumental responding occur when a schedule imposed by the experimenter produces a condition of disequilibrium with respect to baseline levels of responding. The final level of instrumental responding is determined by a balance between the deviations from baseline levels of instrumental and contingent responding. Translated more fully into the language of regulation, the equilibrium approach entails the following set of assumptions: (a) Both the instrumental and contingent response are regulated with respect to their own behavioral set points; that is, in the absence of a schedule, the instrumental and contingent responses are controlled by separate regulatory systems.(b) Behavioral set points for the instrumental and contingent responses can be measured in a free baseline when both responses are relatively unconstrained and simultaneously available. (c) A schedule can be viewed as a constraint function that couples the environmental effects of the outputs of the regulatory systems underlying instrumental and contingent responding. A sched~ ule effective in producing learned performance couples the systems so that the or-

Set Points In psychology, regulatory theory has focused on physiological set points that appear to be involved in such homeostatic mechanisms as salt balance, metabolism, and blood volume (Cannon, 1934). In this article we extend the concept of set points to refer directly to behavior (Tifiberlake, 1980). Instrumental and contingent responding are assumed to be regulated with respect to their total unconstrained responding or, more precisely, with respect to the instigation producing this responding. Undoubtedly, this instigation reflects the existence of more complex underlying physiological processesthat may or may not be"regulatory (cf. Bolles, 1980), but for our purposes it is necessaryonly that the instigating value of any underlying processesbe reliable. Most behavior-regulation theories share the assumption that set points of behavior can be measured by total responding in a paired or multiple baseline in which the instrumental and contingent responsesare relatively unconstrained and simultaneously available. These baseline measures are presumed to indicate the degree of instigation or arousal (cf. Killeen, Hanson, & Osbome, 1978) associatedwith these behaviors during minimal constraint. As constraints are added, responding becomes less related to the fundamentallevel of instigation for a particular response and more determined by its linkage with the other response and by various "expression " characteristics that modify the degreeof change under a particular challenge (e.g., elasticity-Staddon, 1977; resistance-

264

STEPHEN JosE HANSON AND WILLIAM

Nevin, Mandell, & Atak, 1983). Asymptotic responding is determined both by fundamental levels of instigation and by the expression characteristics of a response. However, although expression characteristics can be established under conditions of constraint, the fundamental levels of instigation are more properly measured in the absence of constraint. It might be argued that baseline measures are not really set points because they may vary with the instigating qualities of the circumstances. The important point here is that behavioral set points are useful only if the fundamental levels of instigation they reflect remain present. Thus, experimental procedures are designed to hold instigation constant across sessionsand conditions by fixing variables such as sessionlength, physical environment, time of day, and the subject's physiological state. The only change in conditions typically allowed is the imposition of a schedule constraint following baseline assessment.Note also that altering the eliciting conditions does not invalidate the regulatory approach; it simply requires remeasuring baselinesor, somewhat lesshappily, fitting the baselines from the contingency data. Ultimately, the baseline method of defining set points will give way to a behavioral theory of instigation, but, for the present, the paired baseline serves as a reasonable approximation of instigation levels underlying responding. Constraint {Schedule-Feedback}Functions In regulatory theories a constraint function determines the relation between the output of a system and its resultant input. The constraint function is not an intrinsic part of the regulatory system but depends upon the environment in which the system is Operating. In learned performance there are at least two types of constraint function, one based on the environmental circumstances that control and interact with a particular response, for example, the location, weight, and resistance of a bar (essentially, the "measurement system"), and one connecting the measured performance of two responses.The first type of constraint function is typically invariant acros~ the experiment and is in-

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corporated in the baseline assessmentof set points. The second type of constraint function refers to the relation imposed by the experimenter between the instrumental and contingent responses.It is this constraint that challenges the set points and determines learned performance. Many recent theorists have termed this relation a schedule-feedback function. The schedule-feedback function is a rule relating performance of the instrumental response to the amount of reward. For example, the schedule-feedback function for a fixed-ratio-five schedule is quite simple: Pressing the bar five times produces access to one pellet of food. The overwhelming majority of current theories of learned performance depend on the specification of schedule-feedback functions to generate predictions of absolute levels of performance under schedules. For example, optimality theories have used the schedule-feedback function to constrain the range of molar output so that responding must fall somewhereon this function. The optimality equation then determines where the responding will actually fall. Unfortunately, there are problems with this dependence on the schedule-feedback function. First, implicit in such dependence is an assumption that unique schedule characteristics are critical determinants of molar performance. It is clear that momentary response patterns may be determined by specific schedule characteristics; however, it is possible to consider molar output as independent of these details. For example, an alternative assumption is that organisms adopt strategies based on general constraints that arise in any schedule. Thus, over a session a stable set of behavioral strategiescould emerge that are relatively insensitive to the particular schedule characteristics. The schedule-feedback function adopted by Staddon and Motheral ( 1978) for the variable-interval schedule illustrates how aspects of responding can be independent of specific schedule characteristics. According to their schedulefeedback function, the organism was assumed to produce random interresponse times. The second problem with dependence on the schedule-feedback function is the. disagreement about its precise form. Consider,

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for example, the case of the variable-interval schedule. The schedule constraint in this case is quite weak: Both the number of responses and their exact location in time are left uncontrolled; a response need only occur after a variable interval to produce reward. As a result, the responseoutput can be quite complicated to specify, depending upon the characteristics of interresponse times (cf. Nevin & Baum, 1980). In fact, the exact characterization of responding has been a source of considerable disagreement in specifying the precise form of the schedule-feedback function for variable-interval schedules(Heyman, 1979; Heyman & Luce, 1979; Rachlin, 1978; Staddon & Motheral, 1978, 1979). The use of approximations to obtained rate of reinforcement potentially sidesteps this problem. Even so, it is hard to know which approximation to use. Myerson and Miezen (1980) assumed that obtained rate was reasonably approximated by programmed rate. Staddon and Motheral (1978) assumed that obtained rate was determined by a Poisson process that tracked programmed rate. Nevin and Baum ( 1980) offer a closer approximation to obtained rate based on a complex burst-pause random process.Given thesealternatives, it is possible that predictions from the same model will differ as a function of the approximation. There is a clear danger in basing a model on such a variable cornerstone. If, on the other hand, predictions from a model do not vary under such a range of approximations, then the precise designation of the schedule-feedback function may be unnecessary. In the present approach we have chosen the alternative of eliminating the specific details of the schedule-feedback function. We have.preservedonly the effect of the schedule as a molar constraint that couples or relates the instrumental and contingent responses. Therefore, we specify the schedule-feedback function as an unknown, unique transformation of the schedule imposed by the experimenter. Given a particular allocation of time to one behavior, !2, the level of the other behavior, 11,is given by 11= h(!2; p), (1) where h is the constraint function describing the actual relation of the two behaviors under

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the schedule and p is the rate of exchange between the two behaviors specified by the experimenter. This treatment of the schedule-feedback function has three immediate consequences. First, because the exact nature of the constraint functio~ need not be specified, we may develop a general model of learned performance that should suffice for any schedule. Second, such a general model can provide a framework for exploring proposed schedule-feedbackfunctions. Third, responding cannot be predicted as a function of specific characteristics of the schedule, such as programmed rate of reinforcement or ratio requirement. Control Functions In a regulatory system a control function determines the output by relating the input to a reference input or set point. Differences between the set point and input produce a signed error signal that determines the direction and! amount of change in output. The control function of a system statesthe precise relation between the error signal and changes in output. In the case of learned performance there are two regulatory systems, one each for the instrumental and contingent responses; therefore, there are also two control functions. Because of the experimenter-imposed contingency these control functions are not independent. By definition a contingency references the output level of one response to another. This kind of constraint means that the regulatory actions of the responsesystems are connected or coupled. Specifically, the control functions associated with each response system cross-coupIe the error signals for the two behaviors such that the error signal for one response is related to the output of the other. Thus, the output of the instrumental .responseis coupled to the error signal for the contingentresponse so that moving the contingent response closer to its set point requires an increasein the instrumental response.Similarly, the ou~put of the contingent response is altered by the error signal for the instrumental response so that moving the instrumental response closer to its set point requires a de-

266

STEPHEN JosE. HANSON AND WILLIAM

crease in the contingent response. Stable behavior in such a cross-coupled system depends on compromise between deviations from the two set points involved. A set of differential equations more precisely formulates the coupling in the riext section. The Coupled-Regulation Assumption General Coupled-Regulatory Equations We present a set of two differential equations to capture in a general way the assumptions of the equilibrium approach. The first equation describes the change in one of the 'responseslinked by a schedule, and the second describes the accompanying change in the other response. Although we typically call one of the responsesinstrumental and the other conting.ent,this designation is a decision made by the experimenter, usually on the basis of which response is relatively constrained. A second important point is that becausewe chose to represent the constraint function as an unknown function, we cannot specify changesin responding on a moment by moment basis. Rather the equations specify behavior change over the sessiontime, T. However, our general approach is not limited in principle to this molar level of analysis becausemore specific assumptions about the constraint function may be incorporated into these equations. Given two activities with time allocation t land t2, with respective set points s? and sg, related by a contingency schedule h, in a fixed time sample T, the set of equations is as follows: dtl - f n-

o [S2-

h

.o (t(,p),

tl,

SI]

and

(2)

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the schedule [(h(tl; -p)], (b) the set point for the first behavior (s?), and (c) the level of output of the first behavior (tl). The second equation specifies symmetric relationships for the second activity, that is, the change in the second behavior is a function of the error signal for the first behavior, and the set point and level of output of the second behavior. Note that the inverse of the constraint function, h-l, is used to specify th~ level of out.put of the second b~havior given the first behavior. The equations are said to be coupled because the changes in one activity are referenced to the deviation of the other activity from its set point. This deviation is transformed by the control function, f, which expressesthe relation between the error signal of one activity and the output of the other. If we assume that the function f preserves signed differences and emphasizes the coupled terms of the equations, it can be seen that this general system subsumes the predictive aspects of molar-equilibrium theory with respect to changes in baseline responding under a schedule (Timberlake, 1980). If the constraint function, h, allows the subject to achieve only one set point while it constrains the other response, then the derivative of the behavior linked to the constrained response will be positive, producing an increase in its level of responding over the session (reinforcement). On the other hand, if the constraint forces responding over the set point for one behavior, then the derivative of the other behavior will be negative, producing a decrease in its level of responding over the session (punishment). Finally, if the constraint function allows both behaviors to achieve their set points at once, both derivatives will be zero and responding shouJd remain at set-point levels for both behaviors (no change).

These are the most general regulatory differSpecial Coupled-Regulation Equations ential equations that represent changes in behavior under schedule challenge. The first To develop a more specific set of equations, equation saysthat the change in the time al- we must designate the form of the control located to the first behavior (ti) over the ses- function, f. The development of this function sion time (T) is a function of (a) the error requires consideration of several important signal for the second behavior, that is, the issues. The first is the representation of the difference between the set point for the sec- constraint function, h. Because it represents ond behavior (sg) and its performance under the level of performance of one behavior at

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any moment with respect to the other behavior, we can replace it with the actual level of performance of the behavior in question. Thus, the deviation of the first behavior from its set point (the error signal) can be written as s? -1 I. The second issue is the role of the present level of a behavior in scaling the effects of the coupled-error signal on responding. A reasonable assumption is that the effects of the coupled-error signal on changes in responding are directly proportional to its output level. Thus, the effects of the error signal for the second behavior on the change in the first behavior can be written as (s? -11)12. In other words, the more time an organism invests in a particular behavior, the greater the cbange in that behavior for a particular coupled-error signal. The third issue is the role of the set point of a behavior in modifying its own change in responding. It is important that the change in a behavior be related to its own set point because a fundamental assumption of the equilibrium approach is that each behavior is controlled by a separate regulatory system. Although the coupling in the regulation equations cited clearly adds other determinants of behavior change, it would not be expected to remove entirely the importance of the uncoupled response determinants. We considered a variety of ways in which a response's own set point could modify its responding over time. Many of the forms that we considered served at least partly as a resistanceto increasesin responding that varied directly with the level of output. We eventually chose the ratio of the expression of a response to ItS .. set pOInt ( e.g., 0-11 ) ' In . part S.QI

because of its relative simplicity and tractability. According to this expression the resistance to increase of a response is inversely related to its set point, and is directly related to its current response level. As we show in the equations ahead, this form of including the set point of a behavior means that its contribution to its own change is asymmetric with respect to the direction of movement. Other things being equal, behaviors with high set points will increase more readily and decreaseless readily under a schedule and tend to be more stable over time in a baseline condition. In contrast, re-

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sponses with low set points show larger resistance to increase and less resistance to decrease under a schedule, and tend to drift exponentially toward lower levels of responding in a baseline condition. A final important issue is how to reference the behavior change predicted by these equations. To clarify the ability of the general set of equations (Equation 2) to subsume the predictions of equilibrium theory, we referenced the changes in behavior over the sessiontime to the baseline values. In the current equations we reference the changes at any time 1 to the zero levels of responding at the beginning of the session interval. Combining the general coupled equations above with our specification of the control functions relating the inputs and outputs of the two behaviors produces the set of equations that follows. Given two behaviors with times of expression equal to 1land 12, setpoints of s? and sg, and a constraint function cross-coupling the outputs such that the second behavior is relatively deprived, the change in each behavior at any moment in the session time, 1, can be written as -d11.= dl

b(sg -12)11 -1J. sYal

and dt2 d t

-1 ---

b

(Si

O

-li)t2

-0

t2

'

\.II

s2a2

where ai and a2 represent each behavior's resistance, and the amplification factor or linkage is represented by b. These equations are not as general as the first set in that it is necessary to enter the relatively unconstrained (instrumental) response in .the first equation and the relatively constrained (contingent) responsein the second equation. The negative sign emphasizes that as the performance for one behavior increases the other should decrease (cf. Myerson & Miezen, 1980). It. also causes both the instrumental and contingent responsesto increase through time even though error signals of these responseshave opposite signs. The parameters of this equation have reasonable interpretations as scaling factors. For example, the parameter b modifies the cOIitribution of the error signal from the coupled behavior and appears likely to be related to the strength of the motivational linkage be-

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STEPHEN JosE HANSON AND WILLIAM

tween the two responses. In other words, b is at least partly an index of the motivated association between the responsesand, thus, should vary with characteristics of the schedule. Schedulesthat specify close relations between instrumental and contingent behaviors 'would be more likely to have high values of b. In the present form of the model, we made the simplifying assumption that this linkage is reciprocal for the two responses,that is, if the effect of one coupled-error term is modified by b, the effect of the other is modified by l/b. Whether this simplification is broadly useful remains to be seen. The parameters al and a2 scale the resistance of the two responsesto increase. These values reflect some of what we previously called the expression characteristics of a response.Those aspectsof responding have received recent attention in concepts such as elasticity (Rachlin, Battalio, Kagel, & Green, 1981; Staddon, 1977) and behavioral inertia or momentum (Nevin, 1974; 1979; Nevin et al., 1983). In the present model, expression characteristics should modify changesin behavior under schedule constraint as well as stability of baselines when the responsesare uncoupled. Under uncoupled conditions, according to these equations, the expression of each response will drift exponentially downward under fixed instigation and time toward an output level proportional to its set point. If an increase in instigation occurs, a response should increase therefore in a fashion inversely proportional to its set point and with a degree of hysteresis (relative to its return) that is directly proportional to its set point. Under coupled conditions, this differential system representsthe tension between the set points of the two responsesand the requirements of the schedule-constraint function. The solution to such a system results in a functional relation between the two responses.The plot of this function, the molar output of one responseagainst the other, is called the phase plane of the system. Given any initial conditions, all solutions to the differential equations can be shown in the phase plane. In the present case,therefore, the phase plane represents the predictive space of the theory, and the shape and form of the data in the

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phase plane provide an important initial test of the model. Solutions The intent of this section is to derive and represent several solutions to this system of differential equations. The basic form of a solution is a functional relation between the two behaviors. We first outline three approaches to producing such a solution, and then represent the solutions in a phase plane. The first solution is an exact form; that is, it represents a complete solution to these equations. Unfortunately, the form is not a simple analytic function. The second and third solutions are based on standard approximations to an exact solution. They reveal critical properties of a system of differential equations and are especially useful when no exact form can be found. These other solutions are particularly interesting because they represent current models of Allison (1976) and Staddon (1979). Exact Form Because these equations include the nonlinear term tlt2, they would ordinarily prove quite difficult to solve-requiring some sort of approximation technique (e.g., RungeKutta). Fortunately, this system is similar in form to the well-known Lotka- Volterra system (Lotka, 1956), which has a relatively simple solution and a well-understood behavior. The Lotka- Volterra system has been used with some successin describing intraindividual competition and predator-prey relations. The solution to the system is relatively straightforward and is repeated here. Eliminating the time variable, t, and considering the derivative of t I with respect to t2 yields dl, dl2

= --

.0

b (SI -11)12-

{2

(4)

0 S2a2

This equation is part of the class of separable differential equations (Ross, 1974) and therefore each variable and derivative may be isolated by multiplication or division. After iso-

~

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COUPLED-REGULATION

MODEL

lating each variable we may integrate both sides of the equation:

b log -- s?

t I

+

tlb

-

1

-0-

log t 1 + Cl

S2Q2 =

bsg

log

(2 -b(2

1

-

log h + C2,

(5)

s?aJ

. where Cl and C2 are constants of integration. Factoring this equation, collecting terms on one side, and letting Cl -C2 = c produces the following

)

-.bs~ -+

log

solution: 12-

b12-

slal

0

+

~+ b

~

)

1 2

log

t I -..!

2

t

= c.

(6)

b

Exponentiating (letting ec = k) reveals the complex structure of this function:

=

k.

(7)

The exponential terms on each side of the equation do not allow the isolation of either variable. In fact, these so-called "transcendental equations" (Southworth & Delleeuw, 1965) cannot be reduced to a simple analytic form [neither y = j(x) nor x = j(y)]. This property makes regression analysis difficult. Moreover, the simple behavior of the function in the phase plane has to be determined through some approximation technique. Two other techniques reveal properties of the differential system and phase plane without an analytic solution. The first identifies the stationary behavior of the system (optimal solution) and allows the location of "critical points" around which the phase plane is organized. In nonlinear systems this stationary analysis provides important qualitative information about the system prior to using a numerical technique to reveal its detailed behavior (Burghes & Borrie, 1981). A second technique called linearization may be used to gain an intermediate level of information between a stationary analysis and an analytic solution. By linearizing the differential equations, the intrinsic nonlinearity of the system may be explicitly ig-

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nored, leaving a linear system whose solution provides conservative information about the system. That is, the linearization produces a subset of the behavior of the nonlinear system by providing an approximation of the exact functional form of the solution to the differential equation (Equation 7). In addition to providing more information about the system of equations, both of these techniques lead to special casesof the differential system that are related to other recently proposed models of learned performance. The stationary solution of the coupled-regulation model can be used to derive Allison's ( 1976) linear conservation model, whereas the linearization of the coupled-regulation model can be used to derive Staddon's (1979) minimum-distance model. Stationary Analysis: Linear Models As already mentioned, the phase plane for a set of differential equations is a plot of the two variables independent of time. The critical points organize the phase plane and determine the direction and movement of the system from various' starting points. Critical points are defined as those values of the variables that cause the derivatives simultaneously to vanish. For the present model here are four such cases.The first is a trivial case-when both t land t2 simultaneously vanish. Thus, the origin is a critical point, and initial or starting values of the system close to the origin tend to be drawn into the origin (see Figure 1). The next two cases do not have a simultaneous solution and therefore do not define critical points: (I

=

0

andI

.--'--

1 -=0

b1(sy

-t1)

-sga2

or !2 =

0,

and I

1

vh~{('Q-t~),,\U " ."'

O stat

The last case defines a locus of critical in the phase plane and occurs when bl(sg -(2)

-0-

1

-

=1}_0

(8)

points

= O

stat and 1 S~a2

(9)

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STEPHEN JosE HANSON AND WILLIAM

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by focusing on its linear terms. This approach provides both a simple solution and an approximation of the behavior of the complete nonlinear system. The linearization can be accomplished by considering the Taylor expansion of the nonlinear solutiori (Equation 6) around two arbitrary positive constants, kl and k2: .o bl b;

(2

(-10)

1

bls I + 0, s2a2 (t2 -k2)2 -

)(log k2 + (t2 -k2) k2

, ~

+ (l2-~~r

1-+ 00 0 -blt2 This linear function represents an optimal 2k~ 3k~ compromise relative to the instigation and resistanceof each behavior as it changesalong + some constraint function. As an outcome of the coupled-regulation model, the linear function would be likely to arise when (a) the constraint function is relatively simple as, for example, in the case of -b2t 1 = C. proportionality, (b) the range of each behav(13) ior is restricted so that resistance. of either response has little effect on output, and (c) As t 1 and h are in the neighborhood of the the behaviors have relatively low resistance origin, we may ignore the higher order terms. to change. If enough of these conditions are Letting .. satisfied, the behavior of an organism should k=-1 b2 be predictable from a linear model. Allison ( 1976) proposed a linear model of and molar performance based on the notion that some dimension of the behavior is conserved (14) during baseline and contingency sessions. The optimal solution of the coupled regulation is identical to Allison's conservation model when --

k=!?-! b2'

and in our notation

0 s1a1

-0

,

(11)

s2a2

2[~(b2s~ -i;) ]2

appears as

tl = s? + ks~ -kt2

(12)

Thus Allison's model-assuming we provide appropriate interpretations for the parameters-can be shown to be an optimality model based on the present regulation approach. Linearization: Elliptical Models The behavior of linear differential systems is well understood, and these equations are usually straightforward to solve. Thus, a nonlinear differential system may be simplified

= constant.

( 15)

This is an ellipse with center at kt and k2, major axis at 4kI, and minor axis at 4k2° I We have chosen to write two instigation

parameters

instead of one. All of the following results would be identical except that the instigation parameter would be squared in some cases.

A

COUPLED-REGULATION

MODEL

Thus, the center interacts with the height and width of the ellipse. As the center of the ellipse moves toward the origin, the ellipse grows. Both the height and width of the ellipse increase. Staddon ( 1979) proposed a similar elliptical model of learned performance based on the premise that an organism under schedule constraint attempts to minimize the Euclidean distance from the schedule performance to the baseline level. To derive his model Staddon assumed that the schedule con:. straint had to be introduced as a unique function (we consider only his ratio schedule case) and that the background behavior (13) varied inversely with the two measured behaviors under a contingency. From these assumptions Staddon derived the standard second-degree equation relating response outputs 11and 12:

OF

LEARNED

values, a prediction that differs from the one based on the linearization of our model and that seemsarbitrary. Now consider the general case of b ~ 0. It can be shown that if 2b2 -4(a2 + b2)(~ + c2) < 0,

.k tl

-t2(b2s~

+ b2S? + C2S~)

+ 2b2t)t2 = 0,

2

)

1

a2

1 C2

constant.

k3

( ) ~

2

k1

.!

( )2 ~

+ 4

k3

,

(16)

)

.=

)2

k3

+

= .! 4

1 o t2 -2 S2

+

k

t2 -~

kl

where a, b, and c represent "costs" of engaging in each activity (t 1, t3, and t2) respectively. The linearization of our model was shown above to be an elliptical form. At first glance it is not clear that Staddon's (1979) model is generally elliptical because there are other conics that this second degree equation can define. However, it can be shown generally that Staddon's model is elliptical and therefore a special case of the coupled regulation model. First, consider the special case, b = 0, where background behaviors are not considered or where they have insignifical)t cost for the organism. Staddon's model immediately reduces to a simple elliptical form: . 2 . 2 1 o tl -2 SI

(

)

--1

.kl

+ b2s~ + a2s?)

( 18)

then Staddon's model is always an ellipse as opposed to other second-degree forms like parabolas, hyperbolas, or degenerate conics. This condition is always satisfied unless negative costs are allowed. Playing the b parameter back into the equation allows the ellipse to rotate in the xy plane. The x and y axes can be rotated so that the xy term in Staddon's second-order form disappears, revealing a standard elliptical form:

tT(a2 + b1 + t~(b2 + c1 -t1(b2s?

PERFORMANCE

(17)

In this case, the center of Staddon's ellipse, the point that defines the maximum, is predicted to be exactly one half of the baseline

r = ! arctan 2

(

~ a

-c

)

,

and ifa2

=

C2

1!"

then

r=-. 4

Thus, the center of Staddon's ellipse is a complex function of the cost parameters and baseline values. Further rotation of the ellipse is defined by r, which from this last equation can be seen to be a function of all three cost parameters. Also from this form it is obvious that the center of the ellipse interacts with its height and width. As the center moves to-

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scribing the determinants of learned performance. He referred to the schedule-feedback function as an output-input function that designated the relation between responding and the effect of that responding in the environment. Similarly, he referred to the control function as an input-output function that described the relation between the consequencesof responding and subsequent responding. An important difference from Staddon's analysis is that our model begins with the assumption that there are two separate regulatory systems, one for the instrumental response and one for the contingent response. These two systems are then linked together by the schedule-feedback function to form the final system described by Staddon ( 1980). Despite the similarities with previous work, there are several aspectsof our approach that merit emphasis. First, we have tried to make explicit the assumptions that went into the development of our differential equations~ Most of these assumptions were based on the molar-equilibrium approach to learned performance espoused by Timberlake ( 1980); however, further assumptions were needed as the development of our equations forced us to clarify the precise nature of the control function. We hope the result of our efforts is a general approach that can be scrutinized at both the level of assumptions and the level of functional form. Alternatives within this approach can be readily developed by altering an assumption or the form of expression of that assumption. Second, our model represents a clear break with previous work in its lack of dependence on an explicitly derived schedule-feedback function. The role of the schedule-feedback function in coupling the environmental effects of the outputs of the two regulatorysystems is clearly vital to our model, but the prior knowledge of the precise form of the linkage is not. As we pointed out, there are advantages to stating the schedule-feedback function only in a general form. Most advantages are related to the problem that the relation between the instrumental and contingent responses specified by the experimenter does not accurately capture the empirical relationship worked out by the animal in interaction with its environment.

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On the other hand, there are several potential objections to the absence of a specific schedule-feedbackfunction in our model: (a) Responseoutput cannot be stated as a func-'o ti on of schedule parameters such as programmed rate of reinforcement or ratio size, (b) the moment-to-moment contingencies of responses and reward cannot be specified, and (c) the present work does not continue the long history of research on the local reinforcement determinants of schedule performance. Oddly enough, these objections can be turned to some advantage. Each objection is based partly on the underlying assumption that learned performance is produced by the response-by-response ( or response-interval by responSe-interval) tracking of probability of reward. In fact, the evidence for this fundamental assumption is mixed. The most successfulgeneral approach to fitting learned performance, the matching theory, does not include this assumption in any testable form. Prelec (1982) has recently argued that theories that depend on response-by-response tracking of reward probability may be in difficulty because the local differences between schedules that produce different levels of responding may be too small for the animals to discriminate (cf. Killeen, 1982b}. We think that the framework that we have provided will encourage investigation of the actual form of schedule-feedback functions and of the response or time frame over which they operate. We admit that the absence of a specific schedule-feedback function necessitates the inclusion of parameters that in part serve to estimate its form. However, it is our hope that this general approach can be used to develop a theory of schedule effects that will allow specification of more aspects of the constraint function ahead of time. Many recent theories of learned performance have a heavy investment in optimality assumptions. A third important aspect of our approach is the lack of dependence on such an optimality analysis. To date, the key characteristics of optimality models have included a dependence on specification of the schedule-feedbackfunction, the exclusive use of stationary analysis to realize the model, and the underlying assumption that all responding is reducible to a single dimension

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points, it is necessary to discover the time interval over which instigation is/integrated to affect behavior and then to estimate the level of instigation over these time periods. Ultimately, studies of the way in which feedback functions contact behavior and studies of the timing and nature of the baseline instigation of responding must be combined to produce a coherent study of local effects. A second area of concern and an important question in the development of any model of behavior is the role of free parameters. The present model appears to have a large number of such parameters: the two set points (s? and sg), the index of instigation (b), and the two resistance scales(al and Q2). In most cases,the two set points are fixed by the baseline data prior to imposing the schedule, thus leaving a total of three parameters to be fit from the data. This number of parameters is comparable to the number in most other models that have been proposed to account for learned performance. The number of parameters in a model is not critical as long as the available data contain enough degreesof freedom to estimate these parameters. Further evaluation of the present model and any other model will require a significant number of data points over a large range of schedule values. Such data for interval schedules are notably absent. A third area of concern is the extension of the model to deal with more complex circumstances involving choice situations and specific background responses.For example, it appears possible to model choice data on concurrent schedules by using a matrix of differential equations that describes the regulatory interplay of four behaviors and their interactions. Although the general choice Areas of Concern model has an unwieldly number of parameters, they shrink to a reasonable number As indicated, an important omission in the present model was the accounting for mo- given the typical experimental circumstances mentary changes in responding. Part of the of the same instrumental and contingent rebasis for this omission was to avoid problems sponses for each choice. Finally, we wish to outline some general associated with specifying the constraint ways in which the present model can fail. A function. However, there is another potential model with a flexible form and a number of problem in designating the momentary set points of behavior. The paired baselines serve free parameters may easily assume a life of as measuresof total instigation over the entire its own that is reasonably impervious to dissession; they do not necessarily reflect mo- proof. Fitting a model to data can become ment-to-moment changes in instigation. To a primary focus of attention, with good fits designate the appropriate momentary set somehow offsetting bad fits in any overall

(Prelec, 1982). Our model shares none of these characteristics. A schedule feedback function was treated only as a device for coupling the outputs of two regulatory systems rather than as a necessarymeans for apportioning predicted responding between alternatives. Second, we analyzed the differential system dynamically, considering stationary analysis as a special case. Last, we made no assumptions concerning the reducibility of behaviors to some common dimension. As far as the coupled-regulation model is concerned, no relation among responsesis needed other than their linkage by the schedule. The final important aspect of our model is the absence of dependence on an explicit concept of background responding. Although this may seem to be an important omission, it may actually be relatively unimportant. Background responding referred to in other models has not been developed or tested as an empirical concept but was included largely for its response-limiting and scaling properties. Background responding contributes to predictions by competing with the increases in instrumental responding and by allowing rescaling of the importance of instrumental and contingent responding as a function of changesin variables such as deprivation level (e.g., Herrnstein, 1974; Staddon, 1979). In our model these functions are performed by the parameter modifying the resistance of each response to change and by the instigation or linkage parameter that modifies the importance of the error signals. Thus, we may be able to incorporate background responding in our model in a form more consistent with what is actually present.

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evaluation. We tried to make this model susceptible to disproof by clearly stating the assumptions of the model and by developing parameter interpretations based on our intuitions concerning regulatory processes. The model can be contradicted empirically in many important ways. First, in spite of the flexibility suggestedby Figure 2, the form of the model is not compatible with many different forms of data. The model is always bitonic and the rate of rise is constrained by the rate of decline. The location of the peak of the model is a function of the relationship of the set points and the resistance parameters but is independent of the instigation or linkage parameter. The skew apparent in most fits is a natural response of the model; symmetric data (e.g. a normal curve) violate the model. Second,given a reasonable amount of data from systematic manipulations of schedule parameters, it should be possible to compare precisely the predictions of any similar model with those of our form. For example, although Staddon's ( 1979) model is related to ours, the two models make different predictions becausethe center of each function interacts in opposite ways with its peak value. Finally, it is possible to test the model across situations where the parameters are expected to remain invariant. For example, under the uncoupled conditions of baseline assessment, an estimate of the resistance parameters can be obtained that would be expected to be constant during contingency. Not only would this estimate provide a cross-situational test of the model but it would allow the model to fit contingency data with one free parameter. Summary We have translated the behavior-equilibrium approach of Timberlake. ( 1980) and Timberlake and Allison (1974) into a theory of behavior regulation based on the concepts of control theory. The fundamental tenet of this approach is that under schedule challenge, learned performance is produced by the simultaneous regulation of behavior with respect to the free-baseline set points of both instrumental and contingent responding. From the behavior-regulation approach, we

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developed a general mathematical framework for dealing with coupled regulation. We also developed a specific model of two coupled differential equations that described the changes in instrumental and contingent responding under a schedule. Solution of this set of coupled equations produced a complex and flexible function relating instrumental and contingent responding. It was possible to show that two current models of learned performance are special casesof the general coupled-regulation model. Furthermore, the coupled-regulation model provided an adequate fit to a variety of data from simple schedules. Reference

Notes

I. Timberlake, W., & Peden, B. Peckingfor food in pigeons: Regulation or reinforcement?Paper presented at the meeting of the Midwestern Association for Behavior Analysis, Chicago, May 1977. 2. Lucas, G. Unpublished manuscript, Indiana University, 1980. References Allison, J. Contrast, induction, facilitation, suppression and conservation. Journal of the Experimental Analysis of Behaviot; 1976,25, 185-199. Atkinson, J. W., & Birch, D., The dynamics of action. New York: Wiley, 1970. Baum, W. M. Optimization and the matching law as accounts of instrumental behavior. Journal of the Experimental Analysis of Behaviot; 1981,36, 387-403. Bolles, R. C. Some functionalistic thoughts about regulation. In F. M. Toates& T. R. HaIliday (Eds.), Analysis of motivational processes. London: Academic Press, 1980. Booth, D. A. (Ed.). Hunger models. London: Academic Press, 1978. Burghes, D. N., & Borrie, M. S. Modeling with differential equations. New York: Halsted Press, 1981. Cannon, W. B. Hunger and thirst. In C. Murchison (Ed.), The handbook of general experimental psychology. Worcester, Mass.: Clark University Press, 1934. Catania, A. C., & Reynolds, G. S. A quantitative analysis of the responding maintained by interval schedulesof reinforcement. Journal of the Experimental Analysis of Behaviot; 1968, 11, 327-383. Crozier, W. J., & Hoagland, H. The study of living organisms.In C. Murchison (Ed.), The handbook of general experimental psychology.Worcester,Mass.: Clark University Press, 1934. Curry, R. E. A random search algorithm for laboratory computers. Behavior Research Methods and Instrumentation,-1975, 7, 369-376. Herrnstein, R. J. Relative and absolute strength of response as a function of the frequency of reinforcement. Journal of the Experimental Analysis of Behaviot; 1961,4, 267-272.

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Herrnstein, R. I. On the law of effect. Journal of the Experimental Analysis ofBehavio1; 1970, 13, 243266. Herrnstein, R. I. Formal properties of the matching law. Journal of the Experimental Analysis of Behavio1; 1974,21, 159-164, Herrnstein, R. I. Derivatives of matching. Psychological Review, 1979,86, 486-495. Herrnstein, R. I., & Vaughan, W., lr. Melioration and behavioral allocation. In I. E. R. Staddon (Ed.), Limits to action: The allocation of individual behaviiJ1:New York: Academic Press, [980. Heyman, G. M. Matching and maximizing in concurrent schedules.Psychological Revie~ 1979, 86, 496500. Heyman, G. M., & Luce, R. D. Operant matching is not a logical consequenceof maximizing reinforcement rates. Animal Learning and Behavio1; 1979, 7, 133140. Hull, C. L. Principles of behavio1:New York: Appleton, , 1943. Kelsey, I. E., & Allison, I. Fixed-ratio lever pressing by VMH rats: Work vs. accessibility of sucrose reward. Physiology & Behavio1; 1976,17,749-754. Killeen, P. R. The genesis, modulation and extinction of arousal. In M. D. Zeiler & P. Harzem (Eds.), Advancesin analysis of behavior: Vol. 1. Reinforcement and the organization of behavio1:New York; Wiley, 1979. Killeen, P. R. Incentive theory II: Models for choice. Journal of the Experimental Analysis of Behavio1; 1982, 38, 217-232. (a) Killeen, P. R. Incentive theory. In D. I. Bernstein (Ed.), Nebraska Symposium on Motivation: Val. 29. Responsestructure and organization. Lincoln: University of Nebraska Press, 1982. (b) Killeen, P. R., Hanson, S. I., & Osborne, S. R. Arousal: ~ts genesisand manifestation as response rate. Psychological Review, 1978,85, 571-581. Loeb, I. Forced movements,tropisms, and animal canduct. Philadelphia: Lippincott, 1918. Lotka, A. I. Elements of mathematical biology. New York; Dover, 1956. McDowell, I. I., & Kessel, R. A multivariate rate equation for variable interval performance. Journal of the Experimental Analysis of Behavio1; 1979,31, 267284.

McFarland, D. I. Feedback mechanisms in animal behavio1:London: Academic Press, 1971. McFarland, D. I., & Houston, A. I. The state space approach to behaviou1:London: Pitman, 1980. Myerson, J., & Miezen, F. M. The kinetics of choice: An operant systems analysis. Psychological Review, 1980, 87, 160-174.

Nevin, J. A. Response strength in multiple schedules. Journal of the Experimental Analysis of Behavio1; 1974,21, 389-408. Nevin, J. A. Reinforcement schedules and response strength. In M. D. Zeiler & P. Harzem (Eds.), Advances in analysis of behaviour: Vol. 1. Reinforcement and the organization of behaviou1:New York: Wiley, 1979. Nevin, J. A., & Baum, W. Feedback functions for variable interval reinforcement. Journal of the Experimental Analysis of Behavio1; 1980, 34, 207-217. Nevin, J. A., Mandell, C., & Atak, J. R. The analysis of

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behavioral momentum. Journal of the Experimental Analysis of Behaviol; 1983,39, 49-59. Powers,W. T. Quantitative analysis of purposive systems: Some spadework at the foundations of scientific PSYchology. PsychologicalReview, 1978,85, 417-435. Prelec, D. Matching, maximizing and the hyperbolic reinforcement feedback function. Psychological Revi~ 1982,89, 189-230. Premack, D. Toward empirical behavioral laws. 1. Positive reinforcement. Psychological Revi~ 1959, 66, 219-233. Premack, D. Reinforcement theory. In D. Levine(Ed.), Nebraska Symposium on Motivation (Vol. 13). Lincoin: University of Nebraska Press, 1965. Pring-Mill, A. F. Tolerable feedback: A mechanism for behavioral change.Animal Behaviol; 1979, 27, 226236. Rachlin, H. C. A molar theory of reinforcement schedules. Journalof the Experimental Analysis of Behaviol; 1978, 30, 345-360. Rachlin, H. C., & Burkhard, B. The temporal triangle: Response substitution in instrumental conditioning. PsychologicalReview, 1978,85, 22-47. .Rachlin, H. C., Kagel, J., & Battalio, R. C. Substitutability in time allocation. Psychological Revi~ 1980, 87, 355-374. Rachlin, H. C., Battalio, R. C., Kagel, J., & Green, L. Maximization theory in behavioral psychology. BehavioralandBrain Sciences, 1981,4, 371-418. Ross, S. L. Introduction to ordinary differential equations. Lexington, Mass.: Xerox, 1974. Schneider, B. A two-state analysis of fixed-interval responding in the pigeon. Journal of the Experimental Analysis of Behaviol; 1969, 12,667-687. Southworth, R. W., & Delleeuw, S. L. Digital computation and numerical methods. New York: McGrawHill, 1965. Staddon, J. E. R. On Herrnstein's equation and related forms. Journal of the Experimental Analysis of Behaviol; 1977,28, 163-170. . Staddon, J. E. R. Operant behavior as adaptation to constraint. Journal of Experimental Psychology:General, 1979,108, 48-67. Staddon, J. E. R. Optimality analysesof operant behavior and their relation of optimal foraging. In J. E. R. Staddon (Ed.), Limits to action: The allocation of individual behavio1:New York: Academic Press, 1980. Staddon, J. E. R., & Motheral, S. On matching and maximizing in operant choice experiments. Psychological Revi~ 1978, 85, 436-444. Staddon, J. E. R., & Motheral, S. Response independence, matching and maximizing: A reply to Heyman. Psychological Review, 1979,86, 501-505. Timberlake, W. A molar equilibrium theory of learned performance. In G. H. Bower (Ed.), The psychology of learning and motivation (Vol. 14). New York: Academic Press, 1980. Timberlake, W., & Allison, J. Responsedeprivation: An empirical approach to instrumental performance. PsychologicalRevieK: 1974,81, 146-164. Toates, F. M. Animal behavior-a systems approach. Chichester, Sussex,England: Wiley, 1980.

Received October 20, 1982 Revision received April 4, 1983 .