BACKGROUND ACTIVITIES, INDUCTION, AND BEHAVIORAL ALLOCATION INOPERANT PERFORMANCE

WILLIAM M. BAUM

University of California, DavisAND

MICHAEL DAVISON

University of Auckland

In experiments on operant behavior, other activities, called “background” activities, compete with theoperant activities.Herrnstein’s (1970) formulation of thematching law includedbackground reinforcers inthe form of a parameter rO, but remained vague about the activities (BO) that produce rO. To gain moreunderstanding, we analyzed data from three studies of performancewith pairs of variable-interval schedulesthat changed frequently in the relative rate at which they produced food: Baum and Davison (2014), Belkeand Heyman (1994), and Soto, McDowell, and Dallery (2005). Results sometimes deviated from thematching law, suggesting variation in rO. When rO was calculated from the matching equation, two resultsemerged: (a) rO is directly proportional to BO, as in a ratio schedule; and (b) rO and BO depend on the foodrate, which is to say that BO consists of activities induced by food, as a phylogenetically important event.Other activities unrelated to food (BN) correspond to Herrnstein’s original conception of rO and may beincluded in the matching equation. A model based on Baum’s (Baum, 2012) concepts of allocation,induction, and contingency explained the deviations from the matching law. In the model, operant activity B,BO, and BN competed unequally in the time allocation: B and BO both replaced BN, BO replaced leverpressing (Soto et al.), and key pecking replaced BO (Baum&Davison). Although the dependence of rO andBO on food rate changes Herrnstein’s (1970) formulation, the model preserved the generalized matchinglaw for operant activities by incorporating power-function induction.Key words: matching law, behavioral allocation, induction, phylogenetically important event, induced

activity, background activity, extraneous reinforcers

In any study of operant behavior, otheractivities besides themeasured operant activitiesoccur. These other activities are often called“background” activities, but they are not neutral,because they may compete with the operantactivities. Researchers have occasionally tried toidentify the background activities and the waythat they compete (e.g., Baum, 1981; Davison,1993, 2004;Herrnstein, 1970; Soto,McDowell, &Dallery, 2005), but no definite answers havearrived. In this paper, we extend the inquiry intothe nature of background activities and theirrole in determining operant performance.Herrnstein (1961) originally expressed the

matching law as a relation of proportions:

B1

B1 þ B2¼ r 1

r 1 þ r 2; ð1Þ

where B1 and B2 are rates of responding atAlternatives 1 and 2, and r1 and r2 are thecorresponding reinforcer rates obtained fromAlternatives 1 and 2.

Later, Herrnstein (1970; see also Baum &Rachlin, 1969) generalized the matching law ton alternatives:

BiPB¼ r iP

r; ð2Þ

where the numerators left and right indicateresponse rate at any one alternative andreinforcer rate obtained from that alternative,and the denominators left and right representthe sum of all response rates and the sum of allreinforcer rates.Herrnstein used Equation 2 to derive an

equation for describing performance on singleschedules by assuming the sum of all behavior tobe a constant k and the sum of all reinforcerrates other than that programmed by theschedule to equal rO:

B ¼ kr

r þ r O: ð3Þ

For consistency, we are using thenotation rOhere,even though the same parameter has sometimesbeen denoted other ways, such as re or rb.Subsequent research showed that Equations 1

and 2 were oversimplified, because deviations

Correspond with: William M. Baum, 611 Mason #504, SanFrancisco, CA 94108, Email: wbaum@sbcglobal.netdoi: 10.1002/jeab.100

JOURNAL OF THE EXPERIMENTAL ANALYSIS OF BEHAVIOR 2014, 102, 213–230 NUMBER 2 (SEPTEMBER)

213

from matching in the form of overmatching,undermatching, and bias occur (Baum, 1974).Thus, Equations 1 and 2 are generally presentednow as including power functions that capturebias in the coefficient and sensitivity in theexponent. McDowell (2005) suggested rewritingEquation 3 in a similar way, adding exponents tor and rO. In the present paper, we opt for thetheoretical coherence and simplicity of Equa-tion 3, rather than introduce extra, possiblyunnecessary, fitting parameters that lack a basisin theory.

Following the logic of Equation 3, Herrnsteinderived an equation to describe responding onany one of two concurrent schedules by assum-ing rO to equal the sum of all reinforcer ratesother than those programmed by the twoschedules, expressing the sum of all reinforcerrates as the sum of r1, r2, and rO:

B1 ¼ kr 1

r 1 þ r 2 þ r O: ð4Þ

Belke and Heyman (1994) tested Equations 3and 4 with an experiment that included bothsingle Variable-Interval (VI) schedules andconcurrent VI schedules. Seven single scheduleswere presented, within each session, separatedby brief blackouts, and those same seven werealso presented with a constant VI schedule as aconcurrent alternative. In the concurrentphase, the seven schedules were all presentedunsignaled and in random order within eachsession along with the constant VI, with theresult that each session included seven differentfood ratios. Their results supported Equations 3and 4.

We (Baum & Davison, 2014) studied fivedifferent conditions in which pairs of concurrentVI schedules were arranged within sessionssimilarly to Belke and Heyman’s (1994) concur-rent schedules. In each condition a different VIschedule was held constant on the left key whilethe same seven different VI schedules werepresented in seven different components, sepa-rated by blackouts, within each session. Eachcomponent lasted until the 10th food delivery.Each condition lasted 100 sessions, and weanalyzed the last 85 sessions. To reduce thenumber of figures, we pooled the data of the sixpigeons in the study and showed that the pooleddata were representative of the individualpigeons’ data [see Baum & Davison (2014) formore details].We turn now to those pooled data.

Herrnstein’s Response-Rate Hyperbola

To analyze the response rates when they wereroughly asymptotic within components, we usedthe results from the last three food deliveriesonly, and all figures in this section show onlydata from the last three food deliveries. Tosummarize, we added up all the times betweenthe seventh and eighth food deliveries, betweenthe eighth and ninth food deliveries, andbetween the ninth and tenth food deliveries,and these sums constituted the time bases. Weexcluded the time taken to switch keys, because,in contrast with Belke and Heyman (1994), whoused a changeover delay on switching, werequired two pecks on a changeover key toswitch alternatives. We summed all the pecksbetween the seventh and eighth deliveries,between the eighth and ninth, and betweenthe ninth and tenth. Dividing the three sums ofpecks by the three time bases gave threeresponse rates. We counted all the seventh,eighth, and ninth food deliveries and divided bythe time bases to compute the three food rates.

When Equation 3 is fitted to response rates,both k and rO are usually assumed to be constant(e.g., Belke & Heyman, 1994; Herrnstein, 1970;Soto et al., 2005). The assumption that rO isconstant, however, seems incorrect when viewedin the light of theoretical treatments of otherreinforcer sources. By definition, reinforcers arecontingent on behavior. Just as r depends onB, so we expect rO to depend on BO. Thedependence of reinforcers on the activitiesthat produce them is specified by a feedbackfunction (Baum, 1973). Every schedule ischaracterized by a feedback function (Baum,1973, 1992). For example, the feedback func-tion for a ratio schedule is direct proportionalitybetween reinforcer rate and response rate(Baum, 1973). The feedback function for a VIschedule is curvilinear, because it approachesthe programmed reinforcer rate as an asymp-tote as response rate increases (Baum, 1992).By extension, if rO is the result of activities, BO,we would expect rO to vary with BO (e.g.,Davison, 1993). Do we have any evidence thatbears on the question of whether rO is constantor variable?

Total Pecking on Two Alternatives

We can simplify the inquiry initially byconsidering total responding, including both

214 WILLIAM M. BAUM and MICHAEL DAVISON

alternatives, in relation to total food delivered,including both alternatives. The relevant equa-tion is Equation 3. Figure 1 shows pecks perminute (ppm) plotted against food deliveriesper minute (fpm) in logarithmic coordinates.The line in each graph is the best-fittinghyperbola specified in Equation 3, fitted inarithmetic coordinates using Microsoft Excel’sSolver. The least-squares parameter estimatesare shown in each graph. The asymptote kshowed no trend across conditions, but regres-sion showed some tendency for rO to increasewith food rate (Baum & Davison, 2014; Fig. 3).

In arithmetic coordinates, the fits to thehyperbola looked satisfactory (VAC or r2> .8),but when viewed in these logarithmic coordi-nates, systematic deviations from the hyperbolabecome evident, because they are proportion-ate, rather than absolute, making the deviationsat lower food rates particularly evident. In everygraph, the response rates for the low food rateslay above the curve, the response rates for themiddle food rates fell below the curve, and theresponse rates for the higher food rates layabove the curve. The systematic deviationssuggest some sort of systematic variation, and

Fig. 1. Peck rate versus food rate for five conditions of Baum and Davison’s (2014) study of frequently changingconcurrent VI VI schedules. All points represent total pecking on two keys in the last three interfood intervals of components.Conditions differed in the VI schedule programmedon the left key, which was constant while the VI on the right key changed.Broken lines represent the best fit of Herrnstein’s (1970) hyperbola in arithmetic coordinates. Note logarithmic coordinates.

BACKGROUND ACTIVITIES AND INDUCTION 215

since a hyperbola has only two parameters, oneof them must vary systematically with food rate.Since k specifies asymptotic response rate andshouldn’t change under the present conditions,the most theoretically consistent interpretationis that rO changes. Other possibilities exist, suchas McDowell’s (2005) proposal to add expo-nents. Adding an exponent as a free parameterwould help fit the data in Figure 1, butMcDowell’s equation still assumes constant rO.Since the sum BþBO equals k, if B increases, BOmust decrease (and vice versa). If rO representsreinforcers produced by BO, then rO must varywith BO. Adding an exponent fails to solve thistheoretical problem with Equation 3. We chosethe present approach because adding freeparameters with no specific meaning seemsless useful than pursuing consistent theory.

Possibly rO varies with r; possibly rO varies withBO, the behavior associated with it (Staddon,1977; e.g., magazine approach, non-key peck-ing, grooming, stretching, etc.). If so, thequestion arises, as posed by Davison (1993),“What is the feedback function for rO?”

To answer that question, we calculated rO andBO for each specific pair of response rate andfood rate. Since k represents the totality ofbehavior measured as peck rate, we will assumeit to be constant, so we may take BO to equal k-B,all the behavior apart from pecking at theresponse keys. Then rO may be calculated by

rearranging Equation 3, as follows:

rO ¼ r ðk−BÞB

: ð5Þ

The left-hand graph of Figure 2 shows rOcalculated this way and plotted against r, thetotal food rate. Up to about 15 fpm, the pointsfrom the five conditions could all be describedby a single increasing function. Above 15 fpm, rOfalls precipitously. A gap appears between 9 and16 fpm. If we take only the estimates of rO above15 fpm and plot them against BO (estimated as k-B), we get the result shown in the right-handgraph in Figure 2. At these high food rates, rOwas approximately directly proportional to BO,as in a ratio schedule (Baum, 1973, 1981, 2012a;Davison, 1993). The slope of a regression linefitted to the points equals 0.16 [about equivalentto variable-ratio (VR) 6]. Equation 5 implies thatacross these high food rates, the ratio r/Bremains approximately constant—r and B in-crease together in proportion.

The lines in the left graph of Figure 2 show theresults of a model that accommodates theresults. The model was constructed accordingto the principles laid out by Baum (2012a):allocation, induction, and contingency. The contin-gency between food and key pecking makes keypecking a food-related activity, with the resultthat food induces key pecking along with other

Fig. 2. Calculated rO from Equation 5 across the five conditions of Baum and Davison’s (2014) study. Left: The brokencurve shows the fit of the model based on Baum’s (2012a) induction theory up to the food rate at which BþBO equals kand BN and rN equal zero. The solid line shows the model’s prediction of decline in rO at still higher food rates.Right: Calculated rO plotted against BO calculated as k-B including only the points in the left graph above 15 fpm. The brokenline shows direct proportionality with a slope of 0.16.

216 WILLIAM M. BAUM and MICHAEL DAVISON

food-related activities. The model required fiveassumptions:

1. The total behavior was allocated amongthree activities, key pecking (B), other food-induced activity (e.g., posturing in front ofthe key or pecking elsewhere; BO), and non-food related activity (e.g., running, groom-ing, or resting; BN). The total equals k inunits of responses per unit of time. Forthese data, we set k equal to 172 pecks/min(ppm), at the top of the range of estimatesshown in Figure 1 and also the estimate of kwhen the hyperbola in Equation 4 was fittedto all the response rates on the right key.

2. These activities compete unequally in ahierarchy. As food-related activities BO andkey pecking B increase, they both replacenon-food-related activities BN, but when Band BO sum to the maximum k, thenincreased key pecking B replaces otherfood-related activities BO. Thus, key peckingis at the top of the hierarchy, then otherfood-induced activities, and then non-food-related activities.

3. Food-induced activity BO increases accord-ing to a power function of food per minute(r), as suggested by Staddon (1977):

BO ¼ c � r s ; ð6Þ

where c and s are free-parameter constants.For fitting the points in Figure 2, c wasestimated at 12 and s at 0.7. The function

was chosen on the assumption that rOcovaries with BO, but its importance lies inits statement that BO depends on r.

4. rO (fpm) is directly proportional to food-induced activity BO, as in a ratio schedule.We set the constant of proportionality to0.16.

5. When BN is greater than zero, k-B equalsBOþBN, and BN stands in an interval-likerelation to its value rN, on the assumptionthat the non-food-related activities main-tain the organism and are limited in theirvalue:

rN ¼ 11BN

þ 1V N

: ð7Þ

The constant VN was allowed to be a freeparameter. This assumption implies that rO inEquations 3 and 4 actually equals rOþ rN. For thefit in Figure 2, VN equaled 0.3.Figure 3 shows a graphical representation of

the model. As r increases, at first both BO and B(diamonds and squares) increase, and rOincreases with BO. BN (triangles) decreasesas B and BO replace it. The sum BOþBN(plain line), which equals k-B, decreases slightly.As r increases further, however, a point isreached at which BO and B have replaced allthe non-food-related activity BN, BN and rN bothequal zero, and the sum BþBO equals k. InFigure 2, this occurs at about 15–20 fpm; it isshown by the broken vertical line in Figure 3. Asr increases further, B (key pecking) increasesmore rapidly and replaces BO (other food-related activity), with the result that BO and rOdecrease, as shown in both graphs of Figure 2.Since BO and B sum to k when r exceeds 15 fpm,BO equals k-B in that range, as shown in the right-hand graph of Figure 2.To fit the decrease in rO shown in the

left-hand graph of Figure 2, and to avoidintractable algebraic expressions, we substituted0.16�(k-B) for rO in Equation 3 and solvedthe resulting quadratic equation, of whichonly one root described B less than k. As Bincreased and replaced BO, rO and BO declinedrapidly.The idea that BO replaces BN was suggested

also by Staddon (1977) when he argued thatrats’ wheel running is not an induced activitybecause drinking and food anticipation (activi-ties directed at the food hopper) replacerunning as food rate increases.

Fig. 3. Graphical representation of the induction-basedmodel applied to Baum and Davison’s (2014) results. Keypecking (B) and other food-induced activities (BO) increaseup to 16 fpm, driving out non-food-related activities (BN).When BN equals zero, BþBO equals k, and further increasesin food rate induce more key pecking, which displaces BO.

BACKGROUND ACTIVITIES AND INDUCTION 217

In the light of thismodel, Equation 5 would bean approximation at the relatively lower foodrates, because k-B equals BOþBN, and rOincludes a term rN corresponding to BN. LikelyrN is small even at low food rates and onlygets smaller as BN decreases; if so, rN mightsometimes be considered negligible, althoughthe shape of the increase in rOwould be as shownby the squares in Figure 3. In the calculation of Bfrom Equation 3, we avoided intractable com-putation by treating rN as negligible, but in thecalculation of rN from BN (i.e., k-B-BO), we usedEquation 7 with VN equal to 0.3. If rN is notnegligible, then Equation 5 would be affected asBN diminished if BO were taken to equal k-B.

One implication of the model is that rO isprimarily due to induction of BO by the food.Since no adjustment of parameters acrossconditions was required, food rate is the onlyoperative variable. Food induces both keypecking and other food-related activities. Theother food-related activity produces rO accord-ing to a ratio-like relation, but as peckingincreases with r, it eventually competes withthe other food-related activity and replaces it,causing rO to decrease.

The changes in rO account for the deviationsof high response rates from the hyperbolashown in Figure 1, but they do not account forthe deviations of the low response rates, whichmust involve not only BO but also BN. Thedeviations of the low response rates indicate thatresponse rate failed to decrease as r decreased,and they may be peculiar to this sort of rapid-change procedure, in which a food ratio ispresented for such a brief time that low r wouldoften result in all food deliveries coming fromthe same key (the constant VI here). In otherwords, these brief-exposure procedures mayentail a discriminability threshold that foilsdiscrimination among the lowest food rates.

The model offers a simple answer to thequestion about the feedback function for rO: It isa ratio-like direct proportionality. The storydoesn’t end there, however, because rO turns outto depend on r as a result the food’s inducingkey pecking and other food-related activities.

Pecking on One Alternative

To consider pecking only on the right(varied) key, we fitted the hyperbola shown inEquation 4, taking r1 and B1 to represent fpmand ppm on the right. Although the VI on the

left was fixed in each condition, fpm on the leftvaried inversely with fpm on the right. For eachcomponent, we inserted the measured fpm onthe left into Equation 4. Figure 4 shows theresult. As with Figure 1, the fitting was done inarithmetic coordinates, but the graphs appear inlogarithmic coordinates. The fits appearedsatisfactory in arithmetic coordinates (VAC orr2> .97), but in logarithmic coordinates one cansee systematic deviations of the response ratesfrom the hyperbola. The pattern differs fromthat in Figure 1: Response rates leveled off andlay above the curve at the lower end, as inFigure 1, but generally lay close to the curveabove about 0.2 fpm.

We made the same set of five assumptions asabove. We calculated rO by first using Equation 6to estimate BO, with r equal to total fpm, andallowing c and s to be free parameters. Then rOwas set to 0.16 times BO. We calculated BN bysubtracting left and right peck rates and BO fromk, set equal to 172 as before, and used Equation 7with VN, a free parameter. Equation 8 was usedto derive a predicted peck rate on the rightkey (B1):

B1 ¼ kr 1

r 1 þ r 2 þ rO þ r N: ð8Þ

The graph in Figure 5 shows these predictionsplotted against fpm on the right, r1 in Equation 8.We treated the condition with VI 10 on the leftseparately from the other conditions, becausethe response rates were noticeably lower, andwe treated the other conditions together,because the rates overlapped. The solid lineshows the predictions for VI 10, and the brokenblack line shows the predictions for the rest. Thevertical broken line marks fpm equal to 0.2,below which response rate apparently ceased todecrease. The model was fitted to food ratesabove 0.2 fpm using Excel’s Solver. The fits weregood (r2 equaled .996 and .994). The constantVNin Equation 7 for rN was estimated at 2.5, c and sin Equation 6 were estimated at 9 and 0.162 forboth VI 10 and the other left VI schedules, but kwas set to 135 for VI 10 and 172 for the other leftVI schedules. These different estimates of VN, c,and s compared with those in Figure 2 probablystem from the inclusion of rN in Equation 8,because, although rN decreased as response rateincreased, it was not negligible and thus loweredthe estimates of BO required from Equation 6.The lower estimate of k for VI 10 probably

218 WILLIAM M. BAUM and MICHAEL DAVISON

reflects a change in topography of pecking at thehighest food rates—for example, pausing brieflyfollowing pecks to view the food hopper.

Belke and Heyman (1994)

Can we find any evidence of induction anddependence of rO on BO in the data published byBelke and Heyman (1994)? Although theystudied only one choice condition, they con-ducted two series of single-VI conditions withthe same set of varying schedules as in the choicecondition. We analyzed stable data from thesingle-VI series and the choice condition.

Figure 6 shows the results. The top left graphshows peck rate on the seven single VIschedules, and the top right graph shows totalpeck rate and peck rate on the constantalternative in the choice condition (right graph)plotted against food rate. In the top left graph,the hyperbola fitted in arithmetic coordinates isshown in these logarithmic coordinates by thesolid line; it remained a good fit. In the top rightgraph, the diamonds show total peck rateplotted against total fpm, and the solid lineshows the hyperbola of Equation 3, fitted inarithmetic coordinates and displayed in theselogarithmic coordinates. The squares show peck

Fig. 4. Peck rate on the varied right key versus food rate on the right key across the five conditions of Baum and Davison’s(2014) experiment. The broken curves show the fits of Equation 4. The fits were done in arithmetic coordinates, but aredisplayed here in logarithmic coordinates.

BACKGROUND ACTIVITIES AND INDUCTION 219

rate as a function of fpm on the constant VIschedule; peck rate varied enough to producevariability in the food rate. The broken curveshows a VI feedback function fitted to the points;the equation was:

r ¼ 1t þ a

B

ð9Þ

where t is the average interval, the VI, and a is afree parameter that aids fitting at low responserates (Baum, 1992; Prelec & Herrnstein, 1978).Although the constant schedule was nominallyVI 27 s, setting t to 27 s failed to fit the actual foodrates. The curve shown, fitted by Solver in Excelwith a least-squares criterion, sets the estimateof t at 17.5 s and a at 9.5. Thus, the programresulted in higher food rates for the peck ratesthan was intended. The parameter a appliesparticularly at low food rates, and its exceeding1.0 implies that individual food deliveriesproduced bursts of pecking (Baum, 1992). Athigh food rates, when t approaches zero, amustapproach 1.0 if responding is random in time, asassumed in Equation 7 (Nevin & Baum, 1980).

The middle two panels in Figure 6 showcalculated rO plotted against food rate. In the leftgraph, Belke and Heyman’s results with singleVI schedules, k was estimated according to

Equation 3, and rO was calculated according toEquation 5. Calculated rO increased with foodrate, except for the three low filled points. Thosethree points correspond to the three highestpeck rates. In the lower left panel, the samecalculated rO is plotted against BO (estimated ask-B), and the three points corresponding to thefilled points in themiddle panel are again shownas filled points. They stand away from the trendin the other points and represent low levels ofBO, which could indicate a hint of the depen-dence of rO on BO shown in Figure 2 (right). Ifthe food was inducing food-related activity otherthan key pecking, and key pecking replacedthose activities only when peck rate was highenough, then the three outstanding pointsmight represent that replacement.

The middle right graph in Figure 6 showscalculated rO plotted against total food rate onthe left and right. We estimated k according toEquation 3, and calculated rO according toEquation 5 with B equal to the total peck rate.The lower right graph shows calculated rOplotted against BO estimated as k-B. No trendin rO can be seen.

Soto et al. (2005): Rats’ Total Pressingon Two Levers

In a related experiment with rats, Soto et al.(2005) studied seven different conditions,each with a constant VI schedule or extinction(one condition) on the right lever, rangingfrom VI 10 s to VI 350 s, and the 10 varied VIschedules, ranging from VI 6 s to VI 350 s, thesame 10 in each condition, on the left lever.The 10 pairs of schedules were presented intwo sessions per day, five in each session, andeach pair for 8 minutes, separated by black-outs. Three conditions were replicated; sincethe replications were close to the originalconditions, we averaged each replication withthe corresponding original condition. Theprocedure differed from those of Belke andHeyman (1994) and Baum and Davison (2014)in that the different components were signaledby discriminative stimuli. Most conditionslasted for 20 pairs of sessions, but a few foronly 10. Data from the last 6 days of eachcondition were analyzed. Paul Soto kindly sentthe raw data to us.

We averaged the data across rats and analyzedthose means. Figure 7 shows mean responserate on the two levers combined across all

Fig. 5. The inductionmodel (Eq. 8) fitted to peck rate onthe varied right key across the five conditions of Baum andDavison’s (2014) experiment. The vertical broken linemarks off the low food rates that were hardly discriminatedbecause of the brevity of components in the frequently-changing-ratios procedure. The solid black line shows theresults for schedules paired with the richest schedule, VI 10,on the left key. The broken black line shows the results for allthe other conditions treated together.

220 WILLIAM M. BAUM and MICHAEL DAVISON

components and conditions (70 points) plottedagainst combined food rate, as in Figure 1.Conditions with extinction on the constant rightlever are marked with X. The trend is for pressrate to increase with combined food rate up toabout 3.8 fpm and then to decrease. The lowestcluster of 10 points at the right were all fromcomponents with VI 6 s on the left (varied) lever.Thus, at the highest food rates, press ratedecreased.We fitted thehyperbola of Equation 3to the press rates for food rates below 2 fpm,keeping k and rO constant. The solid curve inFigure 7 shows the best-fitting hyperbola, with k

equal to 80 ppm and rO equal to 0.45 fpm. Thisfit gave us an estimate of k to work with.Next, we calculated rO for all components

according to Equation 5. Figure 8 shows thesecalculated estimates of rO plotted against foodrate, as in the left graph in Figure 2. The pointsfall into four groups: Diamonds denote calcu-lated rO for food rates less than 2 fpm; Xs denoterO for food rates between 2 and 3.8 fpm; squaresdenote rO for food rates between 3.8 and 4.5fpm; and triangles denote rO for food ratesgreater than 4.5 fpm generated by componentswith VI 6 on the varied lever. The diamonds

Fig. 6. Results of Belke and Heyman (1994) reanalyzed. The left column shows results for seven single VI schedules. Theright column shows results for the same seven VI schedules presented with a constant concurrent VI (nominally 27 s, butactually VI 17.5 s). Top: Peck rate versus food rate. The solid black lines show the fits of the hyperbola in Equations 3 and 4.The broken curve is a VI feedback function fitted to the performances on the constant schedule. Middle: Calculated rO fromEquation 5 versus food rate. The filled symbols mark points that appear to diverge from the rest as food rate increased.Bottom: Calculated rO versus BO estimated as k-B.

BACKGROUND ACTIVITIES AND INDUCTION 221

show less change in calculated rO than shown onthe left in Figure 2. Instead of just one verticallyvarying cluster, however, three clusters can beseen: the Xs, the squares, and the triangles.

Figure 9 shows calculated rO as a function ofBO (estimated as k-B, with k equal to 80) for thethree vertically varying clusters, in which,according to ourmodel, BN has been completelyreplaced by lever pressing (B) and other food-induced activity (BO), and in which BO equalsk-B. The Xs show rO directly proportional toBO, as in the right-hand graph in Figure 2.The constant of proportionality equals 0.0448

(equivalent to VR 22), less than the 0.16 ofFigure 2; the difference might be attributed toany of several differences of procedure, such asrats versus pigeons, food pellets versus grain,unsignaled components versus signaled compo-nents, key pecking versus lever pressing, etc.

The squares and triangles in Figure 9 standapart from the Xs, as they do in Figure 8. Thelines drawn through the squares and trianglesindicate, however, that these points also showdirect proportionality between rO and BO. Forthe squares, relatively high food rates between3.8 and 4.5 fpm, the constant of proportionali-ty equals 0.0728 (VR 14), steeper than for foodrates between 2 and 4 fpm. For food ratesabove 4.5 fpm, generated by VI 6 s, directproportionality appears again, but with aconstant of proportionality equal to 0.1241(VR 8), steeper still.

Wemay speculate that the rats’ topography oflever pressing changed as food rate becameextremely high. Beyond rates of 3.8 fpm, forexample, the rats may have spent more timenosing in the food hopper. With this change,the activities induced by the food that com-prised BO differed too, to the point where theyinterfered with lever pressing at the highestfood rates and caused the decrease shown inFigure 7. In all three situations, however, directproportionality—a ratio-like relation—may beseen between rO and BO, supporting theconclusion that the feedback function relatingrO to BO is a ratio-like relation. The model thatapplied to key pecking in pigeons also applies tolever pressing in rats.

Fig. 7. Rates of pressing two concurrent levers combinedfrom the experiment by Soto et al. (2005). The Xs showresponding when the constant schedule on the right leverwas extinction. The line shows the fit of Equation 3 to theconditions with food rate below 2 fpm.

Fig. 8. Calculated rO from Equation 5 versus food ratefrom the experiment by Soto et al. (2005). Different symbolsindicate different ranges of food rate. The broken line showsthe fit of the induction model, assuming calculated rO toequal actual rOþ rN.

Fig. 9. Calculated rO versus BO estimated as k-B from theexperiment by Soto et al. (2005). Different symbols indicateestimates from three different ranges of food rate. The lineswere fitted to the estimates from the different food-rateranges separately. Note logarithmic coordinates.

222 WILLIAM M. BAUM and MICHAEL DAVISON

Figure 8 makes clear, however, a differencebetween rats’ lever pressing and pigeons’ keypecking. It shows what may be called a rolereversal. Instead of decreasing at high food rates,as with key pecking (Fig. 2), rO and BO increasedwhen the operant activity was lever pressing.This increase in BO at the expense of leverpressing appears in Figure 7 as decreased pressrate at high food rates. Instead of lever pressingreplacing other food-induced behavior BO,other food-induced activities replaced leverpressing. Thus, lever pressing affords an exam-ple of what Breland and Breland (1961) called“misbehavior of organisms.” In contrast, keypecking appears to correspond to the kind ofactivity that is facilitated by induction, as in theBrelands’ example of the “dancing chicken.”Food induces pecking in pigeons even withno contingency between pecking and food(Staddon, 1977). Most likely, food with nocontingency has no such effect on leverpressing, and the difference may explain wheninterference occurs and when facilitationoccurs.The broken line in Figure 8 shows the result of

applying the same model as before, includingEquations 6 and 7, but with three changes: (a)the three different constants of proportionalityshown in Figure 9 were used instead of just one;(b) BO was assumed to replace B once BN wasreduced to zero and the sum of BO and Bequaled k (equal to 80); and (c) to avoidintractable computation in calculating B fromEquation 3, rN was represented as a smallconstant (0.2) added to the denominator, butotherwise rN was calculated from Equation 7 asbefore. A graphical depiction like that inFigure 3 would look much the same, exceptthat BOwould increase to the right of the verticalline and B would decrease, and BO would notincrease all the way to equal k, because B couldonly decrease until B equaled r (Fixed Ratio 1).The estimates of c and s in Equation 6 differedfrom those for the pigeons’ key pecking: c was0.1, and s was 3.5, so c was much smaller, and swas much larger.

Soto et al. (2005):Rats’ Pressing on One Lever

As with the pigeons’ key pecking, we analyzedrats’ pressing on the varied lever according tothe (modified)model and Equation 8. Figure 10shows the results. The graphs show the rates of

pressing on the varied lever with different VIschedules on the constant lever. The bottomright panel shows pressing on the varied leverwith either VI 10 or VI 17 on the constant lever.The broken lines show the fits of Equation 4,with rO assumed constant. The solid lines showthe results of fitting the present model, with allparameters carried over from the model used tofit calculated rO in Figure 8, except that k wasallowed to vary across the different constantschedules to help assess the effects of lack ofdiscrimination among components, particularlyfor VI 10 and VI 17.The lackof discriminationappears in Figure 10

as the same tendency for responding to level offat low food rates as appears in Figures 4 and 5.Apparently the putative discriminative stimulifailed to distinguish among the low food rates onthe varied lever. The flattening out occurred athigher and higher food rates, the richer was theschedule on the constant lever. For the singleVIs,it was minimal, for VI 350 and VI 150, the lowesttwo food rates were indistinguishable, for VI 75and VI 50, the lowest three food rates wereindistinguishable, for VI 17, the lowest four foodrates were indistinguishable, and for VI 10, thelowest six food rates were indistinguishable. Mostlikely, the richer the constant schedule, themoreoften all food during the 8-min exposure camejust from that schedule, thereby preventingdiscrimination among the lean schedules onthe varied lever. In fitting Equation 4 (brokenlines) and the model (solid lines), the indistin-guishable food rates were omitted. For most ofthe different constant schedules, the modelcaptured the decrease in pressing at high rates(cf. Fig. 7). It was less successful with VI 17 andleast successful with VI 10, presumably because ofthe lack of discrimination among the differentschedules on the varied lever. The estimates ofk from the model decreased with press rates:unsystematic variation from 75 to 90 (mean equalto 82) for the five leanest, to 70 for VI 17, and65 for VI 10.Figure 11 shows variation of food and press

rates on the constant lever. Since the press rateson the varied and constant levers were inverselyrelated, one would expect that as press rate onthe constant lever varied, food rate might alsovary, so that food rates on the two levers wereinversely related [see Soto et al. (2005); Figs. 1and 2]. The curves fitted to the points for thevarious constant schedules represent Equation 9,the feedback function (RFF) for a VI schedule

BACKGROUND ACTIVITIES AND INDUCTION 223

(Baum, 1992). Estimates of t varied with the VI,and estimates of a varied a bit with the VI, beinglargest (25) for VI 150 and smallest (13) for VI10. Variation in a might reflect topographicalchanges in activities around the lever.

Discussion

Two results stand out. First, the resultssupport Baum’s (1981) and Davison’s (1993)conjecture that the feedback relation betweenBO and rO is a ratio-like relation. Second, these

results show that BO represents food-relatedactivities other than the operant activity that areinduced by the food along with the operantactivity, as proposed by Baum (2012a). Thesuccess of the hierarchical model of inducedactivities, illustrated in Figures 2 and 8 for overallpecking and pressing, and in Figures 5 and 10for pecking and pressing on the varied VI,supports Baum’s (2012a) proposal. Though rOwas originally thought of as independent of foodrate r in Equations 3 and 4, rO turns out todepend on r.

Fig. 10. Rates of pressing the varied left lever in the experiment by Soto et al. (2005). Each graph shows results from onecondition with the schedule on the right held constant, except for the bottom right graph which shows results from the twoconditions with the constant schedule VI 10 and VI 17. Broken lines show the fits of the hyperbola of Equation 4. Solid linesshow the fits of the induction model (Eq. 8).

224 WILLIAM M. BAUM and MICHAEL DAVISON

Herrnstein’s (1961, 1970) original formula-tion of the matching law, which Belke andHeyman (1994) relied on, implicitly assumedthat alternatives were independent sources ofreinforcers; each r depended only on its B andnot on any other activity. The unlooked-forresult, not included in Herrnstein’s conceptionof matching, is that rO depends on r inEquations 3 and 4. The reason rO depends onr is that the food induces both operant activityand other food-related activities BO that dependon r—are induced at different rates by differentfood rates produced by the operant activity—and, as Figures 2 and 9 show, rO in Equation 8 isdirectly proportional to BO. Equations 3 and 4,to be accurate, ought to include a term rN in thedenominator, as in Equation 8. The term rN,however, conforms to the original conception ofrO, because rN arises from activities (BN; e.g.,grooming and stretching) that occur indepen-dently of food.Themodel for right-key pecking incorporates

an effect of food from the left key inducingactivities that affect pecking on the right key(Eq. 7), because food induces the same food-related activities (BO) regardless of the foodsource. Thus, the rO term that appears inEquation 8 is equal to rO in Equations 3 and 4minus rN; the total of induced activities’ value rOappears in the denominator regardless ofwhether we are considering total operantresponding (Eq. 3) or responding only on thevaried VI (Eqs. 4 and 8). In contrast withconcurrent schedules, successive componentsinmultiple schedulesmight differ in both rO andrN. Some researchers have proposed to explain

behavioral contrast by supposing that BO mightshift from a richer component to a leanercomponent, decreasing operant activity inthe leaner component and increasing operantactivity in the richer component (McLean,1992, 1995; McLean & White, 1983; Staddon,1982). According to the present account, likelyBN would be fungible this way, but BO mightbe less so, because BO might tend to remain inthe component with the food that induces it.The fungibility of BNmight explain why essentialbodily maintenance activities like grooming andstretching can decrease to zero, as shown inFigure 3, even in concurrent schedules; theactivities are simply postponed until after thesession.A striking difference appeared between

pigeons’ key pecking and rats’ lever pressing:Key pecking replaced other food inducedactivities BO, whereas those activities replacedlever pressing—a role reversal. The differencecorresponds to the difference in inducedactivities that Breland and Breland (1961)described. When induced activities interferedwith operant activities, as they did for leverpressing, the Brelands called the interference“misbehavior,” but they also noted that inducedactivities could facilitate operant training, asoccurred for key pecking.Even though key pecking replaced other

food-induced activity in the present results,that replacement should not be confused withthe reduction in key pecking that results fromaddition of a second concurrent source of food,as represented by the difference betweenEquations 3 and 4. Equation 3 was meant toapply to a single source of food, and we appliedit here to the combined food from twoalternatives as if they were one source (cf.Baum, 2002, Figure 3), whereas Equation 4 wasmeant to apply to one alternative out of two. Theinclusion of a nonzero term for r2 decreases keypecking B1 below what would occur if B1 werealone. Reduction in BO and reduction in B1 bothresult from competition between activities forthe time available, but the reduction in BO is theresult of dominance of key pecking, whereas thereduction in B1 is the result of adding a secondfood source. That second food source need notdepend on key pecking, as Rachlin and Baum(1972) found. In their experiment, a constant VIschedule was maintained on a response keywhile additional food was presented, varying inrate and amount. When they added response-

Fig. 11. VI feedback functions (Eq. 9) fitted to the pressrates on different constant schedules in the experiment bySoto et al. (2005).

BACKGROUND ACTIVITIES AND INDUCTION 225

independent food, it affected pecking on theconstant VI just like an independent secondsource of food. Consistent with the idea thatmore food usually induces more food-relatedactivities, Rachlin and Baum reported that,when they applied a version of Equation 4 thatincorporated the product of amount times rate(v instead of r), the estimate of vO analogous torO was much higher with the presence of theadded food.

Baum (2005, 2012a) argued that phylogenet-ically important events (PIEs) such as foodinduce activities related to them both byphylogeny and ontogeny. Food delivered inde-pendently of behavior induces several differentactivities in hungry pigeons, including movingabout, wing flapping, and pecking at screws,floor, or in the air (Staddon, 1977). Whenmadecontingent on key pecking, food comes quicklyto induce key pecking too, because the contin-gency makes key pecking a food-related activityalso. Similarly, food induces various activities inrats, such as drinking and nosing in the foodhopper (Staddon, 1977). When food is madecontingent on lever pressing, lever pressing, likekey pecking in pigeons, becomes a food-inducedactivity as a result of its contingency with food.Thus, Baum (2012a) argued, reinforcementmay be recast as induction—operant activitiesare induced by the PIEs to which they are relatedby contingencies.

That rO depends on r doesn’t invalidateEquation 3 or Equation 4. Rather, the depen-dency makes the equations more plausible,because it brings our understanding of rO intoline with our understanding of other PIE-relatedphenomena. That every schedule is character-ized by a feedback function allows us to write amatching equation

BO

B¼ r O

r; ð10Þ

confident that rO depends on BO according to afeedback function—a ratio-like function—justas r depends on B according to a feedbackfunction—a VI function in all the studies wehave considered in the present paper, andwhich appears in Figures 6 and 11 (Eq. 9;Baum, 1992).

If the relation between rO and BO is a ratiorelation, aswehave concluded, thenEquation 10appears to be incorrect. For example, it impliesthat B/r ought tomatch BO/rO. If the ratio BO/rO

is constant, it should fix the point whereresponse rate should fall on a VI feedbackfunction (Baum, 1973, 1992). If BO/rO isconstant, then B/r should be constant, whichwould mean that B is directly proportional to r,contrary to Figures 1, 6, and 7 and all previousresearch. Looked at the other way around, wewould have to conclude that the ratio BO/rOshould change across VI schedules, because theratio B/r changes, but Figure 2 appears to showthat, at least for pigeons’ key pecking, BO/rO isconstant. Possibly the third category BN plays arole yet to be determined at low or moderatefood rates, when k-B equals BOþBN. Possibly thecorrect equation is the one corresponding toEquation 5:

BO þ BN

B¼ rO þ rN

r: ð11Þ

Equation 11 allows the possibility that BN mightchange as the VI schedule changes and mightpreserve the relation that B/r matches (BOþBN)/(rOþ rN) even though BO/rO is constant.

A key to solving this conundrum, at least forpigeons’ key pecking, may lie in Baum’s (1993)study of variable-ratio (VR) and VI schedules.We speculated that topography of lever pressingchanges as food rate gets high, but key peckingat high food rates may change also. Baum(1993) studied multiple schedules in which onecomponent was a VR schedule and the secondcomponent was a VI schedule programmed byplaying back the intervals generated in the VR tomatch the food rate and distribution of intervalsacross the two schedules. VR and VI compo-nents alternated, with blackouts separating thecomponents. Varying the VR over a wide range,all the way to requiring only a single peck,resulted in a wide range of food rates, frommoderate to extremely high. The top panel inFigure 12 shows the results averaged acrosspigeons. As food rate increased, peck rate on theVI increased, started to level off over a middlerange, greatly increased again for higher foodrates and then decreased at the highest foodrates. The high peck rates could not beaccounted for by decreases in rO, because rOwas too small, perhaps because BO and BN werereallocated to the blackouts between compo-nents (McLean, 1992, 1995; McLean & White,1983; Staddon, 1982). The lower panel showsthat the decrease at the highest food rates isaccommodated if one assumes thatB,BO, andBN

226 WILLIAM M. BAUM and MICHAEL DAVISON

sum to a constant k equal to 180 ppm andthat Equation 6 governs BO, with a power s of1.679 and a coefficient c of 0.07. Peck rates forlesser food rates are predicted perfectly, butthat is to be expected, because BN was calculatedas k-B-BO.Baum (1993) speculated that pigeons’ key

pecking contains amix of two topographies: onein which the beak meets the key and then pullsaway, operating the key’s switch once, and onein which the beak swipes the key so as to operatethe key’s switch multiple times. Palya (1992)made the same suggestion, calling the multiple-operation topography a “flick.” As the food rategot high in Baum’s experiment, key swipingmayhave replaced key pecking, resulting in increas-ing measured peck rates (switch operatingrates). He reasoned that the change in units

of pecking changed the correspondence be-tween pecking and time; the higher peck ratemight affect the amount of time taken up,because the time per switch operation wassmaller. The change of unit would be reflectedin a change in k in Equation 3. The presentaccount makes no assumption that k changes,but instead assumes that BN decreased to zero atthe highest peck rate and that BO increasedaccording to the power function. Thus, thedecrease in calculated rO shown in Figure 2might actually be a result of a change intopography and k at the highest food rates,and possibly actual rO did not decrease. Thepossibility of changes in k may need to beincorporated into future research.The leftmost four points for the VI compo-

nent in Figure 12 could be described by ahyperbola like Equation 3, suggesting anapproximately constant value for backgroundreinforcers rOþ rN. Possibly, decreasing rNtended to offset increasing rO at low andmoderate food rates, until BN decreased tozero and B (composed of key pecking and keyswiping) began replacing BO as r increasedpast that point. All these speculations may beaddressed by future research.The absence of pecking below about 0.3 fpm

indicates ratio strain; if pecking ceased in the VRcomponent, the yoking of the componentscaused food rate to drop to zero in the VIcomponent as well as the VR component. If therelation between rO and BO is a ratio relation,ratio strain might be understood as the outcomeof competition between pecking and BO. A VRschedule and the rO schedule would constitute apair of concurrent ratio schedules. Performanceon concurrent ratio schedules moves towardexclusive preference for the richer schedule.The food produced by pecking on a VRschedule, like the food produced by peckingon a VI schedule, induces other food-relatedactivities BO. Whenever the relation between rOand BO is richer than the relation between r andB on the VR schedule, pecking on the VRschedule will cease. That cessation could occurat moderate food rates, if rO/BO were moderate,and it would explain ratio strain. As long as theVR schedule (r/B) was richer than the rO/BOschedule, pecking on the VR would be main-tained and would tend to occur at the highestrate possible (exclusive preference). Thatdoesn’t explain why peck rate increased asfood rate increased, unless ratio strain is a

Fig. 12. Peck rates and food rates averaged across fourpigeons in the experiment by Baum (1993). Top: Rates fromthe VR components and the yoked VI components. Bottom:Rates from the VI components predicted by assuming power-function induction of BO.

BACKGROUND ACTIVITIES AND INDUCTION 227

variable effect and modulates the peck ratebelow the maximum. The decrease at thehighest food rates, however, might be explainedin the same way as the decrease for the VI peckrates—the effect of food-induced activities BO.Future research may address these questionsalso.

The picture for rats’ lever pressing may bemore complicated. Possibly the proportionalitybetween rO and BO might change with food rate,as Figure 9 suggests. Since BO probably repre-sents multiple activities, change in proportion-ality would reflect change in the activitiesconstituting BO. As food rate increased for therats, new activities apparently appeared in BO. Asfood rate increased for the pigeons, someactivities in BO apparently dropped out. Furtherresearch may clarify these issues.

Explaining decreased response rate at highfood rates as competition with induced adjunc-tive activities is not new. For example, Rachlin(1978; Rachlin & Burkhard, 1978) and Staddon(1983) developed optimality models that ex-plained the downturn at high food rates ascompetition. Our approach doesn’t assume thatindividual organisms optimize, but instead seeksmechanisms that might be optimal in thenarrower sense that they were selected bynatural selection. In this respect only, ourapproach resembles Killeen’s (1994), in whichhe developed an equation (his Eq. 12) thatcombined a hyperbola, in which a constant cstands in place of rO, with a subtractive term thatalso would predict a downturn at high foodrates. Rachlin’s, Staddon’s, and Killeen’s expla-nations all entail far more assumptions andparameters than the model we propose here.In particular, because Killeen’s “mathematicalprinciples of reinforcement,” rely on a molecu-lar view of behavior that takes discrete responsesand response-reinforcer contiguity as primary,his approach requires many assumptions andparameters, and appeals to hypothetical con-structs such as memory (in place of extendedactivities) and coupling (in place of a feedbackfunction). The present approach based on amolar view of induction may offer a simpleraccount of the same phenomena.

Equation 6 gives a function for the depen-dence of BO on r—that is, for induction of food-related activities (Baum, 2012a). In the presentcontext, Equation 6 served our purposes, but itmay be inaccurate. Induced activity sometimesfollows a bitonic function of food rate, increas-

ing with r over a range, and then decreasing atrelatively high food rates (Staddon, 1977). Thedecrease at high food rates may be related to thedecrease in rO at high food rates seen in Figure 2.Possibly, with no operant activity present, newconstituent activities of BO are induced at highfood rates that are not measured by the usualmethods, which rely on specific manipulanda,like water spouts (polydipsia) or mirrors (in-duced aggression). For example, if a rat orpigeon began keeping its head in the foodhopper, that might not register as the measuredactivity.

The highest food rates that Soto et al. (2005)studied apparently induced new constituentactivities. Figure 9 suggests that new constituentsarose either in BO or in lever pressing at the highfood rates, because the proportionality betweenrO and BO changed. Possibly, as new constituentactivities begin taking up time, the estimates ofBO in units of presses/min begin systematicallyto fall short, because the correspondencebetween BO in presses/min and time changes.If so, a correction might be derived that wouldmove theXs and triangles in Figure 9 sufficientlyto the right to have them fall in line with thesquares representing lesser food rates.

Modeling the induction of key pecking andlever pressing as we have done here has atheoretical benefit, because it preserves thegeneralized matching law (Baum, 1974, 2012b).If we assume that operant activities in generalare induced by food or other consequences, wemay write:

Bi

Bj¼ ci r i si

c j r j sjð12Þ

for all pairs of concurrent operant activities iand j (i 6¼ j). The ratio ci=cj corresponds to bias.Although si and sj are usually assumed to beequal, they need not be equal, for example, ifthe activities differed in topography or withdifferent PIEs as consequences (Baum, 2012b).Possibly, variation in BO and BN leave Equa-tion 12 intact, because they only affect the totaltime available for the operant activities

In the final analysis, all behavior consists oftime allocation (Baum, 2010, 2012a, 2013; Baum& Rachlin, 1969). Attaching a switch to amanipulandum like a key or lever is an easyway to estimate the time taken in interactingwith the manipulandum. For example, ratstypically lick, bite, chew, and paw a lever that

228 WILLIAM M. BAUM and MICHAEL DAVISON

provides food, and these activities jiggle thelever and operate the switch fairly often(Baum, 1976). The method has its limitations,however, particularly when the activities withrespect to the manipulandum change. As longas the activity time per switch operation remainsthe same, the correspondence between switchoperations and time taken remains the same.When the activities with respect to the manip-ulandum change—say, involving the food hop-permore or swiping the key instead of pecking it—then the conversion from switch operations totime may change, and one may be at a loss as tohow to characterize the new conversion rate.More research on variation in the activities thatcompose operant activities may help to clarifythe behavioral changes that correspond tochanges in time taken up.

Conclusion

The present results support Baum’s (2012a)formulation of operant activity based on alloca-tion, induction, and contingency. The novel resultis that rO, so-called “extraneous” reinforcers,depends on the reinforcers produced by theoperant activities. It does so because theactivities on which rO depends, represented byBO, are induced by the reinforcers produced bythe operant activities. The dependence of rO onBO appears to be simple proportionality, similarto a ratio schedule. The dependence of BO onfood rate wasmodeled here by a power function,but another function might prove more accu-rate in future research. The results raisequestions about the matching law, becausethey appear to challenge the assumption thatall terms in the matching equations areindependent. The generalized matching lawmay remain unaffected, as shown in Equation 12,but the present results suggest some complexitymay arise when non-operant-induced activitiesare incorporated in the matching equations.

References

Baum, W. M. (1973). The correlation-based law of effect.Journal of the Experimental Analysis of Behavior, 20,137–153.

Baum, W. M. (1974). On two types of deviation from thematching law: bias and undermatching. Journal of theExperimental Analysis of Behavior, 22, 231–242.

Baum, W. M. (1976). Time-based and count-based measure-ment of preference. Journal of the Experimental Analysis ofBehavior, 26, 27–35.

Baum, W. M. (1981). Optimization and the matching law asaccounts of instrumental behavior. Journal of theExperimental Analysis of Behavior, 36, 387–403.

Baum, W. M. (1992). In search of the feedback function forvariable-interval schedules. Journal of the ExperimentalAnalysis of Behavior, 57, 365–375.

Baum, W. M. (1993). Performances on ratio and intervalschedules of reinforcement: Data and theory. Journal ofthe Experimental Analysis of Behavior, 59, 245–264.

Baum, W. M. (2002). From molecular to molar: A paradigmshift in behavior analysis. Journal of the ExperimentalAnalysis of Behavior, 78, 95–116.

Baum, W. M. (2005). Understanding behaviorism: Behavior,culture, and evolution (2nd Ed.). Malden, MA: BlackwellPublishing.

Baum, W. M. (2010). Dynamics of choice: A tutorial. Journalof the Experimental Analysis of Behavior, 94, 161–174.

Baum, W. M. (2012a). Rethinking reinforcement: Alloca-tion, induction, and contingency. Journal of the Experi-mental Analysis of Behavior, 97, 101–124.

Baum, W. M. (2012b). Mathematics and theory in behavioranalysis: Remarks on Catania (1981) “The flight fromexperimental analysis.” European Journal of BehaviorAnalysis, 13, 177–179.

Baum, W. M. (2013). What counts as behavior: The molarmultiscale view. The Behavior Analyst, 36, 283–293.

Baum, W. M., & Davison, M. (2014). Choice with frequently-changing food rates and food ratios. Journal of theExperimental Analysis of Behavior, 101, 246–274.

Baum, W. M., & Rachlin, H. C. (1969). Choice as timeallocation. Journal of the Experimental Analysis of Behavior,12, 861–874.

Belke, T. W., & Heyman, G. M. (1994). Increasing andsignaling background reinforcement: Effect on theforeground response–reinforcer relation. Journal of theExperimental Analysis of Behavior, 61, 65–81.

Breland, K., & Breland, M. (1961). The misbehavior oforganisms. American Psychologist, 16, 681–684.

Davison, M. (1993). On the dynamics of behavior allocationbetween simultaneously and successively availablereinforcer sources. Behavioural Processes, 29, 49–64.

Davison,M. (2004). Interresponse times and the structure ofchoice. Behavioural Processes, 66, 173–187.

Herrnstein, R. J. (1961). Relative and absolute strength ofresponse as a function of frequency of reinforcement.Journal of the Experimental Analysis of Behavior, 4, 267–272.

Herrnstein, R. J. (1970). On the law of effect. Journal of theExperimental Analysis of Behavior, 13, 243–266.

Killeen, P. R. (1994). Mathematical principles of reinforce-ment. Behavioral and Brain Sciences, 17, 105–171.

McDowell, J. J. (2005).On the classic andmodern theories ofmatching. Journal of the Experimental Analysis of Behavior,84, 111–127.

McLean, A. P. (1992). Contrast and reallocation ofextraneous reinforcers betweenmultiple-schedule com-ponents. Journal of the Experimental Analysis of Behavior,58, 497–511.

McLean, A. P. (1995). Contrast and reallocation ofextraneous reinforcers as a function of componentduration and baseline rate of reinforcement. Journal ofthe Experimental Analysis of Behavior, 63, 203–224.

McLean, A. P., & White, K. G. (1983). Temporal constrainton choice: Sensitivity and bias in multiple schedules.Journal of the Experimental Analysis of Behavior, 39,405–426.

BACKGROUND ACTIVITIES AND INDUCTION 229

Nevin, J. A., & Baum, W. M. (1980). Feedback functions forvariable-interval reinforcement. Journal of the Experimen-tal Analysis of Behavior, 34, 207–217.

Palya, W. L. (1992). Dynamics in the fine structure ofschedule-controlled behavior. Journal of the ExperimentalAnalysis of Behavior, 57, 267–287.

Prelec, D., &Herrnstein, R. J. (1978). Feedback functions forreinforcement: a paradigmatic experiment. AnimalLearning and Behavior, 6, 181–186.

Rachlin, H. (1978). A molar theory of reinforcementschedules. Journal of the Experimental Analysis of Behavior,30, 345–360.

Rachlin, H., & Baum, W. M. (1972). Effects of alternativereinforcement: Does the source matter? Journal of theExperimental Analysis of Behavior, 18, 231–241.

Rachlin, H., & Burkhard, B. (1978). The temporal triangle:Response substitution in instrumental conditioning.Psychological Review, 85, 22–47.

Soto, P. L., McDowell, J. J., & Dallery, J. (2005). Effects ofadding a second reinforcement alternative: Implica-tions for Herrnstein’s interpretation of re. Journal of theExperimental Analysis of Behavior, 84, 185–225.

Staddon, J. E. R. (1977). Schedule-induced behavior. InW. K. Honig & J. E. R. Staddon (Eds.), Handbook ofoperant behavior (pp. 125–152). Englewood Cliffs, NJ:Prentice-Hall.

Staddon, J. E. R. (1982). Behavioral competition, contrastand matching. In M. L. Commons, R. J. Herrnstein, &H. Rachlin (Eds.),Quantitative analyses of behavior, Vol. II:Matching and maximizing accounts (pp. 243–261). Cam-bridge, MA: Ballinger.

Staddon, J. E. R. (1983). Adaptive behavior and learning.Cambridge: Cambridge University Press.

Received: March 6, 2014Final Acceptance: July 2, 2014

230 WILLIAM M. BAUM and MICHAEL DAVISON