JOURNAL OF THE EXPERIMENTAL ANALYSIS OF BEHAVIOR CHOICE AS TIME ALLOCATION' WILLIAM M. BAUM AND HOWARD C. RACHLIN2 HARVARD UNIVERSITY When pigeons' standing on one or the other side of a chamber was reinforced on two con- current variable-interval schedules, the ratio of time spent on the left to time spent on the right was directly proportional to the ratio of reinforcements produced by standing on the left to reinforcements produced by standing on the right. The constant of proportionality was less than unity for all pigeons, indicating a bias toward the right side of the chamber. The biased matching relation obtained here is comparable to the matching relation obtained with concurrent reinforcement of key pecks. The present results, together with related re- search, suggest that the ratio of time spent in two activities equals the ratio of the "values" of the activities. The value of an activity is the product of several parameters, such as rate and amount of reinforcement, contingent on that activity. Psychology has inherited from reflexology the notion that behavior can be viewed as a mosaic of responses. Skinner (1938) thought of the rat's lever press as a type of reflex. Accord- ingly, his basic measure of behavior was a count of the number of lever presses made during an experimental session. He computed the frequency of lever presses by dividing the number of presses by the duration of the ses- sion. In order to make this computation, he had to treat each lever press as an instanta- neous event, as a point in time, having no duration. This assumption implies that two re- sponses could immediately follow one another, with no time intervening. Since each response requires a certain amount of time, however, the minimum interresponse time is greater than zero. When the actual interresponse times approach the minimum interresponse time, the computation of response rate should include a correction for response duration. Since Skinner's work, experiments on oper- ant behavior have usually treated responses as 'This research was supported by grants from the National Science Foundation and the National Insti- tutes of Health to Harvard University. We are grateful to Lincoln Laboratory of the Massachusetts Institute of Technology for making available the Lincoln Reck- oner, on which most of the data analysis was done. Reprints may be obtained from William M. Baum, Departlnent of Psychology, Harvard University, Wil- liam James Hall, 33 Kirkland Street, Cambridge, Mas- sachusetts 02138. 2Now at Department of Psychology, State University of New York, Stony Brook, Long Island. instantaneous. Indeed, response keys and levers, in conjunction with pulse-formers (that produce short pulses of constant duration) are so commonly used today that available appara- tus tends to enforce response counting as the means of measuring behavior. If we admit that behavior has duration, an alternative scheme of measurement becomes available. Behavior that can be counted can also be timed. Response duration, or time spent responding, can be just as basic a mea- sure of behavior as response frequency. In some situations the two measures are equivalent. If a pigeon's pecks at a key, for example, were approximately constant in dur- ation, then the key-peck time would equal that constant duration multiplied by the num- ber of key pecks. If a rat's holding of a lever is reinforced, on the other hand, then lever-hold- ing time might often vary independent of the number of depressions of the lever. It is usual to select the measure of behavior on the basis of the conditions of reinforce- ment. If we reinforce at a certain point of time, say, at the moment when the lever has been depressed 5 mm, then it seems natural to count the number of such momentary occur- rences as could have produced reinforcement. We might, on the other hand, reinforce while the animal is engaged in some activity, at no particular moment, as when we reinforce being in a certain location and continue reinforce- ment as long as the animal stays in that loca- tion. When such continuous action is rein- 861 1969, 12, 861-874 NUMBER 6 (NOVEMBER)
Transcript
JOURNAL OF THE EXPERIMENTAL ANALYSIS OF BEHAVIOR
CHOICE AS TIME ALLOCATION'
WILLIAM M. BAUM AND HOWARD C. RACHLIN2
HARVARD UNIVERSITY
When pigeons' standing on one or the other side of a chamber was reinforced on two con-current variable-interval schedules, the ratio of time spent on the left to time spent on theright was directly proportional to the ratio of reinforcements produced by standing on theleft to reinforcements produced by standing on the right. The constant of proportionalitywas less than unity for all pigeons, indicating a bias toward the right side of the chamber.The biased matching relation obtained here is comparable to the matching relation obtainedwith concurrent reinforcement of key pecks. The present results, together with related re-search, suggest that the ratio of time spent in two activities equals the ratio of the "values"of the activities. The value of an activity is the product of several parameters, such as rate andamount of reinforcement, contingent on that activity.
Psychology has inherited from reflexologythe notion that behavior can be viewed as amosaic of responses. Skinner (1938) thought ofthe rat's lever press as a type of reflex. Accord-ingly, his basic measure of behavior was acount of the number of lever presses madeduring an experimental session. He computedthe frequency of lever presses by dividing thenumber of presses by the duration of the ses-sion. In order to make this computation, hehad to treat each lever press as an instanta-neous event, as a point in time, having noduration. This assumption implies that two re-sponses could immediately follow one another,with no time intervening. Since each responserequires a certain amount of time, however,the minimum interresponse time is greaterthan zero. When the actual interresponsetimes approach the minimum interresponsetime, the computation of response rate shouldinclude a correction for response duration.
Since Skinner's work, experiments on oper-ant behavior have usually treated responses as
'This research was supported by grants from theNational Science Foundation and the National Insti-tutes of Health to Harvard University. We are gratefulto Lincoln Laboratory of the Massachusetts Instituteof Technology for making available the Lincoln Reck-oner, on which most of the data analysis was done.Reprints may be obtained from William M. Baum,Departlnent of Psychology, Harvard University, Wil-liam James Hall, 33 Kirkland Street, Cambridge, Mas-sachusetts 02138.2Now at Department of Psychology, State University
of New York, Stony Brook, Long Island.
instantaneous. Indeed, response keys andlevers, in conjunction with pulse-formers (thatproduce short pulses of constant duration) areso commonly used today that available appara-tus tends to enforce response counting as themeans of measuring behavior.
If we admit that behavior has duration, analternative scheme of measurement becomesavailable. Behavior that can be counted canalso be timed. Response duration, or timespent responding, can be just as basic a mea-sure of behavior as response frequency.
In some situations the two measures areequivalent. If a pigeon's pecks at a key, forexample, were approximately constant in dur-ation, then the key-peck time would equalthat constant duration multiplied by the num-ber of key pecks. If a rat's holding of a lever isreinforced, on the other hand, then lever-hold-ing time might often vary independent of thenumber of depressions of the lever.
It is usual to select the measure of behavioron the basis of the conditions of reinforce-ment. If we reinforce at a certain point oftime, say, at the moment when the lever hasbeen depressed 5 mm, then it seems natural tocount the number of such momentary occur-rences as could have produced reinforcement.We might, on the other hand, reinforce whilethe animal is engaged in some activity, at noparticular moment, as when we reinforce beingin a certain location and continue reinforce-ment as long as the animal stays in that loca-tion. When such continuous action is rein-
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WILLIAM M. BAUM and HOWARD C. RACHLIN
forced, we tend to use continuous measures ofbehavior, that is, to measure response time,rather than response number.Although experimental procedures often
carry clear implications for choosing measuresof behavior, many experimental situationsdefy such ready decisions. Bullock (1960), forexample, trained pigeons to peck a responsekey that was not connected to a pulse former,but instead produced reinforcement wheneverthe key was operated at the same time that re-inforcement (on a variable-interval schedule)was due. The pigeons eventually came to holdthe key, rather than peck it. Key holding ledto a decrease in response rate. Bullock resolvedthe incompatibility between his measure ofbehavior and his method of recording behaviorby changing his recording, rather than his mea-sure. He found that a pulse former eliminatedkey holding and restored the response rate toa high level. He might, however, have sub-stituted a timer for his response counter.
Ambiguities of measurement have com-monly arisen in the study of responding onfixed-ratio schedules of reinforcement. The as-sumption that a brief response is instantaneousapplies only when the interresponse timesare substantially longer in duration than theresponses themselves. The "internal coher-ence" of the bursts of responding typical ofperformance on fixed-ratio schedules (Mech-ner, 1958a and b) has led to the suggestionthat these bursts themselves be considered asindividual units, or "higher-order" operants(Millenson, 1967, pp. 170-172).A reasonable alternative to this conception
of fixed-ratio performance remains to be ex-plored. Fixed-ratio runs are emitted at analmost constant rate (Ferster and Skinner,1957). The number of responses in a fixed-ratio run determines the duration of the run.As with Bullock's solution to the problem ofkey holding, an alternative to re-defining theunit of behavior is to change to a differentmeasure of behavior: the time spent respond-ing. When we consider variable-ratio sched-ules, measuring response time has a decidedadvantage over counting bursts of respondingas units. Performance on variable-ratio sched-ules includes as high and as constant a re-sponse rate as performance on fixed-ratioschedules. The bursts of responding, however,contain a variable number of responses. Whileit would be difficult to accept response runs of
widely different lengths as equivalent units ofbehavior, it would be easy to think of thesevariable runs as variable times spent respond-ing.The notion that a rate of responding de-
fines a continuous activity can be applied tobehavior other than performance on ratioschedules. Gilbert (1958) has suggested thatlocal response rate on any type of schedulecan be separated from periods of pausing ornon-responding. Some experimental evidencesupports his contention that long-term re-sponse rates are built up from combinations ofpauses and periods of responding at a constantrate. Catania (1961) characterized performanceon a variable-interval schedule as divided intoresponse "runs" and pauses in responding. Hefound that time per run, responses per run,and response rate within a run all remainedconstant in performance on a 3-min VI sched-ule paired with a variety of other schedules inboth multiple and concurrent comparisons.The constancy of the response runs remainedeven when behavioral contrast resulted inchanges in the long-term response rate on theschedule. Catania (1962) found that respond-ing on fixed-interval schedules retains thecharacteristic pattern of accelerating responserate (the Fl "scallop") when paired with a con-current variable-interval schedule. Since re-sponding on the fixed-interval schedules oc-curred in bursts, the pattern of accelerationresulted from a gradual decrease in the periodsbetween bursts, rather than a smooth increasein local rate of responding. Such results arenot peculiar to concurrent schedules. Blough(1963) showed that in a variety of single-keysituations, the majority of interresponse timesfall in the range of 0.3 to 0.5 sec. This basicresponse rate (two to three responses per sec-ond) was insensitive to variations in schedule,rate of reinforcement, and extinction. Bloughfound that variation in long-term responserate, as exhibited in generalization gradientsor extinction curves, results from changes inthe long interresponse times, that is, the pausesbetween bursts of responses at the basic rate.These findings of Catania and Blough sug-
gest that even such brief responses as key peckstend to group into periods of action that alter-nate with periods of inaction. They imply thatthe quantitative relations we have found fornumbers of responses can be reasonably re-interpreted in terms of times spent responding.
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Herrnstein (1961) found that when a pi-geon's pecks on two response keys are rein-forced on two variable-interval schedules, thepigeon distributes its pecks between the keysas follows:
P1 -= (1)PI + P2 r, + r2
where P1 and P2 are the numbers of pecks onKey 1 and Key 2 during the course of a session,and r, and r2 are the rates of reinforcementdelivered by Key 1 and Key 2. Equation (1)states that the bird matches the relative num-ber of emissions of a response to the relativerate of reinforcement for the response.The relative number of pecks in Equation
(1) can be rewritten as follows:
Pl + P2 RIT1 + R2T2 (2)
where R1 and R2 are the rates of respondingon Key 1 and Key 2, and T1 and T2 are thetimes spent responding on Key 1 and Key 2.Herrnstein (1961) reasoned that since bothkeys are always simultaneously available inthe usual concurrent situation, the time basefor calculating the two rates of respondingshould be the same for the two keys. He as-sumed, in other words, that in Equation (2)T1 equals T2. He therefore expressed thematching law as:
R1+R r1 (3)RI + R2 r, + r2
Equation (3) assumes that the matching lawis generated by two simultaneously ongoingresponse rates. An alternative assumption isto suppose that the pigeon divides its time be-tween the two keys, pecking on one and thenthe other for a while, but always pecking atthe same rate at either key. One may assume,in other words, that in Equation (2) R1 equalsR2, but T1 may be different from T2.The results of Catania (1961, 1962) and
Blough (1963) described above lend supportto such an assumption, since they found thatthe response rate while pigeons are respondingis invariant. With a constant response rate,time spent responding determines number ofresponses.According to this line of reasoning, the
matching law would predict relative timespent pecking at the two keys:
T1 -- r+T, + T2 r, + r2 (4)
Although Equations (3) and (4) both predictthe observation of Equation (1), they are in-dependent of each other in the sense thateither equation may apply in a given situationwhile the other does not. Distinguishing be-tween the two experimentally is far fromsimple. In the standard two-key concurrentsituation, there is no easy way to measure thetime spent pecking at each key. There is nodemarcation of time to indicate when thebird is pecking one key, when it is peckingthe other, and when it is pecking neither, butengaged in some other activity altogether. Onemay argue that the time spent pecking a keyis just the collective duration of the pecks, buta key peck undoubtedly requires more timethan the time during which the bird's beak isactually in contact with the key. The lack of aclear beginning and end to each peck makesits duration difficult to measure. Assuming,however, that the time required for a key peckis approximately constant, as Catania's andBlough's results suggest, then the number ofpecks would be an index of the time spentpecking.A technique used by Findley (1958) allows
a more direct approach to measuring the timespent pecking each key in a concurrent sched-ule. The two choice alternatives are repre-sented by two different colors of a single key.The key color changes when the pigeon pecksa second key, called a changeover key. Insteadof changing from one alternative to the otherby moving from side to side, as in the standardtwo-key concurrent situation, the pigeonchanges from one alternative to another inFindley's procedure by pecking the change-over key. Findley found that pigeons behaved,with respect to the two key colors, in the sameway as they behaved with respect to two sepa-rate keys. Catania (1963a) demonstrated thatpigeons in a concurrent situation like Findley'smatch the relative number of pecks on thetwo keys to the relative rate of reinforcementdelivered by the two VI schedules, just asthey do in a standard two-key concurrent situ-ation. Equation (1) holds in both situations.Catania (1966) also showed that the pigeonsmatch the relative time spent in the two com-ponents to the relative rate of reinforcement.The time spent in a component is not the
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WILLIAM M. BAUM and HOWARD C. RACHLIN
same as the time spent pecking in the com-ponent, but if the pigeons spent the sameproportion of time responding in both com-ponents, then their performance would matchrelative time to relative rate of reinforcement.Catania's results, therefore, can be expressedeither in terms of Equation (3) or in terms ofEquation (4).
Brownstein and Pliskoff (1968) showed thatin a concurrent situation like Findley's andCatania's, the matching of relative time spentin a component to relative rate of reinforce-ment occurs in the absence of pecking for rein-forcement. In their experiment, the birds'pecks on the changeover key changed the colorof a stimulus light, but the reinforcers in eachcomponent were delivered independent of thebird's behavior, at the rate determined by theVI schedule. This result presents some dif-ficulty to the interpretation of the matchinglaw as governing relative number of responses.It is difficult to find an appropriate measure ofnumber of responses in Brownstein and Plis-koff's experiment. As noted earlier, it is pos-sible to consider the number of pecks on a keyas an index of the time spent pecking the key.In a like manner, it is possible to consider thetime spent in a component in Brownstein andPliskoff's experiment as an index of the num-ber of emissions of some unspecified response.There would be little empirical basis for suchan assumption, however.The experiment described in this paper
resembles that of Brownstein and Pliskoff inthat it makes use of a non-specific response. Itdiffers from their experiment in the same waythat a standard two-key experiment differsfrom Findley's: the bird changes from onecomponent to another not by pecking a key,but by moving from one position to another.Since the experiment demonstrates the match-ing relation in terms of time spent in twolocations, it supports the interpretation of thematching law as a law of time allocation.
METHOD
SubjectsSix male White Carneaux pigeons were
maintained at 80 to 85% of their free-feedingbody weights. All had been trained previouslywith grain reinforcement to peck a key. Fourbirds, 488, 489, 490, and 496, had a briefperiod of such training. The other two, 334
and 360, had been exposed to a variety of pro-cedures.
ApparatusThe experimental chamber was 9 in. high,
8.75 in. deep, and 19.75 in. long (229 mm by222 mm by 502 mm). Each end wall had a 2-in.by 2-in. (51-mm by 51-mm) opening near thefloor, behind which a standard solenoid-oper-ated food magazine was mounted. The floor ofthe chamber consisted of two separate grids,each pivoted on one side and suspended onthe other side by a spring. The tension of eachspring was sufficient to operate a microswitchwhen no weight was on that side. When a birdstood on either side, the floor dropped about5 mm to release the microswitch on that side.Figure 1 is a diagram of the chamber.Three lights were mounted above the trans-
parent Plexiglas ceiling of the chamber. A redlight was mounted above the left side, a greenlight above the right side, and a white lightabove the center.The chamber was enclosed within a sound-
attenuating box and white noise was con-stantly present. Events in the chamber werecontrolled and recorded by automatic schedul-ing equipment in the next room.
ProcedureThe birds were placed in the experimental
chamber every day for a session that termi-nated when the sum of the reinforcements de-livered by the two magazines equalled 40. Areinforcement on either side lasted 3 sec. Dur-ing reinforcement, the three lights above thechamber were out; the only light on in thechamber was that illuminating the grainhopper. At other times one, and only one, ofthe three lights above the chamber was on.While a bird stood on the left, the red light
alone stayed on. While a bird stood on theRED WHITE SlEEKLOT LIHT UGHT
NGZMFig. 1. The experimental chamber.
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CHOICE AS TIME ALLOCATION
right, the green light alone stayed on. If a birdstood in the center, holding both floors down,the white light alone stayed on. No reinforcerswere delivered while the white light was on.While the red or green lights were on, rein-forcers were delivered only on the side thatwas lit, at variable intervals, according to thedistribution developed by Fleshler and Hoff-man (1962). Separate variable-interval timerscontrolled reinforcements on the two sides. Achangeover delay (COD) of 4.25 sec operatedwhenever a bird changed sides. During theCOD, only the white light was on.These final conditions were gradually ap-
proximated over a period of two weeks. Atfirst, the rates of reinforcement on the twosides were about 200 per hour and the CODwas absent. The rates of reinforcement weregradually decreased to 30 per hour on eachside while the COD was gradually lengthenedto 4.25 sec.The variable-interval timer for each side
ran continuously, except from the moment areinforcement set up for that side to the end
Table 1Summary of Experimental Conditions
Schedule on Schedule on Scheduled RelativeLeft Right Rate of Reinforcement
(in minutes) on Left
a VI 8 VI 2 0.20b VI 4 VI 2 0.33c VI 2 VI 2 0.50d VI I VI 2 0.67e VI 0.5 VI 2 0.80f VI 8 VI 0.5 0.06g VI I VI 8 0.89h VI 4 VI 0.5 0.11i VI 0.5 VI 8 0.94j VI 8 VI I 0.11k VI 0.5 VI 4 0.89
of that reinforcement. The timer for one sidedid not stop during reinforcement on theother side. Reinforcements were scheduled forboth magazines no matter where the birdswere standing. For a reinforcer to be actuallydelivered by either of the magazines, however,the bird had to be standing on the side cor-
responding to that magazine. Thus, in orderto produce available reinforcements from bothmagazines, a bird would have to spend some
time standing on each side of the chamber.The two principal data, time on the left andright, were cumulated on two running-time
meters. These timers did not run during theCOD, during reinforcement, or while the birdheld down both floors; that is, they ran onlywhen the red or green light was on.The variable-interval (VI) schedules studied
averaged 8, 4, 2, 1, and 0.5 min. Table 1 showsthe situations studied. The first five situationsin Table 1 were presented in a repeating cycle,conditions changing every seven days. The rel-ative rate of reinforcement on the left wentstep by step from one extreme to the otherand then back again. Eight weeks were neededto complete a cycle. Three of the birds werealways at a position in the cycle opposite tothe position of the other three. When onegroup was at one extreme, the other group wasat the other extreme. The birds were on thiscycle for nine months. Only the data for thelast 13 weeks were analyzed. For half the birds,the last 13 situations occurred in the order:a, b, c, d, e, d, c, b, a, b, c, d, e; for the otherbirds, the situations occurred in the reverseorder.The last six pairs of schedules in Table 1
were studied after the first five pairs. The birdswere exposed to each of these six conditionsfor two weeks, three birds in the order fromf to k (in Table 1), and three in the order:i, j, k, f, g, h. In the transition from the firstfive conditions to the last six, three birds wentfrom a to i, while the other three went frome to f.The data were summarized as follows. The
times and the number of changeovers werefirst computed as medians of the last three daysof exposure to a set of VI schedules. Since eachbird received conditions f through k in Table1 only once, the three-day medians were thefinal form of the data for these conditions foreach bird. The first five conditions (a throughe in Table 1), however, appeared more thanonce: a and e were presented twice to eachbird, and b, c, and d were presented threetimes to each bird. The three-day medians foreach condition were averaged to produce asingle data set for each condition for eachbird. To obtain average data, the measureswere averaged across birds at this stage in theanalysis. All further computations, for theaverage and the individual birds, were madewith the data sets so produced. The indepen-dent variables, number of reinforcements re-ceived on the left and number received on theright, were computed similarly, except that
865
WILLIAM M. BAUM and HOWARD C. RACHLIN
the original summaries were seven-day aver-ages, instead of three-day medians. It was pos-sible to use seven-day samples because thedistribution of reinforcements between thetwo sides was unaffected by changes in thedistribution of time between the two sides.The only interactions occurred during thefirst few days of the two-week exposures to thelast six conditions in Table 1, when the birdswere changing from an extreme preference forone side to an extreme preference for the op-posite side.Some symbols appearing commonly in the
rest of the paper are defined as follows:
T1 is time spent on the left sideT2 is time spent on the right sideT. is session duration (not including time
during which either food magazine wasoperated)
n1 is number of reinforcements delivered onthe left side during a session
n2 is number of reinforcements delivered onthe right side during a session
RESULTSThe data for individual birds, for all con-
ditions, in the order of presentation, appearin the appendix.
Figures 2 and 3 show the principal result ofthe experiment. For each bird (Fig. 2) and theaverage (Fig. 3), the logarithm of the ratioof time spent on the left to time spent on theright is plotted against the logarithm of theratio of the number of reinforcements receivedon the left to the number of reinforcementsreceived on the right. In such coordinates,direct proportionality between the two ratioswill appear as a straight line with a slope ofone. If the ratios match, as they would accord-ing to the matching law, then the line of slopeone would pass through the point (0,0). Thisline, the locus of perfect matching, appears ineach graph in Fig. 2 and 3 as a light line. Theheavier lines were fitted to the data points bythe method of least squares. The equation ofthe fitted line is given in each graph.The slopes of the fitted lines in Fig. 2 vary
both above and below one. Their average is1.00. The slope of the line fitted to the averagedata (Fig. 3) is close to one. Within the limitsof individual variation, therefore, we can con-clude that the ratio of times is directly propor-tional to the ratio of reinforcements.
For all the birds and the average, the linefitted to the data has a negative intercept.Though the line in Fig. 3 may be parallel tothe matching line, it falls below it. The nega-tive intercept means that the birds spent rela-tively less time on the left than the matchinglaw would predict. Since a constant displace-ment like that in the logarithmic coordinatesof Fig. 3 signifies a constant proportion inlinear coordinates, the birds showed a constantproportional preference for the right side overthe left. The result can be expressed in thefollowing equation:
T k n1T2 ............................n2(5)
where k is a constant less than one. If k wereunity, Equation (5) would be identical to thematching law. Since k is not unity, we maysay that a biased matching has been found,with the bias expressed by the departure of kfrom unity. For the average data, k equals 0.60.
Since the birds generally waited for the endof the COD before changing over again, it isreasonable to suppose that the collective timespent during CODs was equal on the two sides.Adding a constant to two variables will neces-sarily decrease the variance of their ratio. Ifthe measures T1 and T2 in Fig. 2 and 3 hadincluded the collective COD time, the slopesof the lines fitted to the data points wouldhave been less than the slopes of the linesfitted to the data points in Fig. 2 and 3. Theslope of the line fitted to the average data withthe COD time included was 0.42. Equation (5)applies, therefore, only when the time spentin the COD is excluded from T1 and T2.
Catania (1963a) and Rachlin and Baum(1969) demonstrated that in a two-key concur-rent situation, the rate of pecking on eitherkey depends only on the relative rate oramount of reinforcement delivered by the key,and is independent of the rate of pecking onthe other key. Although Equation (5) specifiesthe relative time spent on either side of ourapparatus, it provides no information aboutthe analog to the absolute rate of key pecking,the time spent on a side as a proportion of thetotal session time.The session time in this experiment is de-
fined as the sum of the times during which thered, green, or white lights were on, that is, thesum of T,, T2, the COD time, and the timespent straddling the two floors. Since the latter
866
CHOICE AS TIME ALLOCATION
I-
00-a
1.0 -1.5LOG (NI/N2)
867
Fig. 2. Individual data: the logarithm of the ratio of time spent on the left to time spent on the right plottedas a function of the logarithm of the ratio of number of reinforcements received on the left to number ofreinforcements received on the right during an experimental session. Each of the six plots shows data for oneof the six birds. The heavy lines were fitted to the data points by the method of least squares. The equation ofeach regression line appears beside it. The light lines have a slope of one and pass through the origin; theyrepresent the performance of perfect matching.
WILLIAM M. BAUM and HOWARD C. RACHLIN
O.S.I-0.0.-
_ I
0
-1.0/
I . y.. 102 0.22
-1.5 -1.0 -0.5 0.0 0.5 1.0
LOG (NI/N2)Fig. 3. Averaged data: the logarithnm of the ratio of
time spent on the left to time spent on the rightplotted as a function of the logarithm of the ratioof number of reinforcements received on the left tonumber of reinforcements received on the right dur-ing an experimental session. The heavy solid line was
fitted to the data points by the method of least squares.Its equation appears alongside it. The light solid linehas a slope of one and passes through the origin; itrepresents the performance of perfect matching. Thelight broken line is the performance predicted on thebasis of the data in Fig. 5; a full explanation appearsin the text.
was minimal after initial training, the session
time closely approximated the sum of T1, T2,and the COD time.
Because the COD was fairly long in dura-tion and because the pigeons crossed fre-quently from one side to the other, the CODtime was a significant fraction of the session
time (from 16% to 70%,, depending on thesubject and the conditions of the experiment).It would be possible, then, for the fractionT1/T2 to vary as in Fig. 2 and 3, while T1, forinstance, remained constant, all the variationbeing accounted for by variations in T2 andthe COD time. Despite the relative lawfulnessof T1/T2 as a function of the reinforcementsproduced on the two sides of the chamber,there is no a priori necessity that T1 or T2individually vary lawfully with reinforce-ments.
Figures 4 and 5 show plots for individualbirds (Fig. 4) and the averaged data (Fig. 5),of the proportion of the total session dura-tion spent on each side as a function of therelative number of reinforcements for that
.3
U) .-
'I2
o~~~~~~~~~~~~o
0
2.3
0.1
RELATIVE NUMBER OF REINFORCEMENTS
2~~~~~~~~~~~~In
Fig. 4. Individual data: the proportion of the sessiontime spent on the left (filled circles) and on the right(open circles) plotted as functions of the proportionof reinforcements received on the left (filled circles)and on the right (open circles). The lines were fittedto the data points (open or filled circles separately) bythe method of least squares. The equation of eachregression line appears alongside it.
side. The filled circles represent the propor-tion of time on the left, T1/TS, as a function ofthe relative number of reinforcements on theleft, n1/(n1 + n2). The open circles representthe proportion of time on the right, T2/Ts, asa function of the relative number of reinforce-ments on the right,n2/(n( +nc). The two lines
F:1 .5 VR4side. The fillaed data:ctes preporentinotheproporntioneofptnm on the left,(fied,cicls) an functionrightheinfrelaientsmreeve of hrein forcem ien s)ondthleto n/n ,t 2)herih opencircle)Thli es rereefitedntohdthe proointiopn of tilled circes sprigtelyT2Ty theamuntionoflathsures.aTive equation of eahreinfrces-siontline appears eight n2it. n2Tetw ie
2.4
Z. .2 . 4 . 6 . 6 . .
RELATIVE NUMB3ER OF REINFORCEMENTS n
Fig. 5. Averaged data: the proportion of the sessiontime spent on the left (filled circles) and on the right(open circles) plotted as functions of the proportion ofreinforcements received on the left (filled circles) andon the right (open circles). The lines were fitted to thedata points (open or filled circles separately) by themethod of least squares. The equation of each regres-sion line appears beside it.
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CHOICE AS TIME ALLOCATION
in each graph of Fig. 4 and 5, one for timespent on the left (filled circles) and one fortime spent on the right (open circles), werefitted by the method of least squares. Theequation of each regression line appears along-side its graphical representation.The relationships depicted in Fig. 4 and 5
appear approximately linear but slightly con-
cave upward. The data from Bird 489, fortime spent on the left, constitute an exceptionto the general rule. The intercepts of the re-
gression lines for all subjects were small inabsolute value. Some were negative, otherswere positive. The proportion of the sessionspent on either side appears, therefore, to beapproximately proportional to the relativenumber of reinforcements delivered on thatside. The relationships approximated in Fig.4 and 5 are:
T, = cl n+ (6)
Ts=cin + n2
for the left side, and
T2 C2 +n2 (7)
for the right side, where c1 and c2 are constantsof proportionality.
For every bird, the data for time spent on
the right produced a steeper regression linethan the data for the time spent on theleft. In terms of Equations (6) and (7), forevery bird, c2 was greater than cl. This ten-dency to spend a greater proportion of timeon the right than on the left for the same rela-tive rate of reinforcement illustrates again theposition preference that appeared in Fig. 2and 3.
Equations (6) and (7) may be thought of as
more basic than Equation (5), since Equation5 can be derived from Equations (6) and (7).The ratio of Equation (6) to Equation (7)reduces to:
T, cl n1 8
T2 c2 n2 (8)
Comparison of Equation (8) with Equation(5) indicates that c1/c2 should equal k. Becausethe slopes in Fig. 3 were not all equal to unity,but varied around it, and because of the non-
linearity of some of the individual functionsin Fig. 4, only a very rough correspondenceexists between the individual constants, k andc1/c2. For the average curves, however, where
the slope was equal to unity (Fig. 3) the valuesof k and cl/c2 were 0.60 and 0.66. The close-ness of these two values is illustrated by thebroken line in Fig. 3, which shows the pre-dicted biased matching based on Equation (8)with c, /c2 taken from Fig. 5.
Herrnstein (1961) and Brownstein andPliskoff (1968) found that as the difference infrequency of reinforcement between the twoconcurrent components increased, the fre-quency of changeover between components de-creased. In the present experiment, four birds,488, 489, 490, and 496, showed a similar rela-tionship. Bird 334, however, showed no sys-tematic variation in rate of changeover, andBird 360 changed over most frequently whenthe rates of reinforcement were most different.All birds showed a tendency to change overmore often when they preferred the left sidethan when they preferred the right side. Itmay be that the persistent bias toward theright (Fig. 2, 3, 4, and 5), which generallyweakened preferences for the left when theyoccurred, also made these preferences rela-tively unstable.
DISCUSSIONThe present experiment, together with that
of Brownstein and Pliskoff (1968), showed thatin the absence of reinforcement for any specificresponse, the same type of law governs thedivision of an organism's time among theactivities in which it engages as governs thedistribution of responses among choice alter-natives (Herrnstein, 1961; Reynolds, 1963;Catania, 1963a). Catania (1966) found thateven when behavior is defined and measuredin terms of discrete responses (key pecks), thetime allocation matching law still applies. Asnoted in the introduction, the results of severalexperiments (Blough, 1963; Catania, 1961,1962; Mechner, 1958a and b) suggest thatseries of repetitions of a discrete act (a keypeck or a lever press) can be thought of asperiods of engaging in a continuous activity(key pecking or lever pressing). Thus, eventhough behavior in a given situation may bedefined and measured as if it consisted of dis-crete acts, it is still possible to derive continu-ous measures of behavior in that situation.Laws of time allocation, therefore, are likelyto be more widely applicable to behavior thanlaws of response distribution.
869
870 WILLIAM M. BAUM and HOWARD C. RACHLIN
If we accept the idea that the matching lawgoverns time allocation among activities, whatcan we say about experiments that have dem-onstrated matching of relative number ofpecks to variables other than relative rate ofreinforcement? Catania (1963b) found thatpigeons match relative pecks to relativeamount of reinforcement. Chung and Herrn-stein (1967) obtained matching of relativepecks to relative immediacy of reinforcement(reciprocal of delay of reinforcement). We canexpress the three matching laws in terms oftime spent pecking at two keys (T1 and T2)as follows:
T1 rT2 r2T1 = alT2 a2T1 ilTE2 i2
where r1 and r2 are the rates of reinforcement,a1 and a2 are the amounts of reinforcement,and il and i2 are the immediacies of reinforce-ment, produced by pecking at Key 1 and Key2, respectively.We are now led to ask how these three inde-
pendent variables might combine to determinechoice when they are varied together, insteadof one at a time, as Herrnstein, Catania, andChung and Herrnstein varied them. The sim-plest possible relation might be multiplicationof the ratios of independent variables to pro-duce the ratio of times, as follows:
T1 r, a, il (9)T2 r2a2i2(9
The most general form of such a matchinglaw, which would include new variables be-sides the three already known, would be:
n
T,flxu (10)T2 rlX2' 10
where xlj and x2j are the values of variable xjassociated with Key 1 and Key 2, and there aren such variables, instead of just three, asabove. If we define the value, Vi, of Activity ias:
n
Vi =TFxijj = 1
then Equation (10) reduces to:
Tj_ VIT2 (11)
Equation (11) states that pigeons allocate timeto any given pair of activities in such a waythat the ratio of the times allocated equalsthe ratio of the values of the activities.Neuringer (1967) verified Equations (10)
and (11) for two variables: amount and rate ofreinforcement. He found that pigeons in atwo-alternative choice situation matched rela-tive frequency of choice to relative "total ac-cess to reinforcement," the product of amounttimes rate of reinforcement. He found, inother words, Equation (9) with i1 equal to i2.The form of the position preference shown
in Fig. 2 lends further support to Equation(10). We do not know what variables deter-mined the preference for the right side overthe left. Perhaps the right magazine allowedthe birds to eat more during the magazinecycle. Perhaps a greater movement of the leftfloor when stepped on contributed to the bias.Whatever the determinants, however, the posi-tion preference only necessitated multiplyingthe ratio of the rates of reinforcement by aconstant to produce matching. In terms ofEquation (10), this constant is either the ratioof two values of a single variable that differedfrom one side to the other, or, perhaps morelikely, the product of several ratios of thevalues of several variables that differed fromone side to the other. The form of the positionpreference, therefore, suggests that Equation(10) may predict preference with great gen-erality.We can only hope that other variables that
fit into this formulation will be as simple toexpress as rate, amount, and immediacy ofreinforcement. Staddon (1968) suggested thatsuch simplicity may not completely prevail.
REFERENCESBlough, D. S. Interresponse time as a function of con-
tinuous variables: a new method and some data.Journal of the Experimental Analysis of Behavior,1963, 6, 237-246.
Brownstein, A. J. and Pliskoff, S. S. Some effects ofrelative reinforcement rate and changeover delay inresponse-independent concurrent schedules of rein-forcement. Journal of the Experimental Analysis ofBehavior, 1968, 11, 683-688.
Bullock, D. H. Note on key-holding behavior in thepigeon. Journal of the Experimental Analysis ofBehavior, 1960, 3, 274.
CHOICE AS TIME ALLOCATION 871
Catania, A. C. Behavioral contrast in a multiple andconcurrent schedule of reinforcement. Journal ofthe Experimental Analysis of Behavior, 1961, 4,335-342.
Catania, A. C. Independence of concurrent respondingmaintained by interval schedules of reinforcement.Journal of the Experimental Analysis of Behavior,1962, 5, 175-184.
Catania, A. C. Concurrent performances: reinforce-ment interaction and response independence. Jour-nal of the Experimental Analysis of Behavior, 1963,6, 253-263. (a)
Catania, A. C. Concurrent performances: a baselinefor the study of reinforcement magnitude. Journalof the Experimental Analysis of Behavior, 1963, 6,299-300. (b)
Catania, A. C. Concurrent operants. In W. K. Honig(Ed.), Operant behavior: areas of research and ap-plication. New York: Appleton-Century-Crofts, 1966.Pp. 213-270.
Chung, S. and Herrnstein, R. J. Choice and delay ofreinforcement. Journal of the Experimental Analysisof Behavior, 1967, 10, 67-74.
Ferster, C. B. and Skinner, B. F. Schedules of rein-forcemnent. New York: Appleton-Century-Crofts,1957.
Findley, J. D. Preference and switching under con-current scheduling. Journal of the ExperimentalAnalysis of Behavior, 1958, 1, 123-144.
Fleshler, M. and Hoffman, H. S. A progression forgenerating variable-interval schedules. Journal ofthe Experimental Analysis of Behavior, 1962, 5,529-530.
Gilbert, T. F. Fundamental dimensional propertiesof the operant. Psychological Review, 1958, 65,272-285.
Herrnstein, R. J. Relative and absolute strength ofresponse as a function of frequency of reinforce-ment. Journal of the Experimental Analysis ofBehavior, 1961, 4, 267-272.
Mechner, F. Probability relations within response se-quences under ratio reinforcement. Journal of theExperimental Analysis of Behavior, 1958, 1, 109-121. (a)
Mechner, F. Sequential dependencies of the lengths ofconsecutive response runs. Journal of the Experi-mental Analysis of Behavior, 1958, 1, 229-233. (b)
Millenson, J. R. Principles of behavioral analysis. NewYork: Macmillan, 1967.
Neuringer, A. J. Effects of reinforcement magnitudeon choice and rate of responding. Journal of theExperimental Analysis of Behavior, 1967, 10, 417-424.
Rachlin, H. and Baum, W. M. Response rate as afunction of amount of reinforcement for a signalledconcurrent response. Journal of the ExperimentalAnalysis of Behavior, 1969, 12, 11-16.
Reynolds, G. S. On some determinants of choice inpigeons. Journal of the Experimental Analysis ofBehavior, 1963, 6, 53-59.
Skinner, B. F. The behavior of organisms. New York:Appleton Century, 1938.
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Received 24 March 1969.
WILLIAM M. BAUM and HOWARD C. RACHLIN
Appendix: Table of data for individual birds in each condition.
The data appear in the order in which they were
gathered. See Table 1 for the schedules correspondingto each lettered condition. The symbols T1 and T2stand for time spent on the left and time spent on theright, respectively. All data are medians of the last
three days of exposure to the conditions, except thenumber of reinforcements on the left, which are sums
over the last seven days. The total reinforcements de-livered over the last seven days was 280 for each con-
dition.
Bird 488
COD Session Change- Left Rein-Condition Tl(min) T,(min) Time (min) Time (min) overs forcemlents
