log (B!) = a log (R!) + log c - Europe PMC

1983, 40, 15-34

JOURNAL OF THE EXPERIMENTAL ANALYSIS OF BEHAVIOR

NUMBER

I (JULY)

BIAS AND SENSITIVITY TO REINFORCEMENT IN A CONCURRENT-CHAIN SCHEDULE MICHAEL DAVISON UNIVERSITY OF AUCKLAND

Six pigeons were trained on concurrent-chain schedules in which the initial links were either dependent (Experiment 1) or independent (Experiment 2) concurrent variableinterval schedules and the terminal links were fixed-duration delays to reinforcement in blackout. A changeover delay of three seconds operated in the initial links. Both the initialand terminal-link schedule values were varied over 61 experimental conditions. An analysis of the data using the generalized matching law showed that the sensitivity of initial-link response allocation to the frequency of terminal-link production was independent of both the duration of the terminal-link delays and, though less clearly, of the duration of the initial-link schedules. Sensitivity of initial-link response allocation to terminal-link reinforcer-rate ratios was a joint function of the terminal-link durations and the smaller average initial-link duration. The results showed that the generalized matching law is useful in analyzing concurrent-chain schedule performance and that a changeover delay in the initial links eliminates some terminal- to initial-link interactions that have made quantitative predictions for concurrent-chain performances difficult. Key words: concurrent-chain schedules, generalized matching law, sensitivity to reinforcement, response allocation, time allocation, bias, delay of reinforcement, pigeons

In a concurrent-chain schedule, responding on concurrent variable-interval (VI) schedules is reinforced by the production of a period of time in which, typically, a single food reinforcer is gained. The concurrent VI VI initial links are then reinstated. A diagrammatic representation of the procedure is shown in Figure 1. In general, a concurrent chain differs from a concurrent schedule in that, in the former, the reinforcement period is extended over time. It is evident that shortening the terminal-link reinforcement periods in a concurrent-chain schedule moves the procedure closer to a concurrent schedule and that when the terminal links are 0 sec in duration, the procedure is a concurrent schedule. It may thus be helpful to subsume concurrent-chain schedule performances under the heading of concurrent schedules and to deal with the effects of terminal-link schedules as if chang-

ing these was akin to changing the magnitudes of reinforcers in concurrent VI VI schedules. Viewing the concurrent-chain schedule as a concurrent schedule with different magnitudes of reinforcers brings the benefit of the availability of a considerable amount of published research and an existing and well-supported theory-the generalized matching law (Baum, 1974b). The present experiment was designed to answer the question: Can the generalized matching law account for any or all aspects of concurrent-chain schedule performance? The generalized matching law is written:

log (B!) = a log

(R!) + log c,

(1)

where the subscripts 1 and 2 denote the two response alternatives, B denotes pecks, and R denotes reinforcers obtained. The parameter a is called sensitivity to reinforcement and measures the amount of change in the log-response ratio as a function of changes in the log-reinforcer ratio. The parameter log c is called bias and normally measures a constant proportional preference for one alternative, due to some variable, like differing key pressures on the two keys, which is unchanged throughout the experiment.

I thank the New Zealand University Grants Committee for its continued support for my research and the Masters and Doctoral students who helped run these experiments. A brief report of this research was given at the Fifth Harvard Symposium on Quantitative Analyses of Behavior in June, 1982. Reprints may be obtained from Michael Davison, Department of Psychology, University of Auckland, Private Bag, Auckland, New Zealand.

15

16

MICHAEL DAVISON

schedules or delays. These effects, as previous research has shown (Fantino, 1977), are a function of the terminal-link reinforcer rates. Thus, we can write:

KEY INITIAL LINKS TERMINAL LINKS

Ta2

C SR

VI

A:

INOPERATIVE INOPERATIVE

B:

VI Tb2

) sR

Fig. 1. A diagram of the concurrent-chain schedule procedure. Responding on keys A and B (white) in the initial links on nonindependent (Experiment 1) or independent (Experiment 2) schedules occasionally produced one of two delays to reinforcement in blackout. On key A, responding on VI T.1 produced a delay of T82 sec, and on key B, responding on VI Tbl produced a delay of Tb2 sec. These blackouts terminated in the delivery of food and, following this, the initial links were reinstated.

A version of this law, and by implication Equation 1 itself, was extended by Baum and Rachlin (1969) to the situation in which more than one independent variable is manipulated. For example, if both reinforcer frequency and magnitude were varied in an experiment, the following equation would apply: log ("')

=

a

log ('L) + b log

(

)±+ log c,

(2) where M is the reinforcer magnitude. Such an additive-logarithmic model has been shown to account well for combinations of two independent variables (Hunter & Davison, 1982; Schneider, 1973; Todorov, 1973). Equation 2 implies that if M1 and M2 were kept constant, but unequal, in an experiment, and R1 and R2 were varied, the relationship between log (B1/

B2) and log (R1/R2) would be a straight line of slope a with an intercept of b log (Ml /M2) + log c. This is a biased matching line. Equally, if R1 and R2 were kept constant and unequal while M1 and M2 were varied, the relationship between log (BlIB2) and log (Ml IM2) would be a straight line of slope b with an intercept of log (R1/R2) + log c. If we assume that this equation might apply to concurrent-chain performance, the log-magnitude term is replaced by a measure of the effects of the terminal-link

+ b log (Tb2\ + log c, Bal = a log log \Bbl/ Tall Ta2I (3)

using the nomenclature of Figure 1. If the generalized matching law applied to concurrent-chain performances, a should be independent of log (Tb2/ Ta2) and b should be independent of log (Tbl/ Tal). A similar equation should apply when the dependent variable was time spent responding, rather than numbers of responses, in the initial links. Davison (1976) followed this logic. He arranged terminal links of FI 5-sec vs. FI 5-sec, FI 5-sec vs. Fl 15-sec, and Fl 15-sec vs. FI 5-sec, and varied the initial-link VI schedules keeping the smaller always VI 27-sec. The logic of the generalized-matching-law approach requires that the sensitivity (a) of the initial-link response ratio to the ratio of terminal-link entries be the same for each pair of terminallink schedules. They were not. The unequal terminal-link schedule data deviated maximally from the equal terminal-link data when both initial links were VI 27-sec. But the more different were the initial links, the less this deviation became. These results appear to be a fundamental demonstration that the generalized matching law does not apply to concurrent-chain schedule performance. The bias caused by the terminal links was affected by the initial-link schedule values, and such an interaction rules out the Baum-Rachlin (1969) concatination (Equations 2 and 3). If such an interaction does occur, it is of interest to document its development in the transition between concurrent and concurrent-chain schedules. However, a problem immediately arises if an experiment is designed encompassing both concurrent and concurrentchain schedules. It is that, by and large, concurrent-chain schedules have not used changeover delays (Herrnstein, 1961) in the initial-link concurrent VI VI schedule components, whereas concurrent schedules are usually arranged with changeover delays (de Villiers, 1977). Because I wanted to arrange both concurrent and concurrent-chain schedules in the present experiments to investigate the dif-

GENERALIZED MATCHING LAW AND CONCURRENT CHAINS

17

ference between them, I used a changeover de- rent initial links (see Figure 1) were in effect. lay in the initial links throughout. Noninde- A changeover delay of 3 sec was arranged such pendent concurrent VI VI schedules (Stubbs & that a response on a key could not produce a Pliskoff, 1969) were used to overcome the terminal link, even though an entry was arproblem, raised by Davison and Temple ranged by the VI schedules, until 3 sec had (1973), of obtained terminal-link entries being elapsed since the first peck on that key folvery different from those arranged when ini- lowing responses to the other key. The schedtial-link response allocation is strongly biased ules in the initial links were arranged nonby differences in the terminal-link schedules. independently; that is, when one VI schedule The design of the present experiment was had arranged a terminal-link entry, both to use fixed delays-to-reinforcement in the ter- schedules stopped until that entry had been minal links of a concurrent-chain schedule taken. When an initial-link VI schedule had and, for a number of terminal-link delay com- arranged a terminal-link entry and the changebinations, to vary the initial-link VI schedules over delay on that key had elapsed, a peck to over a range of both relative and absolute the appropriate key turned off all the keylights values. and started a delay period. At the end of this period, which was fixed for each key (Table 1), the food magazine operated and, following EXPERIMENT 1 this, the initial links were again available and both initial-link schedules recommenced. This METHOD procedure (Stubbs & Pliskoff, 1969) ensured Subjects that the relative number of terminal-link enSix homing pigeons, numbered 181 to 186, tries obtained on each key would be similar to were maintained at 80% of their free-feeding that arranged by the initial-link VI schedules. body weights. Birds 181 and 185 provided Sessions ended in blackout after a fixed numdata beginning with Condition 9 due to the ber of reinforcers had been obtained. The deaths of two birds that initially started the birds were then returned to their home cages, experiment. where water and grit were available, and were fed the amount of food necessary to maintain Apparatus 80% of their normal body weights. The experimental chamber, which was situIn each condition, the number of responses ated remotely from solid-state experimental emitted on each key in the initial links and the control equipment, contained three response number of entries into each terminal link were keys 2 cm in diameter, 11 cm apart, and 25 cm recorded. Also recorded was the time spent from the grid floor. Each key was operated by responding on each key in the initial links, a peck exceeding about .1 N and could be measured from the first peck on one key to the illuminated with three different-colored lights. first peck on the other. Time-allocation meaThe food magazine, which contained wheat, surement ceased when 5 sec had elapsed withwas situated beneath the center key and 12 cm out an initial-link response or on terminal-link from the grid floor. The food magazine oper- entry. ated for a nominal 3 sec, during which time Table 1 shows the sequence of experimental the hopper was illuminated and the keylights conditions and the number of sessions of trainwere extinguished. There was no other illumi- ing given on each. Apart from Condition 1, in nation in the chamber. Pecks to blacked-out which an immediate reinforcer was arranged keys were always ineffective and were not in both terminal links, the terminal delays counted. An exhaust fan fitted to the chamber were 0 and 3 sec and the initial-link VI schedhelped mask external noise. ules were varied, both relatively and absolutely. From the nine conditions with 0- and Procedure 3-sec terminal delays, there were three condiThe pigeons had extensive experience of tions with smaller initial-link VI schedules of responding on concurrent schedules and were 60 sec and five with smaller initial-link VI placed directly on the first experimental con- schedules of 30 sec. There was also a set of dition (Table 1). Initially, the two side keys data (Conditions 2, 5, and 8) in which the were both illuminated white, and the concur- initial links were kept equal and the absolute

18

MICHAEL DAVISON

Table 1 Experiment 1. Sequence of experimental conditions, initial-link VI schedules, terminal-link delays (both sec), and number of sessions training given in each condition.

Condition 1 2 3 4 5 6 7 8 9 10 11

12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

Initial-link VI Schedules Right Left 60 60 60 120 120 30 120 30 30 60. 60 30 30 30 120 30 60 120 30 120 120 30 60 30 120 30 30 30 30 30 120 60 120 30 60 30 180 30 90 30 240

60 60 120 60 120 120 30 30 60 30 30 120 30 60 30 30 120 30 120 60 30

120 30 60 30 30 120 120 30 120 60 120 30 60 30 180 30 90 30 240 30

Terminal-link Delays Right Left 0

0

0 0 0 0 0 0 0 0 0

3 3 3 3 3 3 3 3 3

0

0

0

0

0

0

0

0

0

0

0 0 0 0 0 0 5 5 5 5 5 15 30 15 15 15 15 15 15 15 15 15 15 15 15 15

10 10 10 10 10 30 5 5 5 5 5 15 30 30 30 30 30 30 30 30 30 30 30 30 30 30

Sessions

28 27 21 20 27 23 27 30 19 18 23 26 24 19 27 29 20 29 24 24 43 23 16 23 23 21 33 26 30 19

30 21 24 24 20 37

29 19 34 28 31

values of these were 30, 60, or 120 sec. In Conditions 1 and 11 to 15, the terminal delays were both 0 sec (concurrent VI VI schedules), and the initial-link VI schedules were varied keeping (apart from Condition 1) the smaller initial link at VI 30-sec. In Conditions 16 to 20, the terminal-link delays were 0 and 10 sec, and the initial-link schedules were varied. In all of these conditions, except 17 and 20, the smaller initial-link VI schedule was VI 30-sec. In Condition 21, the terminal-link delays were

0 and 30 sec. This condition was arranged to complete a data set with initial-link VI schedules of 120 and 30 sec, one terminal link of 0 sec, and the other varied from 0 to 3 to 10 to 30 sec (Conditions 7, 15, 18, and 21). In Conditions 22 to 26, the delays in both terminal links were 5 sec, and the initial-link VI schedules were varied with one kept at VI 30-sec. Equal terminal-link delays were also arranged in Conditions 27 (both 15 sec) and 28 (both 30 sec) with concurrent VI 30-sec VI 120sec schedules in the initial links. These conditions, with Condition 12 (0-sec delays) and Condition 22 (5-sec delays), provided four conditions in which the absolute values of the equal terminal-link delays were varied with constant initial-link schedules. Finally, in Conditions 29 to 41, the terminal delays were 15 and 30 sec while the initial-link schedules were varied. In Conditions 29, 30, and 33 to 41, the smaller initial-link VI schedule was VI 30-sec, and in Conditions 31 and 32 it was 60 sec. Each experimental condition remained in effect until all birds had reached a defined criterion of stability five, not necessarily consecutive, times. The criterion was that the median relative initial-link response numbers over five sessions did not differ by more than .05 from the median of the five sessions prior to these. The number of sessions training in each condition is shown in Table 1. The initial-link VI schedules comprised an irregular sequence of the first 12 terms from an arithmetic progression with the shortest interval being one-twelfth the mean interval. RESULTS AND DISCUSSION Because of the complexity of the present data, it will be easier if they are dealt with not in the order in which the experimental conditions were conducted but in a more logical order. I shall also deal only with matching law analyses of response and time allocation in the initial-link schedules as a function of frequencies of terminal-link entries and the values of terminal-link delays. The full data, which are shown in Appendix 1, are available for other analyses. The logical place to commence the analysis of these data is with Conditions 11 to 15 in which the terminal delays were both 0 sec (that is, concurrent VI VI schedules), and the smaller initial-link schedule was VI 30-sec. In Figure 2, the logarithm of the ratio of initial-

GENERALIZED MATCHING LAW AND CONCURRENT CHAINS

19

mates. The equal initial-link condition (13) always gave data that fell above the fitted lines, and I can offer no explanation for this. But in view of subsequent analyses, it is important to note that this particular condition does not affect the estimate of sensitivity to the initial-link reinforcement ratio (a). In concurrent-schedule performance, the sensitivity of time-allocation ratios to reinforcer ratios is usually greater than the sensitivity of response-allocation ratios (Baum, 1979), and this was true of the present data. Time-allocation sensitivities (Table 3) averaged .80, ranged from .37 to 1.22, and were greater than response-allocation sensitivities for all six birds. (Note that Table 3 in the .4 -.4 General Discussion summarizes all response LOG ENTRY RATIO and time slopes and their standard deviations.) Fig. 2. Log initial-link response ratios (right/left) as Time-allocation biases (Table 3) averaged -.07 a function of log terminal-link entry ratios (i.e., log initial-link reinforcement ratios) when no terminal de- and ranged from -.35 to +.19. They were lays were arranged in either terminal link. Straight lines very similar to response-allocation biases. were fit by the method of least squares; beneath these Figure 3 shows response-allocation data obequations are shown the standard deviations of the tained when the terminal-link delays were slopes and intercepts. The origins of the graphs are shown by a cross. Unfilled and filled data points dis- both 5 sec and the smaller initial-link schedule was VI 30-sec. Straight lines were again fitted tinguish between birds. by least-squares linear regression. The response link responses is shown as a function of the biases averaged -.05 (range -.13 to .06), and ratio of terminal-link entries when the initial- for four of the six birds they were in the same link schedules were varied with one kept at direction as, though generally not as extreme VI 30-sec. In this and subsequent figures, nega- as, those in Figure 2. The slopes, or sensitivitive values on the X axis came from conditions ties, averaged .65, similar to the value for in which fewer right than left reinforcers were obtained and positive values from the reverse TERMINAL LINKS: 5 VS. 5 of this. Straight lines were fit to the data by 181: the method of least squares. The slopes and Y = 60X + .06 intercepts of the best-fitting lines are shown 0 (.05,02) 184: Y= 53X- .12 in Figure 2 with the standard deviations of S -(07,.03)± -' I these estimates. These plots correspond to w 185: ' Equation 3, and the intercepts are a measure 0z .182: 07 Y .57X+ .05 0 (06. 03) aof b log (Tb2/TTa2) + log c. Figure 2 shows a Lnw Y0 .=89X-.07+7, .14, .06) a wide range of intercepts, ranging from -.28 to cr 0,' +.12 and averaging -.08. Clearly, there was no zy consistent bias to either key or terminal delay. * o 183: In concurrent VI VI schedules, response-al' 186: Y = .43X 13 11 0/' * Y = .89Xlocation sensitivities (the slopes in Figure 2, 0-O (.14,06) (09,.04) and a in Equation 1) generally have a mean of JO 0 ,' about .8 and vary from .5 to 1.5 (Baum, 1979). .-4 .4 -*4 0 .4 LOG ENTRY RATIO The slopes obtained here averaged .66 and ranged from .36 to 1.11. Although in view of Fig. 3. Log initial-link response ratios (right/left) as Baum's work, the present slopes were generally a function of log terminal-link entry ratios when both low, the slope for Bird 182 (1.11) shows that terminal delays were 5 sec. Least-squares regression lines, and below these the standard deviations of the this effect was not caused by the procedure parameter estimates, are shown for each bird. Crosses used. The fits were not particularly good, with show the origins of the graphs. Unfilled and filled data rather high standard deviations of slope esti- points distinguish between birds. TERMINAL LINKS: 0 VS. 0

-

11

-

MICHAEL DAVISON

20 TERMINAL LINKS: 0 VS. 3

o o cx

182:

TERMINAL LINKS: 0 VS. 10 0

184: Y= 56X+39 3

Y .54X+ .32

cx

181 Y= *63X + 55

- ,,

82:

LlI

z

o

LI) z 0 0-

0

184:

(.07, 03)

Y

=

.93X + 63

w

0L

>e/) _

/

,

y z

-.4

0

/~Y=81X+ .02 weK ~~~~~~~~~~~(.08,.04)

--4 *4 LOG ENTRY RATIO

0

*4

Fig. 4. Log initial-link response ratios (right/left) as a function of log terminal-link entry ratios when the terminal delays were 0 and 3 sec. Least-squares regression lines, and below these the standard deviations of the parameter estimates, are shown for each bird. Crosses show the origins of the graphs. Unfilled and filled data points distinguish between birds.

_0

,--

o-

/

/

I-z

o

3 0

,183: -69X

0-I ~J0 0. _j

185: 64 o /+y=.56X+ ~~~~(.14, 06)

/

/

-I)

~~~(.0402) zo t