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JOURNAL OF

THE

1979, 32, 269-281

EXPERIMENTAL ANALYSIS OF BEHAVIOR

NUMBER

2

(SEPTEMBER)

MA TCHING, UNDERMA TCHING, AND 0 VERMA TCHING IN STUDIES OF CHOICE WILLIAM M. BAUM UNIVERSITY OF NEW HAMPSHIRE Almost all of 103 sets of data from 23 different studies of choice conformed closely to the equation: log (B1/B2) = a log (r1/r2) + log b, where B, and B2 are either numbers of responses or times spent at Alternatives 1 and 2, r, and r2 are the rates of reinforcement obtained from Alternatives 1 and 2, and a and b are empirical constants. Although the matching relation requires the slope a to equal 1.0, the best-fitting values of a frequently deviated from this. For B1 and B2 measured as numbers of responses, a tended to fall short of 1.0 (undermatching). For B1 and B2 measured as times, a fell to both sides of 1.0, with the largest mode at about 1.O. Those experiments that produced values of a for both responses and time revealed only a rough correspondence between the two values; a was often noticeably larger for time. Statistical techniques for assessing significance of a deviation of a from 1.0 suggested that values of a between .90 and 1.11 can be considered good approximations to matching. Of the two experimenters who contributed the most data, one generally found undermatching, while the other generally found matching. The difference in results probably arises from differences in procedure. The procedural variations that lead to undermatching appear to be those that produce (a) asymmetrical pausing that favors the poorer alternative; (b) systematic temporal variation in preference that favors the poorer alternative; and (c) patterns of responding that involve changing over between alternatives or brief bouts at the alternatives. Key words: matching relation, undermatching, overmatching, choice, conc VIVI

In experiments with concurrent variableinterval schedules, when the ratio of responding or time spent at two alternatives (Bl/B2) is graphed in logarithmic coordinates as a function of the ratio of reinforcement (r1/r2) obtained from the two alternatives, the data points usually conform to a straight line: log B1

(1)

log r + log b = a log

Exponentiating both sides of this equation produces a power function of the type familiar in psychophysics (Stevens, 1957, 1975): B

B2

=

b (r)a

(2)

r

The matching relation, which may be written:

B1 B1+ B2

-

r1 + r2

or

=,

B2

r2

The author's previously unpublished research was supported by grants from the National Science Foundation (GB-28493) and the National Institute of Mental Health (15, 494-07 and 26, 793-01). Reprints may be obtained from William M. Baum, Department of Psychology, Conant Hall, University of New Hampshire, Durham, New Hampshire 03824.

requires that a and b in Equations 1 and 2 equal 1.0. A deviation of b from 1.0 (bias) can arise from unmeasured inequality in amount of reinforcement across the alternatives or a qualitative difference in the activities at the alternatives. The sources of bias were discussed in an earlier paper (Baum, 1974c). This paper concerns deviations of a from 1.0. Analysis of ratios of behavior and reinforcement allows ready identification of departures from the matching relation. When Equation 1 is tested, a line is fitted to the ratios in logarithmic coordinates, and a appears as the slope of the line. The earlier paper showed that bias, which affects only the intercept of the fit based on Equation 1, produces curvilinear patterns in graphs of proportion of behavior versus proportion of reinforcement (Baum, 1974c). Deviations of a from 1.0 also predict curvilinear patterns in the proportions. Setting b to 1.0 and rewriting Equation 2, we obtain: B1

B1 + B2

_

r_a

rla + r2a

Figure 1 shows the sorts of curvature, for several values of a, that this equation predicts.

269

270

WILLIAM M. BAUM

When a is greater than 1.0, the data points should conform to an S-shaped curve. An example appears in Herrnstein's (1961) original report of matching: Bird 231 in his Figure 1. When a is less than 1.0, Equation 2 predicts the other curves shown in Figure 1. These resemble the polynomial curve fitted by Myers and Myers (1977, Fig. 2). Equation 2 predicts their finding of nonlinearity in the proportions. One may question whether there is any advantage in their use of polynomial regression, however, when Equation 1 provides a convenient linear model for assessing deviation from matching. If we make the reasonable assumption that reinforcement of only one alternative produces exclusive preference for that alternative-i.e., that when proportion of reinforcement equals 1.0 or 0.0, so too does proportion of behaviorthen we see that deviations from matching in the coordinates of Figure 1 must appear nonlinear. Since two points determine a straight line, if the extremes (0,0) and (1,1) are fixed, then the matching line is the only possible linear model that can fit data represented as proportions. Since the data are constrained in this way, the results of linear regression on proportions cannot deviate substantially from the matching relation and will only deviate from the matching relation if few points fall near the extremes. For these reasons, it is a

0.0

0.1

0.2

0.5 0.3 0.4 0.6 0.7 PROPORTION OF REINFORCERS

0.8

0.9

1.0

Fig. 1. Appearance of overmatching and undermatching when relative responding and reinforcement are represented as proportions. Different shapes arise from different values of a in Equation 2.

mistake to perform linear regression on proportions, as some authors have done (e.g., Deluty & Church, 1978; Menlove, Moffitt, & Shimp, 1973; Myers & Myers, 1977; Poling, 1978). Equation 1 provides a sensible alternative. Variations in the Exponent Like the exponent of psychophysical power laws, such as that by which intensity of sound governs loudness (Stevens, 1975), a may be expected to vary from one experiment to another as a result of random error. If a were a random variable, it would vary to both sides of its central tendency, which the matching relation requires to equal 1.0. If any common experimental procedures favored deviation from the central tendency in one direction rather than the other, we might expect the frequency distribution of exponents from various experiments to be skewed. If the systematic deviations constituted a relatively small proportion of the total variation, the mode might remain unchanged. For the matching relation, the skewed distribution would retain its mode at 1.0. To test these propositions, Equation 1 was fitted to all the data I had on hand as of March 1976. Table 1 lists the experiments and gives, for each set of data, the results of two linear regression procedures: least squares and the nonparametric method of Hollander and Wolfe (1973). Some experiments produced measures of both responses and time (R or T in column iii) spent at the alternatives; others produced only one or the other. For those experiments that included two few conditions per subject, the data from all the subjects were pooled. Although the experiments varied in species of subject and type of response, the great majority used pigeons and discrete responses (e.g., key pecks and lever presses). No attempt was made, therefore, to examine these factors beyond noting that the ranges of exponents overlapped substantially (pigeons: .38 to 1.50; rats: .79 to 1.01; humans: .94 to 1.13; discrete responses: .57 to 1.35; continuous activities: .38 to 1.50). Analysis of variance showed that none of these differences was statistically significant at the .05 level. The same was true of two variations of procedure: using a changeover key (COK) instead of two keys (Catania, 1966; Findley, 1958) and

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274

WILLIAM M. BAUM

using a single distributed schedule (forced changeover; FCO) instead of two independent schedules (IND). Two studies combined both techniques (Bauman, Shull, & Brownstein, 1975; Stubbs & Pliskoff, 1969). Two studies involved choice among three response keys (3K) (Davison & Hunter, 1976; Pliskoff & Brown, 1976). These were treated as two separate two-alternative choice situations: responding and reinforcement at the left and center keys separate from responding and reinforcement at the center and right keys. Choice between the two side keys (omitted from Table 1) was considered less comparable to two-key choice, because responses at these keys had to follow changeovers either from the center key or across the center key. Two studies that provided different reinforcers for the two alternatives (Fantino, Squires, Delbruck, & Peterson, 1972; Hollard & Davison, 1971) were included, because the reinforcers remained invariant while their frequencies were changed. The asymmetry in reinforcers should affect only the bias b, but not the slope a (Baum, 1974c). The studies in Table 1 used a variety of techniques for averaging data across sessions. Comparable results across experiments might be taken as evidence that method of averaging is of little consequence. No systematic attempt was made to test this proposition, because data from single sessions were generally unavailable. Averaging of my own data in various ways has never produced substantially different results. Unless variation across sessions is highly asymmetrical, all measures of central tendency will be about the same. Since regression techniques like the method of least squares implicitly average, the results of regression on data for individual sessions are unlikely to differ from those for summaries. Whenever an author gave reasons for including or excluding certain data (e.g., when the changeover delay was varied), the data were treated in the manner suggested. Two studies that involved choice between alternatives that themselves were concurrent schedules were omitted (Baum, 1974b; Menlove et al., 1973). In both, choice between alternatives that produced food showed strong undermatching. Choices between the alternatives that were themselves choice situations appeared consistent with the data of Table I. My experiment produced matching. That of Menlove et al. produced undermatching.

Their inclusion would have had no effect on the conclusions one may draw from Table 1.

EVALUATION OF THE FITS Figure 2 shows the proportion of variance accounted for by least-squares regression using Equation 1 (column vi) plotted against the fitted slope a. In this figure, and those to follow, slope is represented on a logarithmic axis, because the inverse of a slope or exponentthe slope or exponent equally different from 1.0 in the opposite direction-is its reciprocal. The logarithmic axis arranges that .5 and 2.0 are equally distant from 1.0. Figure 2 indicates that Equation 1 generally described the data well. The majority of points lie above .9. Out of 103 sets, less than 80% of the variance was accounted for in only eleven sets; less than 70% in only three. The fits to the time ratios and response ratios were equally good. The averages for time and responses were 90.5% and 90.3%. Figure 2 shows also that there was no systematic relation between goodness of fit and slope. Figure 3 shows frequency distributions of the slopes obtained by least squares: the lower for counts of responses, the upper for measures of time. For these distributions and those to follow, the slopes were grouped in equal logarithmic intervals, all .2 in width, and the midpoints used as abscissae. The distributions of Figure 3 show variation to both sides of a single mode. They are 'U

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IS 1 LEAST-SQUARES'SLOPE

I 0.4

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Fig. 2. Proportion of variance accounted for by leastregression using Equation 1 plotted against the fitted slope a. Horizontal lines indicate 80% and 100% of variance accounted for. Note logarithmic x-axis; see text for further explanation.

squares

2

275

MA TCHING, UNDERMA TCHING, AND O VERMA TCHING _

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1 SLOPE FOR RESPONSES

0.4

2

Fig. 4. Scatter plot of slope fitted to ratios of times versus slope fitted to ratios of responses, for all experiments in which the two measurements were made simultaneously. Thirty-eight pairs of data sets are represented. Diagonal line shows the locus of equality. See text for further explanation.

z ILL

w IL.

sets). If local response rates at the two alternatives are equal, we have: PI = T2 SLOPE

Fig. 3. Frequency distributions of the slope a obtained by fitting Equation 1 to the various sets of data. Vertical lines show the location of 1.0, the value required by the matching relation. The two distributions separate studies that measured responses (bottom) and time (top).

slightly skewed in opposite directions. That for responses is weighted toward slopes less than 1.0 (mode = .79; median = .83; geometric mean = .82). Slopes for time varied around 1.0 (mode = 1.00; median = .92; geometric mean = .89). The slopes for time were more variable. The studies in which both types of measure were taken corroborated these differences: times conformed more closely to the matching relation in their slopes' central tendency, but their slopes varied more. These studies produced 38 pairs of data sets and slopes. Figure 4 shows a scatter plot in which each pair's slope for time was graphed against its slope for responses (pecks in all 38

and PI

T

where P1 and P2 are pecks at the two alternatives, and T1 and T2 are the times at the two alternatives. When this condition holds, the slopes for pecks and time must be equal. Figure 4 shows that this equality, sometimes assumed in theoretical treatments of matching (e.g., Rachlin, 1973), commonly fails to apply. That the slope for time was generally greater indicates that local response rates were generally higher at the poorer alternative. In other words, extra responses tended to be made at the poorer alternative for the time spent there. Or, alternatively, extra time was allocated to the better alternative, possibly time spent in activities other than key pecking. On the basis of skew (Figure 3), one could conclude that the measure of time generally conformed to the matching relation, but that one or more common factors make for undermatching. For responses, the skew suggests that undermatching is observed most commonly, but that one or more common factors make for a closer approximation to matching.

276

WILLIAM M. BAUM

linear fits were generally good, the question remains, particularly for the ill-fitted minority of data sets, whether variation around the regression line was in fact unsystematic or was curvilinear. Comparison of the nonparametric fit with the least-squares fit provides a handy way to answer this. Estimation by least squares averages in all data, regardless of whether any outlying points might deviate from the linear model, whereas the nonparametric technique excludes such outliers. As a result, the two fitted slopes differ only when outliers exist that would suggest a curvilinear trend. When variation around the linear model is only unsystematic, the two slopes will be close. Data from a study of mine illustrate the point (Baum, 1973b, Figure 1). One pigeon (496) produced a pattern of points that could be fitted by an S-shaped curve. The data of another pigeon (334) contained considerable unsystematic variance, but revealed no curvilinear trend. Accordingly, the two fitted slopes differ for the first bird (1.11 and 1.28) and are nearly equal for the second (.38 and .39). A search through Table 1 reveals that the ratio of smaller to larger of the two slopes is less than or equal to .9 for only 7 of the 103 data sets. There is little evidence, therefore, of systematic departure from the linear model. Another approach to assessing the deviations of slope from 1.0 focuses on the proportion of variance that the method of least squares 0 can account for (column vi in Table 1). We can ask simply how much we lose in predictability if we assume the slope to equal 1.0. The proportion of variance accounted for by the method of least squares with the slope set * equal to 1.0 appears in column vii of Table 1. 3-J Let us call it H' and the conventional measure 10 (column vi) H. If a line fitted according to 0i 1 has a slope of 1.0, then a leastEquation me squares line with slope set equal to 1.0 must Iceaa CL account for the same proportion of variance as the line with the slope allowed to vary; the loss in predictability (H - H) equals zero. If the slope of the fitted line is less than or greater than 1.0, assuming the slope to equal '.4 1.0 must result in a loss in the proportion of NONPARRAETR IC SLOPE variance accounted for. A negative value of H' Fig. 5. Probability of a true slope of 1.0 (nonpara- indicates that assuming one variable constant metric test of Hollander & Wolfe, 1973, pp. 201-204) (a slope of zero or infinity) produces a better versus the slope estimated by a nonparametric fitting technique (Hollander & Wolfe, 1973, pp. 205-206). The fit than assuming a slope of 1.0. This occurs graph includes 103 points. The horizontal line shows only when the fitted slope is less than .5 or greater than 2.0. probability equal to .05.

SIGNIFICANCE OF THE VARIATIONS The significance of the deviations from 1.0 was examined in two ways: statistical test and estimation of loss in predictability. A nonparametric test was used to assess statistical significance (Hollander & Wolfe, 1973, pp. 200-204). For this, a nonparametric procedure was used to fit a regression line (column vii in Table 1; Hollander & Wolfe, 1973, pp. 205-216). The number of data (column ix) was sometimes less than for the least-square method (column iv) because ordinates with identical abscissae were averaged into a single value. The probability that the slope could equal 1.0 (column x) appears in Figure 5 plotted against the nonparametric-estimated slope. Apart from four or five outliers, the points conform to a roughly symmetrical curve falling away to both sides of a peak at 1.0. A horizontal line indicates the level of probability equal to .05. Points lie above this line for slopes from about .60 to 1.56. No points lie below the line in the range between approximately .90 and 1.20. According to this test, then, slopes between .9 and 1.2 approximate 1.0. Outside this range, unsystematic variation presumably determines whether the slope should be considered distinct from 1.0. Although Figures 2 and 5 suggest that the

a

a

a

a

ma

0-

a

1

MA TCHING, UNDERMA TCHING, AND O VERMA TCHING

277

Figure 6 shows this loss in predictability as a function of the fitted slope a. The close relation between these two arises because the algebraic definitions of a, H, and H' lead to: 1 HI'-HI 2

SOURCES OF UNDERMATCHING Figures 5 and 6 show that, among the slopes clearly different from 1.0, the great majority were less than 1.0. If we seek the source of the variations in slope, our main effort must be a2 a H to account for undermatching. One might take If H is constant-as it is approximately in Ta- the view that slopes generally fall at about .8, ble 1-then H - H' relates to a by a quadratic with random variation to both sides of this value (Myers & Myers, 1977). Alternatively, equation. To interpret Figure 6, one needs a criterion the normal slope may be 1.0, except that cerof acceptable loss. No accepted convention ex- tain systematic factors make for lower values. Figure 7 suggests an origin in systematic ists as it did for Figure 5. Figure 6 can instead serve to test the conclusions drawn from Fig- factors, rather than random variation. It shows ure 5, and the comparison can provide a pos- frequency distributions of the slopes from the sible criterion of acceptable loss. If a best- data of the two major contributors to this fitting slope of .9 (or its reciprocal 1.11) analysis: Davison (and associates) and myself accounts for 90% of the variance (H = .90), an (sometimes with Rachlin). Since Davison's exassumed slope of 1.0 still accounts for 88.9% periments all included measuring both pecks of the variance (H' = .889). If the best-fitting and interchangeover time at each alternative, slope equals 1.2 or .83, the loss grows to .025; Figure 7 shows separate distributions for slopes a line of slope 1.0 accounts for 87.5% of the derived from responses and times. For my data, variance. Assuming that the loss of 1% (H H' = .01) would be generally acceptable, Figures 5 and 6 together suggest that slopes in the range from about .90 to 1.11 can be considered to approximate 1.0. Slopes outside this Li range require further evaluation by techniques z hi like the two suggested. hi 8. U2. a

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Z g,

a hi cc

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a

LEAST-SQUARES SLOPE

Fig. 6. Loss in predictability (proportion of variance accounted for) of the method of least-squares from assuming the matching relation (slope a = 1.0) to be true. See text for further explanation.

SLOPE Fig. 7. Frequency distributions of slopes fitted by the method of least squares. Top: slopes from experiments by Baum (sometimes with Rachlin); middle: slopes fitted to ratios of times from experiments by Davison and associates; bottom: slopes fitted to ratios of responses from experiments by Davison and associates.

278

WILLIAM M. BAUM

only one distribution appears, because the experiments employed one or the other measure: nine slopes represent response counts; thirteen represent times. Davison's norm has been undermatching, whereas mine has been matching. The difference appears systematic, rather than random. Most likely, different customary details of procedure, perhaps unreported, lead to the divergent results. The procedural details may be diverse: level of deprivation, type of reinforcer, construction of the chambers, the intervals and their ordering in the VI schedules, and so on. Yet the behavioral variables through which these affect choice may be few. Three factors that make for undermatching are: (a) asymmetry in pausing at the alternatives, (b) inconstancy of preference through time, and (c) patterns of changeover and brief bouts of re-

sponding. Asymmetrical Pausing A tendency to pause longer after responses at one alternative than another affects the most commonly used measure of time spent at the alternatives: the cumulated interchangeover time. The asymmetry affects only the time measure, unless pausing is tied somehow to extra responses. Asymmetrical pausing adds extra time to the alternative favored for pausing. If this is always the alternative in a particular location, or otherwise physically defined, only bias results. If, however, it is the more highly reinforced alternative, we observe overmatching. In practice, the alternative favored for pausing is usually the one less reinforced. This makes for undermatching. Inconstancy of Preference Signaled changes in contingencies of reinforcement must be expected to occasion corresponding changes in performance. In such procedure, the prototype of which is the multiple schedule, time must be broken into more than one baseline, depending on the number of distinct situations (Baum, 1973a). In FI schedules, although no varied exteroceptive stimuli accompany the varied performances, the temporal regularity of reinforcement suffices to produce a regular alternation of performances. Reinforcement at the end of the FI initiates a period of low response rate that,

after a time, gives way to a period of moderately high responding. Schneider (1969) found that Fl performance could be described as alternating between two states this way, with the result that the Fl schedule functions equivalently to a multiple schedule in which time out (about .67 of the FI) alternates with reinforcement on a VI schedule. A Fl schedule paired concurrently with a VI schedule produces still its characteristic performance (Catania, 1962). Following Schneider's analysis, we expect that conc VIFI will function equivalently to a multiple schedule in which two components alternate: conc VITO and conc VIVI. For performance, we expect that responding will occur exclusively at the VI during the former and will be distributed between the two alternatives during the latter. This characterization can account for undermatching observed in performance on conc VIFI (Lobb & Davison, 1975; Nevin, 1971; Trevett, Davison, & Williams, 1972). The matching relation might hold in both components. In conc VITO, exclusive preference matched with exclusive reinforcement represents a trivial case. In the second component, we might expect the matching relation to hold as well as in any conc VIVI. If one fails to distinguish the two components, however, the combined performance should show both bias and undermatching. If, following each Fl reinforcement, extra time and responses were given to the VI alternative, then the poorer the VI relative to the FI, the more performance will deviate from matching in apparent favor toward the VI. A line fitted to the choices (Fl relative to VI) will lie below the matching line, but converging on it as the VI becomes more favorable relative to the FI. The data bear this out with only one possible exception (Nevin, 1971, Pigeon 59). This line of reasoning applies also to conc VIVI schedules in which the shortest programmed intervals are long enough to engender postreinforcement pausing. If the animal tends to change over after each reinforcer, it will tend to allocate extra time and responses to the poorer schedule. Killeen (1970), studying postreinforcement responding on identical concurrent arithmetic VI schedules, found exactly this tendency: a negative recency effect with respect to the alternative last producing reinforcement. This will produce no bias when

MA TCHING, UNDERMA TCHING, AND O VERMA TCHING

the schedules are identical, but will produce undermatching when they differ. Besides predictable changes in availability of reinforcement, inconstant performance arises also from changes in amount or value of reinforcement. Such changes occur in particular when deprivation varies. When both alternatives produce the same reinforcer, variations that affect its value still allow the matching relation to hold, because changes in overall responding can leave choice unaffected (Baum, 1972, 1974a; Graft, Lea, & Whitworth, 1977). When the alternatives produce different reinforcers, differential satiation leads to deviation from matching. If two qualitatively different reinforcers satiate at the same rate, matching can still occur, because the ratio of their values remains constant. This may have held in Miller's (1976) experiment with different grains, and Hollard and Davison's (1971) with food and brain stimulation. Differing rates of satiation, however, are probably more common than equal ones. In choice between food and water, for example, the availability of either improves the value of the other. Thirsty organisms generally eat less, and hungry ones generally drink less (McFarland & Sibly, 1975). In other words, the more rapidly the animal satiates on water, the more slowly it satiates on food, and vice versa. Increases in the rate of occurrence of one, therefore, will produce increases in responding for the other-the opposite of what occurs in choice between alternatives providing the same reinforcer. The result will be a tendency to reverse the matching relation: as relative rate of reinforcement for (e.g.) water increases, relative responding for water will tend to decrease. Depending on the particular rates and amounts of reinforcement, one can observe any degree of undermatching or even inversion of the relation. I have found such results in as-yet unpublished studies of pigeons, and Hursh (1978) found them with monkeys. Considerable ingenuity will be required to find ways of controlling or correcting for differential satiation.

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alternatives (e.g., Brownstein & Pliskoff, 1968; Pliskoff, 1971; Silberberg & Fantino, 1970). An earlier paper (Baum, 1974b) raised the possibility that any operation that affects rate of changeover affects also the closeness of approximation to the matching relation. If rate of changeover is crucial to choice, it may be so insofar as it reflects the length of bouts of responding at the alternatives. (By a "bout" is meant a period of responding uninterrupted by any other activity.) Higher rates of changeover mean shorter interchangeover intervals. These, in turn, mean briefer bouts of responding. Since the animal can switch to an activity other than the ones that provide programmed reinforcement, the interchangeover interval can only approximate the length of a bout at a programmed alternative. As the interchangeover interval decreases, however, the approximation is likely to improve, because the bout cannot exceed the interchangeover interval; its duration must be less than or equal to the interchangeover interval. Since lower rates of changeover are likely to reflect longer bouts of responding, one might suppose that matching occurs only when the bouts are of adequate duration. In their discussions of the COD, Findley (1958) and Herrnstein (1961) suggested that it affected choice because it reduced the possibility of adventitious reinforcement of chains of responses across alternatives. In other words, too brief a COD may permit patterns of responding that involve two or more alternatives. If a pigeon's pecking at two keys falls into patterns that include both keys (e.g., alternation), the performance effectively combines the two manipulanda into one and reduces the two supposed alternatives to one. By breaking up patterns of responding across alternatives, the COD functionally separates the alternatives and allows choice between them to occur. Indeed, the COD may not only lengthen bouts of responding but also may enforce truly independent bouts at the alternatives. To the extent that the COD fails, the Patterns of Changeover and two alternatives are treated as one, and underBrief Bouts of Responding matching occurs. Longer CODs naturally sucSeveral experiments show that a longer ceed more often than short ones. changeover delay (COD) makes for a better CONCLUSION approximation to matching (e.g., Shull & Pliskoff, 1967). Lengthening the COD also Doubtless other factors than the three disreduces the rate of changing over between cussed will emerge in time. If we inquire

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further into the conditions that make for undermatching, we may come not only to understand the requirements for matching, but we may also discover how to model choice under laboratory conditions as it occurs in nature. REFERENCES Baum, W. M. Choice in a continuous procedure. Psychonomic Science, 1972, 28, 263-265. Baum, W. M. The correlation-based law of effect. Journal of the Experimental Analysis of Behavior, 1973, 20, 137-153. (a) Baum, W. M. Time allocation and negative reinforcement. Journal of the Experimental Analysis of Behavior, 1973, 20, 313-322. (b) Baum, W. M. Choice in free-ranging wild pigeons. Science, 1974, 185, 78-79. (a) Baum, W. M. Chained concurrent schedules: Reinforcement as situation transition. Journal of the Experimental Analysis of Behavior, 1974, 22, 91-

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