Instrumental Responding383Several studies have attempted to test the prediction of
melioration that local reinforcement rates determine prefer-
ence by arranging two pairs of concurrent schedules within
each session and then testing preference for stimuli between
pairs from different concurrent schedules in probe tests. For
example, B. A. Williams and Royalty (1989) conducted
several experiments in which probes compared stimuli corre-
lated with different local and overall reinforcement rates.
However, they found that the overall, not local, reinforce-
ment rates correlated with stimuli-determined preference in
the probes. In a similar study, Belke (1992) arranged a pro-
cedure with VI 20-s, VI 40-s schedules in one component
andVI 40-s, VI 80-s schedules in the other component.
Afterbaseline training, pigeons’preference approximately
matched relative reinforcement rate in both components (i.e.,
a 2:1 ratio). Belke then presented the two VI 40-s stimuli
together in occasional choice probes. The pigeons demon-
strated a strong (4:1) preference for the VI 40-s stimulus
paired with the VI 80-s. This result is contrary to the predic-
tions of melioration, because the VI 40-s paired with VI 20-s
is correlated with a greater local reinforcement rate (see also
Gibbon, 1995).
Gallistel and Gibbon (2000) have argued that the results of
Belke (1992) pose a serious challenge not only to meliora-
tion, but also to the matching law as empirical support for the
law of effect. They described a model for instrumental choice
that was based on Gibbon (1995; see also Mark & Gallistel,
1994). According to their model, pigeons learn the interrein-
forcement intervals for responding on each alternative and
store these intervals in memory. Decisions to switch from one
alternative to another are made by a sample-and-comparison
process that operates on the stored intervals. They showed
that their model could predict Belke’s (1992) and Gibbon’s
(1995) probe results. However, these data may not be deci-
sive evidence against melioration, or indeed against any the-
ory of matching. According to Gallistel and Gibbon, when
separately trained stimuli are paired in choice probes, the
same changeover patterns that were established in baseline
training to particular stimuli are carried over. If carryover of
baseline can account for probe preference, then the probes
provide no new information beyond baseline responding. The
implication is that any theory that can account for matching
in baseline can potentially explain the probe results of Belke
(1992) and Gibbon (1995).
Extensions of the Matching Law
Generalized Matching.Since Herrnstein’s (1961) orig-
inal study, the matching law has been extended in several
ways to provide a quantitative framework for describing data
from various procedures. Baum (1974) noted that some devi-
ations from the strict equality of response and reinforcement
ratios required by the matching law could be described by
Equation 13.2, a power function generalization of Equa-
tion13.1:b
a
(13.2)Equation 13.2 is known as the generalized matching
law.There are two parameters: bias (b), which represents a
constant proportionality in responding unrelated to rein-
forcement rate (e.g., position preference); and an exponent
(a), which represents sensitivity to reinforcement rate. Typi-
cally, a logarithmic transformation of Equation 13.2 is used,
resulting in a linear relation in which sensitivity and bias
correspond to the slope and intercept, respectively. Baum
(1979) reviewed over 100 data sets and found that the gener-
alized matching law commonly accounted for over 90% of
the variance in behavior allocation (for a review of compara-
ble human research, see Kollins, Newland, & Critchfield,
1997). Thus, in the generalized form represented in Equation
13.2, the matching law provides an excellent description of
choice in concurrent schedules. Although undermatching
(i.e.,a<1) is the most common result, this may result from
a variety of factors, including imperfect discriminability of
the contingencies (Davison & Jenkins, 1985).Matching in Single and Multiple Schedules. If the law
of effect is a general principle of behavior, and the matching
law is a quantitative expression of the law of effect, then the
matching principle should apply to situations other than con-
current schedules. Herrnstein (1970) proposed an extension
of the matching law that applied to single and multiple
schedules. His starting point was Catania and Reynolds’
(1968) data showing that response rate was an increasing,
negatively accelerated function of reinforcement rate on
single VI schedules (see Figure 13.3).
Herrnstein (1970) reasoned that when a single schedule
was arranged, a variety of behaviors other than the target
response were available to the organism (e.g., grooming, pac-
ing, defecating, contemplation). Presumably, these so-called
extraneous behaviors were maintained by extraneous (i.e.,
unmeasured) reinforcers. Herrnstein then made two assump-
tions: (a) that the total amount of behavior in any situation
was constant—that is, the frequencies of target and extrane-
ous behaviors varied inversely; and (b) that “all behavior is
choice” and obeys the matching law. The first assumption
implies that the target and extraneous response rates sum to aRL
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