Handbook of Psychology, Volume 4: Experimental Psychology

384 Conditioning and Learning

constant (B+Be= k), and are maintained by rates of sched-
uled and extraneous reinforcement (RandRe), respectively.
Based on the second assumption,

 ⇒ B (13.3)

Equation 13.3 defines a hyperbola, with two parameters, k
andRe. The denominator represents the context of reinforce-
mentfor a particular response—the total amount of rein-
forcement in the situation. De Villiers and Herrnstein (1976)
fit Equation 13.3 to a large number of data sets and found that
it generally gave an excellent description of response rates
under VI schedules. Subsequent research has generally con-
firmed the hyperbolic relation between response rate and
reinforcement rate, although lower-than-predicted response
rates are sometimes observed at very high reinforcement
rates (Baum, 1993). In addition, Equation 13.3 has been
derived from a number of different theoretical perspectives
(Killeen, 1994; McDowell & Kessel, 1979; Staddon, 1977).
Herrnstein (1970) also developed a version of the match-
ing law that was applicable to multiple schedules. In a multi-
ple schedule, access to two (or more) different schedules
occur successively and are signaled by discriminative stim-
uli. A well-known result in multiple schedules is behavioral
contrast:Response rate in a component that provides a con-
stant rate of reinforcement varies inversely with the rein-
forcement rate in the other component (see B. A. Williams,
1983, for review). Herrnstein suggested that the reinforce-
ment rate in the alternative component served as part of
the reinforcement context for behavior in the constant

kR



RRe

R



RRe

B



BBe

component. However, the contribution of alternative compo-

nent reinforcement was attenuated by a parameter (m), which

describes the degree of interaction at a temporal distance,

B 1  (13.4)

with subscripts referring to the components of the multiple

schedule. Equation 13.4 correctly predicts most behavioral

contrast, but has difficulties with some other phenomena (see

McLean & White, 1983, for review). Alternative models for

multiple-schedule performance also based on the matching

law have been proposed that alleviate these problems

(McLean, 1995; McLean & White, 1983; B. A. Williams &

Wixted, 1986).

Matching to Relative Value. The effects of variables

other than reinforcement rate were examined in several early

studies, which found that response allocation in concurrent

schedules obeyed the matching relation when magnitude

(i.e., seconds of access to food; Catania, 1963) and delay of

reinforcement (Chung & Herrnstein, 1967) were varied.

Baum and Rachlin (1969) then proposed that the matching

law might apply most generally to reinforcement value,with

value being defined as a multiplicative combination of rein-

forcement parameters,

   (13.5)

withMbeing reinforcement magnitude, Dbeing delay, and V

being value.

Equation 13.5 represents a significant extension of the

matching law, enabling it to apply to a broader range of

choice situations (note that frequently a generalized version

of Equation 13.5 with exponents, analogous to Equa-

tion 13.2, has been used here; e.g., Logue, Rodriguez, Pena-

Correal, & Mauro, 1984). One of the most important of these

isself-control, which has been a major focus of research

because of its obvious relevance for human behavior. In a

self-control situation, subjects are confronted with a choice

between a small reinforcer available immediately (or after a

short delay), and a larger reinforcer available after a longer

delay. Typically, overall reinforcement gain is maximized by

choosing the delayed, larger reinforcer, which is defined as

self-control (Rachlin & Green, 1972; see Rachlin, 1995, for

review). By contrast, choice of the smaller, less delayed rein-

forcer is termed impulsivity. For example, if pigeons are

given a choice between a small reinforcer (2-s access to

grain) delayed by 1 s or a large reinforcer (6-s access to grain)

delayed by 6 s, then Equation 13.5 predicts that 67% of

the choice responses will be for the small reinforcer (i.e., the

VL



VR

ML



MR

1 DL



1 DR

RL



RR

BL



BR

kR 1



R 1 mR 2 Re

120

80

40

0

80

40

0

80

40

0

0 50 100 150 200 250 300 0 50 100 150 200250300

RESPONSES/MINUTE

REINFORCEMENTS/HOUR

115, 8.5 100, 300

70, 4 75, 2

69, 11 68, 5

Figure 13.3 Response rate as a function of reinforcement rate for six
pigeons responding under VI schedules. The numbers in each panel are the
estimates of kandRefor fits of Equation 13.3. Source:From Herrnstein
(1970).