An analytic comparison of Herrnstein's equations and a multivariate rate equation

Herrnstein's equations are approximations of the multivariate rate equation at ordinary rates of reinforcement and responding. The rate equation is the result of a linear system analysis of variable-interval performance. Rate equation matching is more comprehensive than ordinary matching because it predicts and specifies the nature of concurrent bias, and predicts a tendency toward undermatching, which is sometimes observed in concurrent situations. The rate equation contradicts one feature of Herrnstein's hyperbola, viz., the theoretically required constancy of k. According to the rate equation, Herrnstein's k should vary directly with parameters of reinforcement such as amount or immediacy. Because of this prediction, the rate equation asserts that the conceptual framework of matching does not apply to single alternative responding. The issue of the constancy of k provides empirical grounds for distinguishing between Herrnstein's account and a linear system analysis of single alternative variable-interval responding.     

Undermatching on concurrent variable-interval schedules and the power law.

The phenomenon of undermatching on concurrent variable-interval schedules is shown to be derivable by transforming the individual interreinforcement intervals of each variable-interval schedule and averaging the transformed values to produce an "estimate" of the rate of reinforcement the schedules deliver. If the transformation is based on a power function with a fractional exponent, such as is found in many studies of temporal control in animals, matching response rations to the ratios of these estimated rates of reinforcement yields undermatching. If the concurrent variable-interval schedules are arranged such that the individual intervals in each schedule have a constant proportionality (a procedure found in many commonly used variable-interval schedules) the slope of the line relating logarithms of response ratios and of programmed reinforcement ratios is identical to the exponent of the power transformation applied to the individual time intervals in the variable-interval schedules. In other cases this simple relation does not hold but the degree of undermatching is greater the lower the value of the exponent of the power function. This account of undermatching predicts values similar to those typically observed.     

Choice between response units: The rate constancy model

In a conjoint schedule, reinforcement is available simultaneously on two or more schedules for the same response. The present experiments provided food for key pecking on both a random-interval and a differential-reinforcement-of-low-rate (DRL) schedule. Experiment 1 involved ordinary DRL schedules; Experiment 2 added an external stimulus to indicate when the required interresponse time had elapsed. In both experiments, the potential reinforcer frequency from each component was varied by means of a second-order fixed-ratio schedule, and the DRL time parameter was changed as well. Response rates were described by a model stating that time allocation to each component matches the relative frequency of reinforcement for that component. When spending time in a given component, the subject is assumed to respond at the rate characteristic of baseline performance. This model appeared preferable to the absolute-rate version of the matching law. The model was shown to be applicable to multiple-response concurrent schedules as well as to conjoint schedules, and it described some of the necessary conditions for response matching, undermatching, and bias. In addition, the pigeons did not optimize reinforcer frequency.     

Choice: Some quantitative relations

Six pigeons responded in fifty-six conditions on a concurrent-chains procedure. Conditions included several with equal initial links and unequal terminal links, several with unequal initial links and equal terminal links, and several with both unequal initial and terminal links. Although the delay-reduction hypothesis accounted well for choice when the initial links were equal (mean deviation of .04), it fit the data poorly when the initial links were unequal (mean deviation of .18). A modification of the delay-reduction hypothesis, replacing the rates of reinforcement with the square roots of these rates, fit the data better than either the unmodified delay-reduction equation or Killeen's (1982) model. The modified delay-reduction equation was also consistent with data from prior studies using concurrent chains. The absolute rates of responding in each terminal link were well described by the same hyperbola (Herrnstein, 1970) that describes response rates on simple interval schedules.     

Relative sensitivity to reinforcer amount and delay in a self-control choice situation.

Rats were exposed to concurrent-chains schedules in which a single variable-interval schedule arranged entry into one of two terminal-link delay periods (fixed-interval schedules). The shorter delay ended with the delivery of a single food pellet; the longer day ended with a larger number of food pellets (two under some conditions and six under others). In Experiment 1, the terminal-link delays were selected so that under all conditions the ratio of delays would exactly equal the ratio of the number of pellets. But the absolute duration of the delays differed across conditions. In one condition, for example, rats chose between one pellet delayed 5 s and six pellets delayed 30 s; in another condition rats chose between one pellet delayed 10 s and six pellets delayed 60 s. The generalized matching law predicts indifference between the two alternatives, assuming that the sensitivity parameters for amount and delay of reinforcement are equal. The rats' choices were, in fact, close to indifference except when the choice was between one pellet delayed 5 s and six pellets delayed 30 s. That deviation from indifference suggests that the sensitivities to amount and delay differ from each other depending on the durations of the delays. In Experiment 2, rats chose between one pellet following a 5-s delay and six pellets following a delay that was systematically increased over sessions to find a point of indifference. Indifference was achieved when the delay to the six pellets was approximately 55 s. These results are consistent with the possibility that the relative sensitivities to amount and delay differ as a function of the delays.     

Increasing and signaling background reinforcement: effect on the foreground response-reinforcer relation.

Herrnstein's (1970) hyperbolic matching equation describes the relationship between response rate and reinforcement rate. It has two estimated parameters, k and Re. According to one interpretation, k measures motor performance and Re measures the efficacy of the reinforcer maintaining responding relative to background sources of reinforcement. Experiment 1 tested this interpretation of the Re parameter by observing the effect of adding and removing an additional source of reinforcement to the context. Using a within-session procedure, estimates of Re were obtained from the response-reinforcer relation over a series of seven variable-interval schedules. A second, concurrently available variable-interval schedule of reinforcement was added and then removed from the context. Results showed that when the alternative was added to the context, the value of Re increased by 107 reinforcers per hour; this approximated the 91 reinforcers per hour obtained from this schedule. Experiment 2 investigated the effects of signaling background reinforcement on k and Re. The signal decreased Re, but did not have a systematic effect on k. In general, the results supported Herrnstein's interpretation that in settings with one experimenter-controlled reinforcement source, Re indexes the strength of the reinforcer maintaining responding relative to uncontrolled background sources of reinforcement.     

Ethical reflections on the status of the preimplantation embryo leading to the German embryo protection act

Ethical conflicts have always been connected with new techniques of reproductive medicine such as in-vitro fertilization. The fundamental question is: When does human life begin and from which stage of development should the embryo be protected? This question cannot be solved by scientific findings only. In prenatal ontogenesis there is no moment during the development from the fertilized oocyte to a human being which could be recognized as an orientation point for all ethical problems connected with the question of the right to dispose of prenatal life (interruption of pregnancy, research on embryos). The protection of an individual human life must be valid for all in the same manner and from the very beginning on. It must not depend on value judgments or on stages of development, so-called grades of humanity, because these would then become selection criteria. Procreated life must have the right to be born. These were the essential ethical arguments which led to the German Embryo Protection Law.     

How to teach a pigeon to maximize overall reinforcement rate

In two experiments deviations from matching earned higher overall reinforcement rates than did matching. In Experiment 1 response proportions were calculated over a 360-response moving average, updated with each response. Response proportions that differed from the nominal reinforcement proportions, by a criterion that was gradually increased, were eligible for reinforcement. Response proportions that did not differ from matching were not eligible for reinforcement. When the deviation requirement was relatively small, the contingency proved to be effective. However, there was a limit as to how far response proportions could be pushed from matching. Consequently, when the deviation requirement was large, overall reinforcement rate decreased and pecking was eventually extinguished. In Experiment 2 a discriminative stimulus was added to the procedure. The houselight was correlated with the relationship between response proportions and the nominal (programmed) reinforcement proportions. When the difference between response and reinforcement proportions met the deviation requirement, the light was white and responses were eligible for reinforcement. When the difference between response and reinforcement proportions failed to exceed the deviation requirement, the light was blue and responses were not eligible for reinforcement. With the addition of the light, it proved to be possible to shape deviations from matching without any apparent limit. Thus, in Experiment 2 overall reinforcement rate predicted choice proportions and relative reinforcement rate did not. In contrast, in previous experiments on the relationship between matching and overall reinforcement maximization, relative reinforcement rate was usually the better predictor of responding. The results show that whether overall or relative reinforcement rate better predicts choice proportions may in part be determined by stimulus conditions.