The Faulty Magnitude Detector: Why SNARC‐Like Tasks Cannot Support a Generalized Magnitude System

Do people represent space, time, number, and other conceptual domains using a generalized magnitude system (GMS)? To answer this question, numerous studies have used the spatial‐numerical association of response codes (SNARC) task and its variants. Yet, for a combination of reasons, SNARC‐like effects cannot provide evidence for a GMS, even in principle. Rather, these effects support a broader theory of how people use space metaphorically to scaffold their understanding of myriad non‐spatial domains, whether or not these domains exhibit variation in magnitude.     

Fixed-ratio pausing: Joint effects of past reinforcer magnitude and stimuli correlated with upcoming magnitude

Pigeons responded on fixed-ratio schedules ending in small or large reinforcers (grain presentations of different duration) interspersed within each session. In mixed-schedule conditions, the response key was lit with a single color throughout the session, and pausing was directly related to the past reinforcer (longer pauses after large reinforcers than after small ones). In multiple-schedule conditions, different colors accompanied the ratios ending in small and large reinforcers, and pausing was affected by the upcoming reinforcer as well as the past one. Pauses were shorter before large reinforcers than before small ones, but they continued to be longer after large reinforcers than after small ones. The influence of the past reinforcer was modulated by the magnitude of the upcoming reinforcer; in the presence of the stimulus before the small reinforcer, the effect of the past reinforcer was enhanced relative to its effect in the stimulus before the large reinforcer. These results show that pausing between ratios is jointly determined by two competing factors: past conditions of reinforcement and stimuli correlated with upcoming conditions.     

Handling time and choice in pigeons

According to optimal foraging theory, animals should prefer food items with the highest ratios of energy intake to handling time. When single items have negligible handling times, one large item should be preferred to a collection of small ones of equivalent total weight. However, when pigeons were offered such a choice on equal concurrent variable-interval schedules in a shuttlebox, they preferred the side offering many small items per reinforcement to that offering one or a few relatively large items. This preference was still evident on concurrent fixed-cumulative-duration schedules in which choosing the alternative with longer handling time substantially lowered the rate of food intake.     

Choice between reliable and unreliable outcomes: mixed percentage-reinforcement in concurrent chains.

Pigeons' choices between alternatives that provided different percentages of reinforcement in mixed schedules were studied using the concurrent-chains procedure. In Experiment 1, the alternatives were terminal-link schedules that were equal in delay and magnitude of reinforcement, but that provided different percentages of reinforcement, with one schedule providing, reinforcement twice as reliably as the other. All pigeons preferred the more reliable schedule, and their level of preference was not systematically affected by variation in the absolute percentage values, or in the magnitude of reinforcement. In Experiment 2, preference for a schedule providing 100% reinforcement over one providing 33% reinforcement increased systematically with increases in the duration of the terminal links. In contrast, preference decreased systematically with increases in the duration of the initial links. Experiment 3 examined choice with equal percentages of reinforcement but unequal delays to reinforcement. Preference for the shorter delay to reinforcement was not systematically affected by variation in the absolute percentage of reinforcement. The overall pattern of results supported predictions based on an extension of the delay-reduction hypothesis to choice procedures involving mixed schedules of percentage reinforcement.     

Unit price as a useful metric in analyzing effects of reinforcer magnitude.

In this paper, we applied the behavioral-economic concept of unit price to the study of reinforcer magnitude in an attempt to provide a consistent account of the effects of reinforcer magnitude on behavior. Recent research in the experimental analysis of behavior and in behavioral pharmacology suggests that reinforcer magnitude interacts with the schedule of reinforcement to determine response rate and total consumption. The utility of the unit-price concept thus stems from its ability to quantify this interaction as a cost-benefit ratio (i.e., unit price = characteristics of the schedule of reinforcement divided by magnitude of reinforcement). Research employing the unit-price concept has shown that as unit price increases, a positively decelerating function exists for consumption (i.e., a function with an increasingly negative slope, when plotted on log coordinates) and a bitonic function exists for response rate. Based on these findings, the present analysis applied the unit-price concept to those studies of reinforcer magnitude and drug self-administration that examined the effects of reinforcer magnitude on response rate using simple schedules of reinforcement (e.g., fixed-ratio schedule). This resulted in three findings: (a) Reinforcer-magnitude manipulations and schedule manipulations interact in a manner that can be quantified in terms of unit price as benefit and cost factors, respectively; (b) different reinforcer-magnitude manipulations are functionally interchangeable as benefit factors in the unit-price ratio; and (c) these conclusions appear warranted despite the differences in reinforcers (food or drug), species (dogs, monkeys, or rats), and schedules (interval or ratio), and despite the fact that these studies were not designed for a unit-price analysis. In methodological terms, these results provide further evidence that employing the unit-price concept is a parsimonious method for examining the effects of reinforcer magnitude. In theoretical terms, these results suggest that a single process may underlie the effect of combined reinforcer-magnitude and schedule manipulations.     

Violations of stochastic transitivity on concurrent chains: Implications for theories of choice

The concurrent-chains procedure has been used to measure how choice depends on various aspects of reinforcement, such as its delay and its magnitude. Navarick and Fantino (1972, 1974, 1975) have found that choice in this procedure can violate the condition of stochastic transitivity that is required if a unidimensional scale for reinforcements is to be possible. It is shown in this paper that two simple unidimensional models of choice on concurrent chains can produce violations of stochastic transitivity. It is argued that such violations may result from the complex contingencies of the concurrent-chains procedure.     

Behavioral economics of drug self-administration. II. A unit-price analysis of cigarette smoking.

In behavioral economics, the ratio of response requirement to reinforcer magnitude is referred to as unit price. Previous research with nonhuman subjects has demonstrated that (a) comparable amounts of food are consumed at the same unit price even though different response requirements and reinforcer magnitudes comprise that unit price and (b) increases in unit price decrease food consumption in a positively decelerating fashion. The present study assessed the generality of these findings to the cigarette smoking of 5 human volunteers. During approximately 18 3-hr sessions, various combinations of response requirement (fixed-ratio 200, 400, and 1,600) and reinforcer magnitude (1, 2, and 4 puffs per bout) were arranged. Consumption (i.e., the number of puffs) generally was comparable at the same unit price independent of the response requirement and reinforcer magnitude comprising that unit price. In addition, increasing unit price generally decreased consumption in a positively decelerating fashion. These results extend the generality of the unit-price analysis to human cigarette smoking. Moreover, these results further support the position that reinforcer magnitude and response requirement are functionally equivalent and interact to determine consumption. The concept of unit price, by integrating and summarizing the effects of those two operations, provides a more parsimonious explanation of the results than do separate evaluations of the effects of response requirement and reinforcer magnitude.     

Reinforcement magnitude and the inhibiting effect of reinforcement

In a two-key concurrent variable-interval schedule (using pigeons), if the reinforcement frequency for one response is held constant while that for the other is increased, the rate of response on the constant key decreases. The immediate reinforcement for key pecking can usually be conceptualized as the change from a condition in which the key light is on and the food hopper light is off to one in which the key light is off and the hopper light is on. The prechange condition is associated with a delay to food of one-half the average interreinforcement interval in effect during this condition. The postchange condition is associated with a delay to food of about .5 seconds. The programming of additional reinforcement results in a decrease in the delay to food associated with the prechange stimulus condition, and thus a decrease in the value of the improvement that results from the change. This would appear to be analogous to a decrease in the amount of reinforcement, and thus sufficient explanation for the decrease in the rate of the response.     

Temporal control of behavior and the power law

The performance of rats and pigeons under fixed-interval schedules was studied in two experiments. The duration of postreinforcement pause was a declining proportion of fixed-interval duration. For pigeons this was true both when the duration of the reinforcer was fixed and when it was increased in direct proportion to increases in fixed-interval duration; the longer reinforcer durations did, however, lengthen the postreinforcement pause at higher schedule values. A quantitative analysis of data from Experiments 1 and 2 and from other studies showed that fractional exponent power functions described the relationship between postreinforcement pause and fixed-interval value; similar functions have previously been observed in studies of temporal differentiation. It was concluded that power functions reflect a direct causal, rather than artifactual, relationship between performance and the temporal requirements of reinforcement schedules.     

Effects of food deprivation and reinforcement magnitude on conditioned suppression

In Experiment I, the responding of rats lever pressing on a variable-interval schedule for sucrose solution was partially suppressed by a variable duration conditioned stimulus followed by shock. When food deprivation was increased, response rates during and before the conditioned stimulus increased monotonically. Varying the concentration of sucrose across blocks of sessions or from session to session in a semi-random sequence had little effect on response rates either before or during the conditioned stimulus. With a fixed sequence of increasing concentrations across a five-session block, increased concentration produced much more rapid increases in response rates before than during the conditioned stimulus. In Experiment II, rats were presented with the same sequence of increasing concentrations across a five-session block. When tested at 80% body weight, response rates increased rapidly as concentration increased, but at 100%, body-weight rates increased only slightly. The effect of a change in body weight in Experiment II thus mimicked the effect of the conditioned stimulus in the latter part of Experiment I. These findings support the view that the effect of a pre-aversive conditioned stimulus is similar to that of a change in food deprivation, but unlike that of a change in reinforcement magnitude.