Search

Three mathematical models of choice--the contextual-choice model (R. Grace, 1994), delay-reduction theory (N. Squires & E. Fantino, 1971), and a new model called the hyperbolic value-added model--were compared in their ability to predict the results from a wide variety of experiments with animal subjects. When supplied with 2 or 3 free parameters, all 3 models made fairly accurate predictions for a large set of experiments that used concurrent-chain procedures. One advantage of the hyperbolic value-added model is that it is derived from a simpler model that makes accurate predictions for many experiments using discrete-trial adjusting-delay procedures. Some results favor the hyperbolic value-added model and delay-reduction theory over the contextual-choice model, but more data are needed from choice situations for which the models make distinctly different predictions.

Two experiments with pigeons used concurrent-chain procedures with variable-interval schedules as initial links and different delays to food as terminal links. Two schedules were present in all sessions, but a 3rd schedule was alternately present and absent in successive sessions. When the 3rd schedule delivered food with no terminal-link delay, the presence of this schedule led to an increase in preference for the schedule with the shorter terminal link of the 2 unchanged schedules. When the terminal-link delay for the 3rd schedule was 30 s, the presence of this schedule led to a decrease in preference for the schedule with the shorter terminal link of the 2 unchanged schedules. These results are inconsistent with the predictions of R. Grace's (1994) contextual-choice model, but they are consistent with 2 other mathematical models--delay-reduction theory and the hyperbolic value-added model.

Pigeons were presented with a concurrent-chains schedule in which terminal-link entries were assigned to two response keys on a percentage basis. The terminal links were fixed delays that sometimes ended with food and sometimes did not. In most conditions, 80% of the terminal links were assigned to one key, but a smaller percentage of the terminal links ended with food for this key, so the number of food reinforcers delivered by the two alternatives was equal. When the same terminal-link stimuli (orange houselights) were used for both alternatives, the pigeons showed a preference for whichever alternative delivered more frequent terminal links. When different terminal-link stimuli (green vs. red houselights) were used for the two alternatives, the pigeons showed a preference for whichever alternative delivered fewer terminal links when terminal-link durations were long, and no systematic preferences when terminal-link durations were short. This pattern of results was consistent with the predictions of Grace's (1994) contextual choice model. Preference for the alternative that delivered more frequent terminal links was usually stronger in the first few sessions of a condition than at the end of a condition, suggesting that the conditioned reinforcing effect of the additional terminal-link presentation was, in part, transitory.

In Experiment 1, pigeons' pecks on a green key led to a 5-s delay with green houselights, and then food was delivered on 20% (or, in other conditions, 50%) of the trials. Pecks on a red key led to an adjusting delay with red houselights, and then food was delivered on every trial. The adjusting delay was used to estimate indifference points: delays at which the two alternatives were chosen about equally often. Varying the presence or absence of green houselights during the delays that preceded possible food deliveries had large effects on choice. In contrast, varying the presence of the green or red houselights in the intertrial intervals had no effects on choice. In Experiment 2, pecks on the green key led to delays of either 5 s or 30 s with green houselights, and then food was delivered on 20% of the trials. Varying the duration of the green houselights on nonreinforced trials had no effect on choice. The results suggest that the green houselights served as a conditioned reinforcer at some times but not at others, depending on whether or not there was a possibility that a primary reinforcer might be delivered. Given this interpretation of what constitutes a conditioned reinforcer, most of the results were consistent with the view that the strength of a conditioned reinforcer is inversely related to its duration.

Two experiments studied the phenomenon of procrastination, in which pigeons chose a larger, more delayed response requirement over a smaller, more immediate response requirement. The response requirements were fixed-interval schedules that did not lead to an immediate food reinforcer, but that interrupted a 55-s period in which food was delivered at random times. The experiments used an adjusting-delay procedure in which the delay to the start of one fixed-interval requirement was varied over trials to estimate an indifference point--a delay at which the two alternatives were chosen about equally often. Experiment 1 found that as the delay to a shorter fixed-interval requirement was increased, the adjusting delay to a longer fixed-interval requirement also increased, and the rate of increase depended on the duration of the longer fixed-interval requirement. Experiment 2 found a strong preference for a fixed delay of 10 s to the start of a fixed-interval requirement compared to a mixed delay of either 0 or 20 s. The results help to distinguish among different equations that might describe the decreasing effectiveness of a response requirement with increasing delay, and they suggest that delayed reinforcers and delayed response requirements have symmetrical but opposite effects on choice.

Choice responding refers to the manner in which individuals allocate their time or responding among available response options. In this article, we first review basic investigations that have identified and examined variables that influence choice responding, such as response effort and reinforcement rate, immediacy, and quality. We then describe recent bridge and applied studies that illustrate how the results of basic research on choice responding can help to account for human behavior in natural environments and improve clinical assessments and interventions.

In two experiments with pigeons, a single variable-interval schedule assigned reinforcers to two response keys on a percentage basis. The percentage of reinforcers assigned to each key was changed every few sessions, and subjects' choice responses were recorded before and after each change. In Experiment 1, the overall rate of reinforcement was varied across conditions. The pigeons' choice responses adapted more quickly to a change in the reinforcement percentages when the overall reinforcement rates were higher, but acquisition rates varied by only about a factor of 3, whereas reinforcement rates were varied by about a factor of 9. In Experiment 2, the reinforcement percentages changed about every 8 sessions in Phases 1 and 3, but every 1 or 2 sessions in Phase 2. Pigeons' choice responses adapted to a change in reinforcement percentages more quickly in Phase 2 than in Phases 1 and 3. The results from both experiments pose difficulties for several prominent models of transitional choice behaviour. The results suggest that each successive reinforcer has more impact on a subject's subsequent choice behaviour when the overall rate of reinforcement is lower and when the reinforcement contingencies have changed frequently in the recent past.

A discrete-trials adjusting-delay procedure was used to investigate the conditions under which pigeons might show a preference for partial reinforcement over 100% reinforcement, an effect reported in a number of previous experiments. A peck on a red key always led to a delay with red houselights and then food. In each condition, the duration of the red-houselight delay was adjusted to estimate an indifference point. In 100% reinforcement conditions, a peck on a green key always led to a delay with green houselights and then food. In partial-reinforcement conditions, a peck on the green key led either to the green houselights and food or to white houselights and no food. In some phases of the experiment, statistically significant preference for partial reinforcement over 100% reinforcement was found, but this effect was observed in only about half of the pigeons. The effect was largely eliminated when variability in the delay stimulus colors was equated for 50% reinforcement conditions and 100% reinforcement conditions. Idiosyncratic preferences for certain colors or for stimulus variability may be at least partially responsible for the effect.

In an adjusting-delay choice procedure, pigeons could peck on either a red key or a green key. A peck on the red key always led to a delay associated with red houselights and then food. The delay was adjusted over trials to estimate an indifference point--a delay at which the two keys were chosen about equally often. In some conditions, a peck on the green key led to food on all trials after delays of either 10 s or 30 s, and green houselights were lit during the delays. In other conditions, food was presented on only half of the green-key trials. If the green houselights continued to occur on both reinforcement and nonreinforcement trials, preference for the green key always decreased. Preference for the green key also decreased if half of the trials had 30-s houselights followed by food and the other half had no green houselights and no food. However, preference for the green key actually increased if half of the trials had 10-s green houselights followed by food and the other half had no green houselights followed by no food. The latter condition therefore demonstrated a case in which preference for an alternative increased when food was removed from half of the trials. The results suggest that the red and green houselights served as conditioned reinforcers. A hyperbolic decay model (Mazur, 1989) provided good predictions for all conditions by assuming that the strength of a conditioned reinforcer is inversely related to the total time spent in its presence before food is delivered.

In three experiments, pigeons chose between a small amount of food delivered after a short delay and a larger amount delivered after a longer delay. A discrete-trial adjusting-delay procedure was used to estimate indifference points--pairs of delay-amount combinations that were chosen about equally often. In Experiment 1, when additional reinforcers were available during intertrial intervals on a variable-interval schedule, preference for the smaller, more immediate reinforcer increased. Experiment 2 found that this shift in preference occurred partly because the variable-interval schedule started sooner after the smaller, more immediate reinforcer, but there was still a small shift in preference when the durations and temporal locations of the variable-interval schedules were identical for both alternatives. Experiment 3 found greater increases in preference for the smaller, more immediate reinforcer with a variable-interval 15-s schedule than with a variable-interval 90-s schedule. The results were generally consistent with a model that states that the impact of any event that follows a choice response declines according to a hyperbolic function with increasing time since the moment of choice.

In Experiment 1, the choice responses of 8 pigeons were observed during 50 periods of transition. Each condition began with equal probabilities of reinforcement on 2 response keys and switched to unequal probabilities. With the ratio of the 2 probabilities held constant, preference for the higher probability developed more rapidly when the 2 probabilities were high than when they were low. In Experiment 2, each condition began with 2 equal variable-interval schedules, but later 1 key delivered 60%, 75%, or 90% of the reinforcers. The rate of approach to asymptotic performance was roughly the same with all 3 reinforcement percentages. These and previous results pose difficulties for some well-known models of acquisition, but the results are well described by a simple model that states that the strength of each response is independently increased by reinforcement and decreased by nonreinforcement.

Two experiments measured pigeons' choices between probabilistic reinforcers and certain but delayed reinforcers. In Experiment 1, a peck on a red key led to a 5-s delay and then a possible reinforcer (with a probability of .2). A peck on a green key led to a certain reinforcer after an adjusting delay. This delay was adjusted over trials so as to estimate an indifference point, or a duration at which the two alternatives were chosen about equally often. In all conditions, red houselights were present during the 5-s delay on reinforced trials with the probabilistic alternative, but the houselight colors on nonreinforced trials differed across conditions. Subjects showed a stronger preference for the probabilistic alternative when the houselights were a different color (white or blue) during the delay on nonreinforced trials than when they were red on both reinforced and nonreinforced trials. These results supported the hypothesis that the value or effectiveness of a probabilistic reinforcer is inversely related to the cumulative time per reinforcer spent in the presence of stimuli associated with the probabilistic alternative. Experiment 2 tested some quantitative versions of this hypothesis by varying the delay for the probabilistic alternative (either 0 s or 2 s) and the probability of reinforcement (from .1 to 1.0). The results were best described by an equation that took into account both the cumulative durations of stimuli associated with the probabilistic reinforcer and the variability in these durations from one reinforcer to the next.

Ten acquisition curves were obtained from each of 4 pigeons in a two-choice discrete-trial procedure. In each of these 10 conditions, the two response keys initially had equal probabilities of reinforcement, and subjects' choice responses were about equally divided between the two keys. Then the reinforcement probabilities were changed so that one key had a higher probability of reinforcement (the left key in half of the conditions and the right key in the other half), and in nearly every case the subjects developed a preference for this key. The rate of acquisition of preference for this key was faster when the ratio of the two reinforcement probabilities was higher. For instance, acquisition of preference was faster in conditions with reinforcement probabilities of .12 and .02 than in conditions with reinforcement probabilities of .40 and .30, even though the pairs of probabilities differed by .10 in both cases. These results were used to evaluate the predictions of some theories of transitional behavior in choice situations. A trial-by-trial analysis of individual responses and reinforcers suggested that reinforcement had both short-term and long-term effects on choice. The short-term effect was an increased probability of returning to the same key on the one or two trials following a reinforcer. The long-term effect was a gradual increase in the proportion of responses on the key with the higher probability of reinforcement, an increase that usually continued for several hundred trials.

This experiment measured pigeons' choices between delayed reinforcers and fixed-ratio schedules in which a force of approximately 0.48 N was needed to operate the response key. In ratio-delay conditions, subjects chose between a fixed-ratio schedule and an adjusting delay. The delay was increased or decreased several times a session in order to estimate an indifference point--a delay duration at which the two alternatives were chosen about equally often. Each ratio-delay condition was followed by a delay-delay condition in which subjects chose between the adjusting delay and a variable-time schedule, with the components of this schedule selected to match the ratio completion times of the preceding ratio-delay condition. The adjusting delays at the indifference point were longer when the alternative was a fixed-ratio schedule than when it was a matched variable-time schedule, which indicated a preference for the matched variable-time schedules over the fixed-ratio schedules. This preference increased in a nonlinear manner with increasing ratio size. This nonlinearity was inconsistent with a theory that states that indifference points for both time and ratio schedules can be predicted by multiplying the choice response-reinforcer intervals of the two types of schedules by different multiplicative constants. Two other theories, which predict nonlinear increases in preference for the matched variable-time schedules, are discussed.

Optimization theory states that organisms behave in a way that maximizes reinforcement or "value." In a two-response situation, pigeons' response proportions approximately equaled reinforcement proportions, even when this behavior pattern substantially decreased the rate of reinforcement. Optimization or reinforcement maximization was not supported as the basic mechanism underlying choice behavior.

This experiment tested for transitivity in pigeons' choices between variable-time (VT) and fixed-time (FT) schedules. In a discrete-trials procedure, a subject chose between two alternatives by making a single key peck. Each choice was between a "standard alternative," which was the same schedule throughout a condition, and an "adjusting alternative," in which the delay to reinforcement was systematically increased or decreased many times a session. These adjustments enabled an approximate indifference point to be identified--the value of the adjusting delay at which the subject chose each alternative about equally often. Each test of transitivity involved four conditions. In one, the standard alternative was a variable-time schedule with a 2-s reinforcer, and the adjusting alternative also delivered a 2-s reinforcer. A second condition was similar except that the adjusting alternative delivered a 5-s reinforcer. The indifference point from each of these conditions was then converted to a fixed-time schedule for subsequent comparisons in the third and fourth conditions, respectively. Each of these last two conditions compared one of the fixed-time schedules (based upon the previous conditions and including their different reinforcer durations) with an adjusting schedule that delivered the alternative reinforcer duration, to determine whether the obtained indifference points would be those predicted from the prior alternative-duration comparisons with the VT schedule. There was little evidence for intransitivity of choice: Averaged across subjects and replications, the obtained indifference points deviated from perfect transitivity by less than 8%, and these deviations were not statistically significant. These results contrast with those of Navarick and Fantino (1972), who found frequent violations of transitivity between periodic and aperiodic schedules using a concurrent-chains procedure with variable-interval schedules in the initial links.

A discrete-trial procedure was used to measure pigeons' choices between fixed and variable ratio schedules and between fixed and variable delays before reinforcement. A peck at a green key produced a reinforcement schedule that was constant within a condition but varied across conditions. A peck at a red key produced a ratio schedule (or, in other conditions, a simple delay) whose size was increased or decreased many times a session, depending on the subject's previous choices. The purpose of these adjustments was to estimate an indifference point--a ratio size (or delay duration) at which the subject chose each key about equally often. The results were used to test a simple "equivalence rule" for choices between fixed and variable schedules (Mazur, 1984). This rule, which was applied without using free parameters, predicted the major trends in the obtained indifference points from both ratio and delay conditions. However, some small but consistent deviations from the predictions were apparent. Better predictions were generated with a more complex equation, which included parameters reflecting the subjects' sensitivities to delay of reinforcement and to events of different probabilities. It was concluded that a successful equivalence rule must include parameters that can be adjusted to describe the effects of delay and probability in a given experimental setting. Once these parameters are estimated, however, choices involving both fixed and variable delays and fixed and variable ratios can be accurately predicted with the same equation.

In a series of conditions, pigeons chose between 1.5 s and 3 s of access to grain, each preceded by some delay. The delay that preceded the small reinforcer was constant throughout a condition. The delay that preceded the large reinforcer was increased or decreased a number of times each session in order to estimate an "indifference point," a delay at which the subject chose each alternative about equally often. The experiment was designed to determine whether variations in any of four features of this adjusting-delay procedure would systematically alter the estimated indifference points. The four features were the total trial duration, the number of center-key responses necessary to begin a trial, the number of choice trials that preceded each change in the adjusting delay, and step size--the size of each increment and decrement in the delay. Manipulation of the first three features had no systematic effects on the indifference points. As step size was increased from 0.5 s to 6 s, within-session variability of the adjusting delay steadily increased, and the 6-s step size produced larger indifference-point estimates for some subjects. The results suggest that, within certain limits, these procedural features can be altered without affecting the indifference-point estimates, but that the use of a large step size can distort the estimates. Some theoretical implications of the relative constancy of indifference points across these procedural variations are discussed.

Pigeons chose between alternatives that differed in the number of reinforcers and in the delay to each reinforcer. A peck on a red key produced the same consequences on every trial within a condition, but between conditions the number of reinforcers varied from one to three and the reinforcer delays varied between 5 s and 30 s. A peck on a green key produced a delay of adjustable duration and then a single reinforcer. The green-key delay was increased or decreased many times per session, depending on a subject's previous choices, which permitted estimation of an indifference point, or a delay at which a subject chose each alternative about equally often. The indifference points decreased systematically with more red-key reinforcers and with shorter red-key delays. The results did not support the suggestion of Moore (1979) that multiple delayed reinforcers have no effect on preference unless they are closely grouped. The results were well described in quantitative detail by a simple model stating that each of a series of reinforcers increases preference, but that a reinforcer's effect is inversely related to its delay. The success of this model, which considers only delay of reinforcement, suggested that the overall rate of reinforcement for each alternative had no effect on choice between those alternatives.

Pigeons chose between two alternatives that differed in the probability of reinforcement and the delay to reinforcement. A peck on the red key always produced a delay of 5 s and then a possible reinforcer. The probability of reinforcement for responding on this key varied from .05 to 1.0 in different conditions. A response on the green key produced a delay of adjustable duration and then a possible reinforcer, with the probability of reinforcement ranging from .25 to 1.0 in different conditions. The green-key delay was increased or decreased many times per session, depending on a subject's previous choices. The purpose of these adjustments was to estimate an indifference point, or a delay that resulted in a subject's choosing each alternative about equally often. In conditions where the probability of reinforcement was five times higher on the green key, the green-key delay averaged about 12 s at the indifference point. In conditions where the probability of reinforcement was twice as high on the green key, the green-key delay at the indifference point was about 8 s with high probabilities and about 6 s with low probabilities. An analysis based on these results and those from studies on delay of reinforcement suggests that pigeons' choices are relatively insensitive to variations in the probability of reinforcement between .2 and 1.0, but quite sensitive to variations in probability between .2 and 0.

In a discrete-trials procedure, pigeons chose between a fixed-ratio 81 schedule and a progressive-ratio schedule by making a single peck at the key correlated with one or the other of these schedules. The response requirement on the progressive-ratio schedule began at 1 and increased by 10 each time the progressive-ratio schedule was chosen. Each time the fixed-ratio schedule was chosen, the requirement on the progressive-ratio schedule was reset to 1 response. In conditions where there was no intertrial interval, subjects chose the progressive-ratio schedule for an average of about five consecutive trials (during which the response requirement increased to 41), and then chose the fixed-ratio schedule. This ratio was larger than that predicted by an optimality analysis that assumes that subjects respond in a pattern that minimizes the response-reinforcer ratio or one that assumes that subjects respond in a pattern that maximizes the overall rate of reinforcement. In conditions with a 25-s or 50-s intertrial interval, subjects chose the progressive-ratio schedule for an average of about eight consecutive trials before choosing the fixed-ratio schedule. This change in performance with the addition of an intertrial interval was also not predicted by an optimality analysis. On the other hand, the results were consistent with the theory that choice is determined by the delays to the reinforcers delivered on the present trial and on subsequent trials.