Delay reduction and optimal foraging: variable-ratio search in a foraging analogue.

The present study investigated conditions under which the conditioned reinforcement principles of delay-reduction theory and views based on simple maximization of reinforcement rate make ordinally opposing predictions with respect to foraging-related choice behavior. The use of variable-ratio schedules in the choice phase also represents an extension of delay-reduction theory to schedules that may better mimic the effort involved in searching. Pigeons responded on modified concurrent-chains schedules in which equal variable-ratio schedules led to unequal variable-interval outcomes and unequal reinforcer amounts. All 4 subjects completed a minimum of two replications of conditions for which the predictions of delay-reduction theory and a simple rate-maximizing theory were opposed. Results were consistent with delay reduction's ordinal predictions in 11 of 11 replications of the divergent predictions favoring the smaller, more immediate alternative. The predictions of rate maximization were upheld only when they were consistent with those of delay reduction. Results are discussed in terms of conditioned reinforcement, sensitivity to reductions in delay to food, and possible rules of thumb that may be useful in characterizing foraging.     

Psychological distance to reward: Effects of S+ duration and the delay reduction it signals

A concurrent-chains procedure was used to examine choice between segmented (two-component chained schedules) and unsegmented schedules (simple schedules) in terminal links with equal inter-reinforcement intervals. Previous studies using this kind of experimental procedure showed preference for unsegmented schedules for both pigeons and humans. In this study, two changes in the experimental procedure were made relative to the usual experimental procedure reported in the literature. The first was that in the segmented schedule the second terminal-link stimulus appeared close to reinforcement presentation and the second was that the change in stimulus was brief, allowing the first stimulus to reappear and to be contiguous with reinforcement. With these changes, preference occurred for the segmented schedule. This result is consistent with principles of conditioned reinforcement, including delay-reduction theory.     

Choice and conditioned reinforcement.

A potential weakness of one formulation of delay-reduction theory is its failure to include a term for rate of conditioned reinforcement, that is, the rate at which the terminal-link stimuli occur in concurrent-chains schedules. The present studies assessed whether or not rate of conditioned reinforcement has an independent effect upon choice. Pigeons responded on either modified concurrent-chains schedules or on comparable concurrent-tandem schedules. The initial link was shortened on only one of two concurrent-chains schedules and on only one of two corresponding concurrent-tandem schedules. This manipulation increased rate of conditioned reinforcement sharply in the chain but not in the tandem schedule. According to a formulation of delay-reduction theory, when the outcomes chosen (the terminal links) are equal, as in Experiment 1, choice should depend only on rate of primary reinforcement; thus, choice should be equivalent for the tandem and chain schedules despite a large difference in rate of conditioned reinforcement. When the outcomes chosen are unequal, however, as in Experiment 2, choice should depend upon both rate of primary reinforcement and relative signaled delay reduction; thus, larger preferences should occur in the chain than in the tandem schedules. These predictions were confirmed, suggesting that increasing the rate of conditioned reinforcement on concurrent-chains schedules may have no independent effect on choice.     

Delay-reduction theory--the case for temporal context: comment on Grace and Savastano (2000)

R. C. Grace and H. I. Savastano (2000) have devised a procedure to determine whether the effectiveness of a stimulus as a conditioned reinforcer depends on the temporal context of reinforcement. Although they interpret their results in terms of the contextual choice model, which maintains that value is independent of context, the results also support delay-reduction theory, which maintains that value depends on temporal context. It is argued that the delay-reduction view of the role of temporal context is also intuitively more plausible and more consistent with the way choice responds to changes in conditions.     

Matching, delay-reduction, and maximizing models for choice in concurrent-chains schedules.

Models of choice in concurrent-chains schedules are derived from melioration, generalized matching, and optimization. The resulting models are compared with those based on Fantino's (1969, 1981) delay-reduction hypothesis. It is found that all models involve the delay reduction factors (T - t2L) and (T - t2R), where T is the expected time to primary reinforcement and t2L, t2R are the durations of the terminal links. In particular, in the case of equal initial links, the model derived from melioration coincides with Fantino's original model for full (reliable) reinforcement and with the model proposed by Spetch and Dunn (1987) for percentage (unreliable) reinforcement. In the general case of unequal initial links, the model derived from melioration differs from the revised model advanced by Squires and Fantino (1971) only in the factors affecting the delay-reduction terms (T - t2L) and (T - t2R). The models of choice obtained by minimizing the expected time to reinforcement depend on the type of feedback functions used. In particular, if power feedback functions are used, the optimization model coincides with that obtained from melioration.     

Conditioned reinforcement value and choice.

The delay-reduction hypothesis of conditioned reinforcement states that the reinforcing value of a food-associated stimulus is determined by the delay to primary reinforcement signaled by the onset of the stimulus relative to the average delay to primary reinforcement in the conditioning situation. In contrast, most contemporary models of conditioned reinforcement strength posit that the reinforcing strength of a stimulus is some simple function only of the delay to primary reinforcement in the presence of stimulus. The delay-reduction hypothesis diverges from other conditioned reinforcement models in that it predicts that a fixed-duration food-paired stimulus will have different reinforcing values depending on the frequency of its presentation. In Experiment 1, pigeons' key pecks were reinforced according to concurrent-chains schedules with variable-interval 10-second and variable-interval 20-second terminal-link schedules. The initial-link schedule preceding the shorter terminal link was always variable-interval 60 seconds, and the initial-link schedule requirement preceding the longer terminal link was varied between 1 second and 60 seconds across conditions. In Experiment 2, the initial-link schedule preceding the longer of two terminal links was varied for each of three groups of pigeons. The terminal links of the concurrent chains for the three groups were variable-interval 10 seconds and 20 seconds, variable-interval 10 seconds and 30 seconds, and variable-interval 30 seconds and 50 seconds. In both experiments, preference for the shorter terminal link was either a bitonic function or an inverse function of the initial-link schedule preceding the longer terminal-link schedule. Consistent with the predictions of the delay-reduction hypothesis, the relative values of the terminal-link stimuli changed as a function of the overall frequency of primary reinforcement. Vaughan's (1985) melioration model, which was shown to be formally similar to Squires and Fantino's (1971) delay-reduction model, can be modified so as to predict these results without changing its underlying assumptions.     

Response-dependent prechoice effects on foraging-related choice

Three experiments explored the influence of prechoice events on pigeons' preference. In two of three studies, a fixed-interval 200-second prechoice period preceded the initial links of a concurrent chain in which outcomes differed either (a) in terms of the delay to food or (b) in terms of amount of food and delay to food. In Experiment 3, the prechoice period preceded the initial links that provided a choice between a small single food presentation and two identical, more delayed food presentations. In all three cases, obtained choice proportions did not vary as a function of prechoice duration. These results suggest that a local-contextual view adequately describes the foraging context; they also have implications for the appropriate formulation of the delay-reduction theory of conditioned reinforcement and rate-maximizing views of optimal foraging theory.     

Temporal context and conditioned reinforcement value

The effectiveness of a stimulus as a conditioned reinforcer depends on the temporal context of reinforcement, that is, the overall rate of reinforcement in the situation. The dominant view has been that context determines the learned value of a stimulus directly, according to delay-reduction theory. By contrast, the contextual choice model (CCM) maintains that value is independent of context and incorporates the effects of context on choice in the framework of the matching law. The authors report 2 experiments with pigeons as subjects that use transfer tests to assess the value of stimuli in the concurrent-chains procedure. Results strongly support the assumption of CCM that pigeons learn the temporal relations between events independently of context but that context modulates the expression of that learning as choice.     

Differences, not ratios, control choice in an experimental analogue to foraging

In choice between outcomes with different delays to reinforcement, most theories require that choice be governed by the ratio of the delays, not by the difference between them, a requirement also consistent with Weber's law. Instead, delay-reduction theory and optimal-foraging theory stipulate, under conditions of the present experiments, that the difference between the delays, and not the ratio between them, controls choice. This prediction was assessed using a procedure, widely used in foraging experiments, in which pigeons chose between accepting and rejecting either of two delays when offered. Across conditions, the delays either differed by a constant amount, with the ratio between the delays varying, or differed by changing amounts, with the ratio between the delays constant. In each of six experiments, rate of acceptance of the longer delay depended only on the difference between the two delays and not on the ratio between them, supporting delay-reduction and foraging theory.     

Evidence against a constant-difference effect in concurrent-chains schedules

Savastano and Fantino (1996) reported that in concurrent-chains schedules, initial-link choice proportions remained constant as terminal-link durations increased as long as the subtractive difference between the two terminal-link schedules remained constant. Two experiments with pigeons were conducted to examine this constant-difference effect. Both experiments used equal variable-interval schedules as initial links. The terminal links were fixed delays to reinforcement in Experiment 1 and variable delays to reinforcement in Experiment 2. The durations of the terminal links were varied across conditions, but the difference between pairs of terminal links was always 10 s. In both experiments, preference for the shorter terminal link became less extreme as terminal-link durations increased, so a constant-difference effect was not found. It is argued, however, that this choice situation does not provide clear evidence for or against delay-reduction theory versus other theories of choice.     

Hyperbolic value addition and general models of animal choice

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.     

Choice As A Function Of Reinforcement Ratios In Delayed Matching-to-sample

Pigeons were studied in two experiments using a delayed matching-to-sample task. In Experiment 1, 4 subjects were exposed to a task in which the proportion of reinforcement associated with matching and nonmatching, and the overall proportion of reinforcement associated with selecting each choice, regardless of the sample stimulus, were varied. Choice was sensitive to both proportions. A least squares regression analysis showed that Wixted's (1989) proportions of reinforcement model closely fit the data from Experiment 1; however, the model failed to make accurate qualitative predictions for some test conditions. In Experiment 2, 4 subjects were exposed to a delayed matching-to-sample task in which the retention intervals and the reduction in delay to reinforcement signaled by the onset of the sample stimulus were independently varied. When the retention interval was short and when the delay-reduction value of the sample stimulus was high, the sample exerted greater control over choice; the control by the overall proportion of reinforcements for selecting each choice stimulus was correspondingly low. Conversely, when the retention interval was long and the delay-reduction value of the sample stimulus was low, the sample exerted relatively less control over choice; control by the overall proportion of reinforcements obtained for selecting each choice stimulus was correspondingly high. A signal detection analysis found that sensitivity to reinforcement varied directly with retention interval. Data were also consistent with misallocation models. No evidence was found to suggest that pigeons ignore the rate at which selecting individual choice stimuli is reinforced, as has been reported in studies with human subjects.     

Preference for mixed versus constant delays of reinforcement: Effect of probability of the short, mixed delay

Preference for mixed versus constant delays of reinforcement was studied with a concurrent-chain procedure. Lever pressing by rats in concurrently available variable-interval 60-second initial links occasionally produced mutually exclusive terminal-link reinforcement delays. A constant delay of reinforcement (either 15 seconds or 30 seconds) composed one terminal link and mixed delays (.2 second and twice the value of the constant delay) were arranged in the other terminal link. The proportion of .2-second delays in the mixed-delay terminal link took on values of 0, .1, .25, .5, .75, .9, and 1.0 over experimental conditions. Based on relative rates of responding in the initial links, preference for the mixed delays was a negatively accelerated function of the proportion of short, mixed delays. Three of five rats preferred the mixed delays to the constant delays when the proportion of short, mixed delays was .1 or higher, and all five rats preferred the mixed delays when the proportion of short, mixed delays was .25 or higher. Neither Squires and Fantino's (1971) delay-reduction model of choice nor a model based on the harmonic mean reinforcement delay provided a close estimate of choice proportions over the range of short-delay proportions studied. The delay-reduction model underestimated choice for the mixed delays at low and intermediate proportions of short delays, and the harmonic-mean-delay model overestimated choice for the mixed delays at intermediate and high proportions of short delays.     

The dynamics of operant conditioning

Existing models of operant learning are relatively insensitive to historical properties of behavior and applicable to only limited data sets. This article proposes a minimal set of principles based on short-term and long-term memory mechanisms that can explain the major static and dynamic properties of operant behavior in both single-choice and multiresponse situations. The critical features of the theory are as follows: (a) The key property of conditioning is assessment of the degree of association between responses and reinforcement and between stimuli and reinforcement; (b) the contingent reinforcement is represented by learning expectancy, which is the combined prediction of response-reinforcement and stimulus-reinforcement associations; (c) the operant response is controlled by the interplay between facilitatory and suppressive variables that integrate differences between expected (long-term) and experienced (short-term) events; and (d) very-long-term effects are encoded by a consolidated memory that is sensitive to the entire reinforcement history. The model predicts the major qualitative features of operant phenomena and then suggests an experimental test of theoretical predictions about the joint effects of reinforcement probability and amount of training on operant choice. We hypothesize that the set of elementary principles that we propose may help resolve the long-standing debate about the fundamental variables controlling operant conditioning.     

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.     

The delay-reduction hypothesis: Effects of informative events on response rates and choice

The present study replicated a prior one by Pearce and Collins (1985) in which informative events displayed greater reinforcing strength than did uninformative ones despite higher rates of reinforcement on the uninformative alternative, both in a choice test and in a test that presented the events successively. The delay-reduction hypothesis of choice and conditioned reinforcement is consistent with results from the successive test but cannot account for the choice results. As the original study conducted the choice test following the successive test for all subjects, and as no reversals of the choice procedure were carried out, the present study replicated Pearce and Collins (1985) while controlling for order effects. Pigeons' relative rate of responding on the informative side was significantly greater in the successive procedure than in the choice procedure (as in the prior study); however, the uninformative side was significantly preferred to the informative side in the choice procedure when order of exposure to the two types of procedures was controlled. Both findings are consistent with the delay-reduction hypothesis.     

Influence of temporal context on value in the multiple-chains and successive-encounters procedures

This set of studies explored the influence of temporal context across multiple-chain and multiple-successive-encounters procedures. Following training with different temporal contexts, the value of stimuli sharing similar reinforcement schedules was assessed by presenting these stimuli in concurrent probes. The results for the multiple-chain schedule indicate that temporal context does impact the value of a conditioned reinforcer consistent with delay-reduction theory, such that a stimulus signaling a greater reduction in delay until reinforcement has greater value. Further, nonreinforced stimuli that are concurrently presented with the preferred terminal link also have greater value, consistent with value transfer. The effects of context on value for conditions with the multiple-successive-encounters procedure, however, appear to depend on whether the search schedule or alternate handling schedule was manipulated, as well as on whether the tested stimuli were the rich or lean schedules in their components. Overall, the results help delineate the conditions under which temporal context affects conditioned-reinforcement value (acting as a learning variable) and the conditions under which it does not (acting as a performance variable), an issue of relevance to theories of choice.     

Preference for unpredictable food rewards occurs with high proportion of reinforced trials or alcohol if rewards are not delayed

Organisms typically prefer situations where reward and nonreward are predictable rather than unpredictable. Although many theories can account for this result (e.g., information theory and delay-reduction theory), a recently developed mathematical model (DMOD) also predicts that subjects prefer the unpredictable reward situation under conditions that substantially decrease aversiveness of unpredictable nonreward (Daly & Daly, 1982). Because a high proportion of reinforced trials (lenient schedule) and alcohol injections decrease aversive conditioning, these variables were tested with rats in five E-maze experiments. A choice to one side of the maze resulted in a stimulus uncorrelated with reward outcome (unpredictable situation). A choice to the other side resulted in stimuli correlated with reward and nonreward (predictable situation). The stimuli were not visible until after the choice was made. A lenient reinforcement schedule resulted in preference for the unpredictable reward situation if rewards were not delayed. Alcohol resulted in preference for the unpredictable reward situation if a medium five-pellet reward was given. A lenient reinforcement schedule combined with an alcohol injection resulted in faster acquisition of the preference for the unpredictable reward situation than did a lenient schedule combined with a saline control injection. These results pose a major challenge to most theories, yet were predicted by DMOD.     

Melioration revisited: a systematic replication of Vaughan (1981)

Organisms that behave so as to forfeit a relatively higher overall rate of reinforcement in favor of a relatively lower rate are said to engage in suboptimal choice. Suboptimal choice has been linked with maladaptive behavior in humans. Melioration theory offers one explanatory framework for suboptimal choice. Melioration theory suggests behavior is controlled by differences in local reinforcer rates between alternatives. Vaughan (1981) arranged two experimental conditions in which maximizing the overall rate of reinforcement required behavior that was compatible, or incompatible, with melioration. Vaughan found pigeons allocated more time to a locally richer alternative even when doing so resulted in suboptimal choice. However, Vaughan did not show whether these effects could systematically reverse and did not provide within‐session data to show that choice across short time spans remains under the control of differences in local reinforcer rates. The present study used pigeons to replicate and extend Vaughan's findings. We investigated shifts in overall‐ and within‐session choice across repeated conditions, according to arranged local contingencies. Behavior systematically followed changes in local contingencies for most pigeons. Within‐session data suggests that, providing differences in local reinforcer rates are discriminated, pigeons will allocate more time to a locally richer alternative, even if this leads to suboptimal choice. These findings facilitate the more confident use of similar procedures that investigate how melioration contributes to suboptimal choice.