Relative reinforcer rates and magnitudes do not control concurrent choice independently

One assumption of the matching approach to choice is that different independent variables control choice independently of each other. We tested this assumption for reinforcer rate and magnitude in an extensive parametric experiment. Five pigeons responded for food reinforcement on switching-key concurrent variable-interval variable-interval schedules. Across conditions, the ratios of reinforcer rates and of reinforcer magnitudes on the two alternatives were both manipulated. Control by each independent variable, as measured by generalized-matching sensitivity, changed significantly with the ratio of the other independent variable. Analyses taking the model-comparison approach, which weighs improvement in goodness-of-fit against increasing number of free parameters, were inconclusive. These analyses compared a model assuming constant sensitivity to magnitude across all reinforcer-rate ratios with two alternative models. One of those alternatives allowed sensitivity to magnitude to vary freely across reinforcer-rate ratios, and was less efficient than the common-sensitivity model for all pigeons, according to the Schwarz-Bayes information criterion. The second alternative model constrained sensitivity to magnitude to be equal for pairs of reinforcer-rate ratios that deviated from unity by proportionately equal amounts but in opposite directions. This model was more efficient than the common-magnitude-sensitivity model for 2 of the pigeons, but not for the other 3. An analysis of variance, carried out independently of the generalized-matching analysis, also showed a significant interaction between the effects of reinforcer rate and reinforcer magnitude on choice. On balance, these results suggest that the assumption of independence inherent in the matching approach cannot be maintained. Relative reinforcer rates and magnitudes do not control choice independently.     

A decision model for steady-state choice in concurrent chains

Grace and McLean (2006) proposed a decision model for acquisition of choice in concurrent chains which assumes that after reinforcement in a terminal link, subjects make a discrimination whether the preceding reinforcer delay was short or long relative to a criterion. Their model was subsequently extended by Christensen and Grace (2008, 2009a, 2009b) to include effects of initial- and terminal-link duration on choice. We show that an expression for steady-state responding can be derived from the decision model, which enables a model for choice that provides an account of archival data that is equal or superior to the contextual choice model (Grace, 1994) and hyperbolic value-added model (Mazur, 2001) in terms of goodness of fit, parsimony, and parameter invariance. The success of the steady-state decision model validates the strategy of understanding acquisition phenomena as a bridge toward explaining choice at the molar level.     

Nonstable concurrent choice in pigeons

Six pigeons were trained on concurrent variable-interval schedules in which the arranged reinforcer ratios changed from session to session according to a 31-step pseudorandom binary sequence. This procedure allows a quantitative analysis of the degree to which performance in an experimental session is affected by conditions in previous sessions. Two experiments were carried out. In each, the size of the reinforcer ratios arranged between the two concurrent schedules was varied between 31-step conditions. In Experiment 1, the concurrent schedules were arranged independently, and in Experiment 2 they were arranged nonindependently. An extended form of the generalized matching law described the relative contribution of past and present events to present-session behavior. Total performance in sessions was mostly determined by the reinforcer ratio in that session and partially by reinforcers that had been obtained in previous sessions. However, the initial exposure to the random sequence produced a lower sensitivity to current-session reinforcers but no difference in overall sensitivity to reinforcement. There was no evidence that the size of the reinforcer ratios available on the concurrent schedules affected either overall sensitivity to reinforcement or the sensitivity to reinforcement in the current session. There was also no evidence of any different performance between independent and nonindependent scheduling. Because of these invariances, this experiment validates the use of the pseudorandom sequence for the fast determination of sensitivity to reinforcement.     

Choice between rewards differing in amount and delay: Toward a choice model of self control

A concurrent-chain procedure was used to study pigeons' choices between rewards differing in both amount and delay. The shorter delay terminated with a 2-second access to grain whereas the longer delay terminated with a 6-second access to grain. The ratio of the delays was constant within a given condition while their absolute values were varied. Over conditions, ratios of 6:1, 3:1, and 3:2 were studied. As the absolute values of the delays to reinforcement increased, preference for the longer-delayed but larger reward decreased under both the 6:1 and 3:1 ratios, but increased under the 3:2 ratio. These results are inconsistent with choice models predicting no change in preference when the ratios of delays and amounts are held constant. In addition, the change in preference under the 3:1 ratio is inconsistent with a simple multiplicative interaction of the trade off between reinforcer amount and delay, and suggests that delay is a more potent determinant of choice than is amount. These results have implications for models that view choice between small immediate rewards and large but delayed rewards as underlying the behavior commonly called self control.     

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.     

Independence of reinforcement delay and magnitude in concurrent chains

A three-component concurrent-chains procedure was used to investigate preference between terminal-link schedules that differed in delay and magnitude of reinforcement. Response and time allocation data were well described by a generalized matching model. Sensitivity to delay appeared to be lower when reinforcement magnitudes were unequal than when they were equal, but when obtained rather than programmed time spent responding in the initial links was used in the model, the difference vanished. The results support independence of delay and magnitude as separate dimensions of reinforcement value, as required by the matching law, and the assumption of the contextual choice model (Grace, 1994) that sensitivities to delay and magnitude are affected similarly by temporal context. Although there was statistical evidence for interaction between successive components, the effects were small and transient. The multiple-component concurrent-chains procedure should prove useful in future research on multidimensional preference, although it may be necessary to control obtained initial-link time more precisely.     

Preference for signaled versus unsignaled reinforcement delay in concurrent-chain schedules

A concurrent-chain schedule was employed to examine pigeons' preferences for signaled versus unsignaled delay of reinforcement in which the delay durations ranged from zero to ten seconds. In general, pigeons preferred signaled delay over unsignaled delay especially when a variable-interval 30-second schedule operated in each initial link; when a variable-interval 90-second schedule operated in each initial link, these preferences tended toward indifference or were attenuated. In addition, prior training seemed to exert partial control over behavior. Responding in the terminal link was higher under signaled delay than unsignaled delay in a majority of the cases. Moreover, response rates under signaled delay remained fairly constant whereas responding under unsignaled delay was initially high, but decreased systematically with delay durations as short as 2.5 seconds. These results are consistent with a number of other studies demonstrating the significant role of a signal for impending positive stimuli.     

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.     

Variation, repetition, and choice

Experiment 1 investigated the controlling properties of variability contingencies on choice between repeated and variable responding. Pigeons were exposed to concurrent-chains schedules with two alternatives. In the REPEAT alternative, reinforcers in the terminal link depended on a single sequence of four responses. In the VARY alternative, a response sequence in the terminal link was reinforced only if it differed from the n previous sequences (lag criterion). The REPEAT contingency generated low, constant levels of sequence variation whereas the VARY contingency produced levels of sequence variation that increased with the lag criterion. Preference for the REPEAT alternative tended to increase directly with the degree of variation required for reinforcement. Experiment 2 examined the potential confounding effects in Experiment 1 of immediacy of reinforcement by yoking the interreinforcer intervals in the REPEAT alternative to those in the VARY alternative. Again, preference for REPEAT was a function of the lag criterion. Choice between varying and repeating behavior is discussed with respect to obtained behavioral variability, probability of reinforcement, delay of reinforcement, and switching within a sequence.     

Contiguity and conditioned reinforcement in probabilistic choice

In a baseline condition, pigeons chose between an alternative that always provided food following a 30-s delay (100% reinforcement) and an alternative that provided food half of the time and blackout half of the time following 30-s delays (50% reinforcement). The different outcomes were signaled by different-colored keylights. On average, each alternative was chosen approximately equally often, replicating the finding of suboptimal choice in probabilistic reinforcement procedures. The efficacy of the delay stimuli (keylights) as conditioned reinforcers was assessed in other conditions by interposing a 5-s gap (keylights darkened) between the choice response and one or more of the delay stimuli. The strength of conditioned reinforcement was measured by the decrease in choice of an alternative when the alternative contained a gap. Preference for the 50% alternative decreased in conditions in which the gap preceded either all delay stimuli, both delay stimuli for the 50% alternative, or the food stimulus for the 50% alternative, but preference was not consistently affected in conditions in which the gap preceded only the 100% delay stimulus or the blackout stimulus for the 50% alternative. These results support the notion that conditioned reinforcement underlies the finding of suboptimal preference in probabilistic reinforcement procedures, and that the signal for food on the 50% reinforcement alternative functions as a stronger conditioned reinforcer than the signal for food on the 100% reinforcement alternative. In addition, the results fail to provide evidence that the signal for blackout functions as a conditioned punisher.     

Independence of terminal-link entry rate and immediacy in concurrent chains

In Phase 1, 4 pigeons were trained on a three-component multiple concurrent-chains procedure in which components differed only in terms of relative terminal-link entry rate. The terminal links were variable-interval schedules and were varied across four conditions to produce immediacy ratios of 4:1, 1:4, 2:1, and 1:2. Relative terminal-link entry rate and relative immediacy had additive and independent effects on initial-link response allocation, and the data were well-described by a generalized-matching model. Regression analyses showed that allowing sensitivity to immediacy to vary across components produced only trivial increases in variance accounted for. Phase 2 used a three-component concurrent-schedules procedure in which the schedules were the same as the initial links of Phase 1. Across two conditions, the relative reinforcer magnitude was varied. Sensitivity to relative reinforcer rate was independent of relative magnitude, confirming results of prior studies. Sensitivity to relative reinforcer rate in Phase 2 did not vary systematically across subjects compared to sensitivity to relative entry rate in Phase 1, and regression analyses confirmed again that only small increases in variance accounted for were obtained when sensitivities were estimated independently compared with a single estimate for both phases. Overall, the data suggest that conditioned and primary reinforcers have functionally equivalent effects on choice and support the independence of relative terminal-link entry rate and immediacy as determiners of response allocation. These results are consistent with current models for concurrent chains, including Grace's (1994) contextual choice model and Mazur's (2001) hyperbolic value-added model.     

Pure timing in temporal differentiation

Temporal control appears to depend on whether the critical durations are those of stimuli or those of responses. Stimulus timing (temporal discrimination) supports Weber's law, whereas response timing (temporal differentiation) indicates decreasing relative sensitivity with longer time intervals. The two types of procedure also yield different conclusions in scaling experiments designed to study the functional midpoint of two or more durations (temporal bisection procedures). In addition, the fractional-exponent power relation between emitted and required duration usually found with animals in differentiation experiments conflicts with deductions from formal analyses. The experiment reported here derived from considering differentiation arrangements as schedules of reinforcement. When analyzed from this perspective, the procedures are tandem schedules involving a required pause followed by a response, and it is the pause alone that involves temporal control. A choice procedure separated timing from responding, and enabled observations of pause timing in isolation. Pure temporal control in differentiation consisted of linear overestimation of the standard duration, and Weber's law described sensitivity. These results indicate that the two problems, the fractional-exponent power relations and the apparently different nature of sensitivity in differentiation and discrimination, disappear when temporal control is observed alone in differentiation.     

Temporal context in concurrent chains: I. Terminal-link duration

Two experiments are reported in which the ratio of the average times spent in the terminal and initial links (Tt/Ti) in concurrent chains was varied. In Experiment 1, pigeons responded in a three-component procedure in which terminal-link variable-interval schedules were in constant ratio, but their average duration increased across components by a factor of two. The log initial-link response ratio was a negatively accelerated function of Tt/Ti. Overall, the data were well described by Grace's (1994) contextual choice model (CCM) with temporal context represented as (Tt/Ti)k or 2Tt/(Tt + Ti), and by Mazur's (2001) hyperbolic value-added model (HVA), with each model accounting for approximately 93% of the variance. In Experiment 2, fixed-parameter predictions for each model were generated, based on the data from Experiment 1, for conditions in which Tt/Ti was varied over a more extreme range. Data were consistent with the predictions of CCM with temporal context represented as 2Tt/(Tt + Ti) and to a lesser extent as (Tt/Ti)k, but not with HVA. Overall, these results suggest that preference increases as a hyperbolic function of Tt/Ti when terminal-link duration is increased relative to initial-link duration, with the terminal-link schedule ratio held constant.     

Residence time and choice in concurrent foraging schedules

Five pigeons were trained on a concurrent-schedule analogue of the “some patches are empty” procedure. Two concurrently available alternatives were arranged on a single response key and were signaled by red and green keylights. A subject could travel between these alternatives by responding on a second yellow “switching” key. Following a changeover to a patch, there was a probability (p) that a single reinforcer would be available on that alternative for a response after a time determined by the value of λ, a probability of reinforcement per second. The overall scheduling of reinforcers on the two alternatives was arranged nonindependently, and the available alternative was switched after each reinforcer. In Part 1 of the experiment, the probabilities of reinforcement, ρ_red and ρ_green, were equal on the two alternatives, and the arranged arrival rates of reinforcers, λ_red and λ_green, were varied across conditions. In Part 2, the reinforcer arrival times were arranged to be equal, and the reinforcer probabilities were varied across conditions. In Part 3, both parameters were varied. The results replicated those seen in studies that have investigated time allocation in a single patch: Both response and time allocation to an alternative increased with decreasing values of λ and with increasing values of ρ, and residence times were consistently greater than those that would maximize obtained reinforcer rates. Furthermore, both response- and time-allocation ratios undermatched mean reinforcer-arrival time and reinforcer-frequency ratios.     

Choice, foraging, and reinforcer duration.

Pigeons were exposed to a foraging schedule characterized by three different states, beginning with a search state in which completion of a variable interval on a white key led to a choice state. In the choice state the subject could, by appropriate responding on a fixed ratio of three, either accept or reject the schedule offered. If the subject accepted the schedule, it entered a handling state in which the appropriate reinforcer amount was presented according to a variable-interval schedule. In Experiment 1 the shorter duration reinforcer was more likely to be accepted the longer the duration of the search state and the shorter the equal durations of the handling states. In Experiment 2 the shorter duration reinforcer was more likely to be accepted the longer the handling time preceding the longer duration reinforcer. All of the results were in qualitative--and some were in quantitative--agreement with those predicted by the delay-reduction hypothesis and the optimal-diet model.     

Travel time and concurrent-schedule choice: retrospective versus prospective control

Six pigeons were trained on concurrent variable-interval schedules in which two different travel times between alternatives, 4.5 and 0.5 s, were randomly arranged. In Part 1, the next travel time was signaled while the subjects were responding on each alternative. Generalized matching analyses of performance in the presence of the two travel-time signals showed significantly higher response and time sensitivity when the longer travel time was signaled compared to when the shorter time was signaled. When the data were analyzed as a function of the previous travel time, there were no differences in sensitivity. Dwell times on the alternatives were consistently longer in the presence of the stimulus that signaled the longer travel time than they were in the presence of the stimulus that signaled the shorter travel time. These results are in accord with a recent quantitative account of the effects of travel time. In Part 2, no signals indicating the next travel time were given. When these data were analyzed as a function of the previous travel time, time-allocation sensitivity after the 4.5-s travel time was significantly greater than that after the 0.5-s travel time, but no such difference was found for response allocation. Dwell times were also longer when the previous travel time had been longer.     

A molecular analysis of choice on concurrent-chains schedules.

Six pigeons responded on concurrent-chains schedules with either independent or interdependent equal variable-interval schedules in the initial links and unequal variable-interval schedules, always in a 2:1 ratio, in the terminal links. Relative response rates in the initial links increased across conditions as initial-link duration was shortened and decreased across conditions as terminal-link duration was shortened, replicating previous findings. Responses in the initial links were recorded in 5-s bins, and local or molecular relative response rates were calculated in order to ascertain how relative response rate varied as a function of time since the onset of the initial links. Two distinct molecular patterns were found. With interdependent initial links, relative response rates for the preferred key were elevated for the first 10 or 20 s of the initial links and then declined to an asymptotic value. With independent initial links, a negative recency effect was found similar to that reported by Killeen (1970). These two molecular patterns were related to the different momentary reinforcement probabilities resulting from independent and interdependent scheduling.     

Context effects on choice.

Four pigeons responded on a concurrent-chains schedule in four experiments that examined whether the effectiveness of a stimulus as a conditioned reinforcer is best described by a global approach, as measured by the average interreinforcement interval, or by a local contextual approach, as measured by the onset of the stimulus preceding the conditioned reinforcer. The interreinforcement interval was manipulated by the inclusion of an intertrial interval, which increased the overall time to reinforcement but did not change the local contingencies on a given trial A global analysis predicted choice for the richer alternative to decrease with the inclusion of an intertrial interval, whereas a local analysis predicted no change in preference. Experiment 1 examined sensitivity to intertrial intervals when each was signaled by the same houselight that operated throughout the session. In Experiment 2, the intertrial interval always was signaled by the stimulus correlated with the richer terminal link. In Experiment 3, the intertrial interval was signaled by the keylights correlated with the initial links and two novel houselights. Experiment 4 provided free food pseudorandomly during the intertrial interval. In all experiments, subjects' preferences were consistent with a local analysis of choice in concurrent chains. These results are discussed in terms of delay-reduction theory, which traditionally has failed to distinguish global and local contexts.     

Choice and reinforcement delay

Previous studies of choice between two delayed reinforcers have indicated that the relative immediacy of the reinforcer is a major determinant of the relative frequency of responding. Parallel studies of choice between two interresponse times have found exceptions to this generality. The present study looked at the choice by pigeons between two delays, one of which was always four times longer than the other, but whose absolute durations were varied across conditions. The results indicated that choice is not uniquely determined by the relative immediacy of reinforcement, but that absolute delays are also involved. Models for concurrent chained schedules appear to be more applicable to the present data than the matching relation; however, these too failed to predict choice for long delays.     

Reporting contingencies of reinforcement in concurrent schedules

Five pigeons were trained on concurrent variable-interval schedules in which two intensities of yellow light served as discriminative stimuli in a switching-key procedure. A conditional discrimination involving a simultaneous choice between red and green keys followed every reinforcer obtained from both alternatives. A response to the red side key was occasionally reinforced if the prior reinforcer had been obtained from the bright alternative, and a response to the green side key was occasionally reinforced if the prior reinforcer had been obtained from the dim alternative. Measures of the discriminability between the concurrent-schedule alternatives were obtained by varying the reinforcer ratio for correct red and correct green responses across conditions in two parts. Part 1 arranged equal rates of reinforcement in the concurrent schedule, and Part 2 provided a 9:1 concurrent-schedule reinforcer ratio. Part 3 arranged a 1:9 reinforcer ratio in the conditional discrimination, and the concurrent-schedule reinforcer ratio was varied across conditions. Varying the conditional discrimination reinforcer ratio did not affect response allocation in the concurrent schedule, but varying the concurrent-schedule reinforcer ratio did affect conditional discrimination performance. These effects were incompatible with a contingency-discriminability model of concurrent-schedule performance (Davison & Jenkins, 1985), which implies a constant discriminability parameter that is independent of the obtained reinforcer ratio. However, a more detailed analysis of conditional discrimination performance showed that the discriminability between the concurrent-schedule alternatives decreased with time since changing over to an alternative. This effect, combined with aspects of the temporal distribution of reinforcers obtained in the concurrent schedules, qualitatively predicted the molar results and identified the conditions that operate whenever contingency discriminability remains constant.