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Grace et al. (2018) showed that humans could estimate ratios and differences of stimulus magnitudes by feedback and without explicit instruction in a nonsymbolic ‘artificial algebra’ task, but that responding depended on both operations even though only one was trained. Here we asked whether control by the trained operation would increase over several sessions, that is, if perceptual learning would occur. Observers (n = 16) completed four sessions in which feedback was based on either ratios or differences for stimulus pairs that varied in brightness (Experiment 1) or line length (Experiment 2). Results showed that control by the trained and untrained operations increased and decreased, respectively, over the sessions, indicating perceptual learning. For about two thirds of individual sessions, regressions indicated significant control by both differences and ratios, suggesting that the perceptual system automatically computes two operations. The similarity of results across experiments with both intensive (brightness) and extensive (line length) stimulus dimensions suggests that differences and ratios are computed centrally, perhaps as part of a general system for processing magnitudes (cf. Walsh, 2003 ).

Concurrent chains is widely used to study pigeons’ choice between terminal links that can vary in delay, magnitude, or probability of reinforcement. We review research on the acquisition of choice in this procedure. Acquisition has been studied with a variety of research designs, and some studies have incorporated no-food trials to allow for timing and choice to be observed concurrently. Results show that: Choice can be acquired rapidly within sessions when terminal links change unpredictably; under steady-state conditions, acquisition depends on both initial- and terminal-link schedules; and initial-link responding is mediated by learning about the terminal-link stimulus-reinforcer relations. The cumulative decision model (CDM) proposed by Christensen and Grace (2010) and Grace and McLean, 2006 , Grace and McLean, 2015 provides a good description of within-session acquisition, and correctly predicts the effects of initial and terminal-link schedules in steady-state designs ( Grace, 2002a ). Questions for future research include how abrupt shifts in preference within individual sessions and temporal control of terminal-link responding can be modeled.

Concurrent chains is widely used to study pigeons' choice between terminal links that can vary in delay, magnitude, or probability of reinforcement. We review research on the acquisition of choice in this procedure. Acquisition has been studied with a variety of research designs, and some studies have incorporated no-food trials to allow for timing and choice to be observed concurrently. Results show that: Choice can be acquired rapidly within sessions when terminal links change unpredictably; under steady-state conditions, acquisition depends on both initial- and terminal-link schedules; and initial-link responding is mediated by learning about the terminal-link stimulus-reinforcer relations. The cumulative decision model (CDM) proposed by Christensen and Grace (2010) and Grace and McLean (2006, 2015) provides a good description of within-session acquisition, and correctly predicts the effects of initial and terminal-link schedules in steady-state designs (Grace, 2002a). Questions for future research include how abrupt shifts in preference within individual sessions and temporal control of terminal-link responding can be modeled.

Four pigeons were exposed to a concurrent-chains procedure in which the initial link was constant (variable-interval 10s) while the terminal-link delays changed across sessions according to a pseudorandom binary series. In the short condition, the left terminal link was a fixed-interval 8s and the right terminal link was either fixed-interval 4s or fixed-interval 16s; in the long condition, the left terminal link was fixed-interval 16s and the right terminal link was either fixed-interval 8s or fixed-interval 32s. Multiple regression showed that response allocation was sensitive to the current session immediacy ratio, with greater sensitivity in the long than the short condition. This replicates the terminal-link effect [MacEwen, D., 1972. The effects of terminal-link fixed-interval and variable-interval schedules on responding under concurrent-chained schedules. J. Exp. Anal. Behav. 18, 253-261] in a pseudorandom binary series design. Grace and McLean's [Grace, R.C., McLean, A., 2006. Rapid acquisition in concurrent chains: evidence for a decision model. J. Exp. Anal. Behav. 85, 181-202] decision model provided a good description of the data and predicted the terminal-link effect. Estimates of the criterion parameter increased in the long condition but less than proportionally, and were close to the expected values based on the programmed schedule durations, as predicted by the decision model. These results provide additional support for the decision model as an account of choice in concurrent chains.

To compare the generalized matching law (Baum, W.M., 1974b. On two types of deviation from the matching law: bias and undermatching. J. Exp. Anal. Behav. 22, 231–242) and contingency discriminability model (Davison, M., Jenkins, P.E., 1985. Stimulus discriminability, contingency discriminability, and schedule performance. Anim. Learn. Behav. 13, 77–84) as accounts of concurrent schedule performance, we conducted a residual meta-analysis of response- and time-allocation data from 20 studies ( n 's = 886 and 774, respectively). Both models were fitted to the individual-subject data from each study, and residuals were obtained. Polynomial regressions were then performed on the pooled residuals to determine whether systematic trends were present as a function of predicted values. For the contingency discriminability model, the cubic coefficients were positive and statistically significant for both response- and time-allocation data. By contrast, no statistically significant systematic trend was obtained in the residuals for the generalized matching law. These results suggest that the relationship between log response allocation and log reinforcer allocation does not deviate significantly from linearity over an approximate range of +1.25 to −1.25 log units, consistent with the generalized matching law. Although qualitative criteria are also important in comparing models of behavioral phenomena, residual meta-analysis provides a powerful quantitative methodology for model selection and should prove useful in future research.

Theories of timing have been applied to choice between delayed rewards by assuming that delays are represented in memory and that subjects sample from memory when choosing between alternatives. To search for covariation in single-trial measures of performance that might confirm this assumption, we used a procedure that allowed for convergent measurement of choice and timing behavior. Four pigeons responded in a concurrent chains/peak procedure in which the terminal links were fixed-interval (FI) 8 s and FI 16 s, across conditions the duration of the initial-link schedule was either short or long, and one quarter of the terminal links lasted for 48 s and ended without reinforcer delivery. Preference for the FI 8-s alternative was stronger with shorter initial links, replicating the ‘initial-link effect’. Responding on no-food trials was unaffected by initial-link duration, and aggregated across trials, was typical of the peak procedure: response distributions were approximately Gaussian, with modes near the FI schedule durations, and variance was greater for the FI 16-s terminal link. Analysis of local measures of initial-link performance (e.g., pause to begin responding, time spent responding, number and duration of visits to each alternative, etc.) found that the initial-link effect was associated with an increase in the number and duration of visits per cycle to the nonpreferred alternative. Regression analyses showed that local initial-link measures contributed relatively little additional variance in predicting performance on individual no-food trials beyond that accounted for by FI schedule. Our results provide no clear evidence that initial- and terminal-link responding in concurrent chains are mediated by a common representation of terminal-link delays.

Preference in concurrent chains for the richer terminal-link schedule becomes more extreme as the schedule values increase with their ratio held constant, a result known as the terminal-link effect. We report two experiments that attempt to determine whether this effect is related to terminal-link duration or the overall rate of reinforcement. These variables have been confounded in prior studies, but can be separated by comparing variable-duration schedules that end after a single reinforcer has been earned, with constant-duration schedules during which a variable number of reinforcers may be earned. In Experiment 1, the terminal-link effect was obtained with variable-duration schedules when duration and overall reinforcement rate were manipulated, but not with constant-duration schedules when overall reinforcement rate was changed with duration held constant. In Experiment 2, the terminal-link effect was obtained with constant-duration schedules when duration was manipulated with overall reinforcement rate held constant. Taken together, these results show that the terminal-link effect depends on changes in terminal-link duration, not overall reinforcement rate (or equivalently, average time to reinforcement). This accords with the account of the terminal-link effect provided by the contextual choice model [J. Exp. Anal. Behav. 61 (1994) 113] but not delay-reduction theory [J. Exp. Anal. Behav. 12 (1969) 723].

To investigate the role of timing processes in choice, we used a new procedure that provided simultaneous measures of ongoing choice and timing behavior. Pigeons responded in a peak procedure in which the delays to reinforcement signaled by red and green center-key stimuli were 10 and 20, or 20 and 40 s. After 25 sessions of training, the peak procedure was embedded within concurrent chains: The inter-trial interval was replaced by a choice phase in which the two side keys were illuminated white; responses to the left and right keys occasionally changed the center-key to red or green, respectively; and the terminal links signaled by the center-key stimuli were identical to the trials of the peak procedure. The temporal control of responding on no-food trials was the same regardless of whether the no-food trials occurred in the peak procedure or as the terminal links of concurrent chains. After an intervening condition with the peak procedure in which the delay for the 10 s stimulus was changed to 40 s (or vice versa), the pigeons were returned to concurrent chains. Choice responding did not reflect the changed delay, despite the fact that the pigeons timed the delays in both terminal links accurately as indexed by responding on no-food trials. This result challenges current accounts of choice based on timing processes, such as scalar expectancy theory, which assume that choice responding is mediated by a representation of terminal link delays to reinforcement. Apparently, pigeons' choice and timing behavior in a single session can be controlled by temporal information from different temporal epochs.

We report two experiments which test whether resistance to prefeeding and satiation for a variable-interval (VI) schedule that delivers a constant rate of reinforcement varies inversely with the reinforcement rate for an alternative schedule. In Experiment 1, eight pigeons responded in a multiple schedule in which the red key was always associated with a VI 90-s schedule and the green key with either a richer (VI 18 s) or leaner (VI 540 s) schedule in different conditions. After baseline training in each condition, prefeeding test sessions were conducted in which 10 g, 20 g, 30 g, 40 g, and 50 g food were provided one-hour prior to test. Additional baseline training was given between each test session. In Experiment 2, two groups of pigeons responded in a multiple schedule similar to Experiment 1. After baseline training, pigeons were exposed to a 5-h satiation test session in which the VI 90-s schedule was available continuously. Test sessions were conducted when pigeons were maintained at 85%, 95%, and 85% of their body weights in an ABA design. Results of both experiments showed that responding in the VI 90-s schedule that alternated with a leaner schedule during baseline was more resistant to prefeeding and satiation. These data rule out alternative explanations for results of previous studies, and confirm that resistance to change varies inversely with reinforcement context.

We conducted a residual meta-analysis to test the assumptions of the generalized matching law that effects of relative reinforcer magnitude on response allocation in concurrent schedules can be described by a power function and are independent from the effects of relative reinforcer rate. We identified five studies which varied magnitude ratios over at least four levels and six studies in which reinforcer rate and magnitude ratios were varied factorially. The generalized matching law provided a reasonably good description of the data, accounting for 77.1% and 90.1% of the variance in the two sets of studies. Results of polynomial regressions showed that there were no systematic patterns in pooled residuals as a function of predicted log response ratios for data sets in which relative magnitude was varied. For data sets in which relative rate and magnitude were varied factorially, there was a significant negative cubic pattern in the pooled residuals, suggesting that obtained response allocation was less extreme than predicted for conditions with extreme predicted values. However, subsequent analyses showed that this result was associated with results from conditions in one study in which the product of the rate and magnitude ratios was 63:1, and in which response allocation may have been attenuated by a ceiling effect. When data from these conditions were omitted, there were no significant components in the residuals. Although the number of available studies was small, results provide tentative support for the assumptions of the generalized matching law that effects of reinforcer magnitude ratios on choice can be described by a power function and are independent from reinforcer rate ratios.

Four pigeons responded in a concurrent-chains procedure in which terminal-link schedules were fixed-interval (FI) 10 s and FI 20 s. Across sessions, the location of the shorter terminal-link changed according to a pseudorandom binary sequence. Each session, the variable-interval initial-link schedule value was sampled from a uniform distribution that ranged from 0.01 to 30 s. On some terminal links, food was withheld to obtain measures of temporal control. Terminal-link delays determined choice (log initial-link response ratios) and timing (start and stop times on no-food trials) measures, which stabilized within the 1st half of each session. Preference for the shorter terminal-link delay was a monotonically decreasing function of initial-link duration. There was no evidence of control by initial-link durations from previous sessions.

Pigeons were trained in a concurrent chains procedure in which the terminal-link schedules in each session were either fixed-interval (FI) 10 s FI 20 s or FI 20 s FI 10 s, as determined by a pseudorandom binary series. The initial-link was a variable-interval (VI) 10-s schedule. Training continued until initial-link response allocation stabilized about midway through each session and was sensitive to the terminal-link immediacy ratio in that session. The initial-link schedule was then varied across sessions between VI 0.01 s and VI 30 s according to an ascending and descending sequence. Initial-link response allocation was a bitonic function over the full range of durations. Preference for the FI 10-s terminal-link at first increased as programmed initial-link duration varied from 0.01 to 7.5 s, and then decreased as initial-link duration increased to 30 s. The bitonic function poses a potential challenge for existing models for steady-state choice, such as delay-reduction theory (DRT) [Fantino, E., 1969. Choice and rate of reinforcement. J. Exp. Anal. Behav. 12, 723–730], which predict a monotonic function. However, an extension of Grace and McLean's [Grace, R.C., McLean, A.P., 2006. Rapid acquisition in concurrent chains: evidence for a decision model. J. Exp. Anal. Behav. 85, 181–202] decision model predicted the bitonic function, and may ultimately provide an integrated account of choice in concurrent chains under both steady-state and dynamic conditions.

Eight pigeons responded in a multiple variable-interval (VI) schedule in which a constant component always delivered 40 rft/h, and an alternated component was either rich (200 rft/h) or lean (6.67 rft/h) in different conditions. Four tests of resistance to change were conducted in each condition: prefeeding, full extinction, constant-component-only extinction, and response-independent food. Resistance to both prefeeding and full extinction in the constant component varied inversely with the reinforcement rate in the alternated component, but resistance to response-independent food did not. The extinction and response-independent food results were consistent with [J. Exp. Psychol.: Anim. Behav. Proc. 25 (1999) 256] behavioral momentum model. Maintaining reinforcement in the alternated component increased resistance to extinction in the constant component, as predicted by the behavioral momentum model but not accounts of multiple-schedule performance based on [J. Exp. Anal. Behav. 13 (1970) 243] equation. Overall, the momentum model gave a good account of the results with the exception of the prefeeding data. Possible ways to reconcile the prefeeding results with behavioral momentum theory are considered.