Strategic capacity sharing between two tasks: evidence from tasks with the same and with different task sets

Abstract

The goal of the present study was to investigate the costs and benefits of different degrees of strategic parallel processing between two tasks. In a series of experiments with the dual-task flanker paradigm, participants were either instructed to process the tasks serially or in parallel, or—in a control condition—they received no specific instruction. Results showed that the participants were able to adjust the degree of parallel processing as instructed in a flexible manner. Parallel processing of the two tasks repeatedly led to large costs in performance and to high crosstalk effects compared to more serial processing. In spite of the costs, a moderate degree of parallel processing was preferred in the condition with no specific instruction. This pattern of results was observed if the same task set was used for the two tasks, but also if different ones were applied. Furthermore, a modified version of the central capacity sharing (CCS) model (Tombu and Jolicoeur in J Exp Psychol Hum Percept Perform 29:3–18, 2003) was proposed that accounts also for crosstalk effects in dual tasks. The modified CCS model was then evaluated by fitting it successfully to the present data.

Appendices

Appendix 1

Formal description of the CCS model

In this appendix we briefly describe the CCS model in formal terms. The formalization is important to understand how we modified the model to fit it to the data of Experiments 1A and 1B, which is described in “Appendix 2”. In Fig. 9, which shows typical PRP situations (they correspond to type B in Tombu & Jolicoeur, 2003), the assumed processing stages are represented by areas, whose size corresponds to the work (W) necessary to accomplish the processing at the respective stage. W is measured in units of capacity (c). Unless the task as such is modified, the work necessary for a given task at a stage is fixed. The vertical extensions of the areas correspond to the momentary processing rate r, which can vary between 0 and the maximum rate r _max. Usually, and without loss of generality, the maximum rate can be set to 1. The unit of r is capacity per second (c/s).

Fig. 9

Predictions of the CCS model are illustrated at short (I) and long SOA (II). Areas A1 and A2 represent the perceptual stages, whereas the areas C1 and C2 refer to the motor execution stages for Task 1 and Task 2, respectively. B1 and B2 denote the corresponding central stages with limited capacity. I a Central capacity is equally distributed between Task 1 and Task 2; I b more capacity is allocated on Task 1. I c The CB situation where all capacity is concentrated on Task 1. RT1 increases the more of the central capacity is allocated to Task 2 (compare I a–c). RT2 decreases with increasing SOA (compare I and II), whereas it is independent from the degree of capacity sharing (compare I a–c)

Because it is assumed that the stages A and C have unlimited capacity, the height of their corresponding areas is constantly r _max. Therefore, the duration of these stages can simply be calculated by dividing the necessary work by the maximum rate (i.e., T = W/r _max). More specifically, for two tasks, we have as processing times for the stages A1, C1, A2, and C2: T _A1 = W _A1/r _max, T _C1 = W _C1/r _max, T _A2 = W _A2/r _max, and T _C2 = W _C2/r _max, respectively. Interpreted in graphical terms, this means that the horizontal extensions of the areas in Fig. 9 directly reflect their relative duration of processing. The crucial point is to calculate the duration for the stages B1 and B2. According to the CCS model, the capacity for the central stage is limited, but can be divided between the tasks.

During the overlap of the central stages, the capacity limit implies that the processing rates for B1 and B2 have to sum up to r _max at each point in time. Let us denote the rate for stage B1 at time t by r _B1(t). Then, the rate for B2 at that time is r _max − r _B1(t). Consequently, the durations T _B1 and T _B2 depend on the central overlap between the tasks and on the degree of capacity sharing. If we consider the special case where the rate for B1 is constantly r _max, then the situation corresponds to a strict CB. Therefore, all results supporting a CB can also be accounted for by the CCS model. Additionally, however, the CCS model makes specific predictions for situations in which capacity is shared between the central stages. Two such situations with different sharing proportions are shown in Fig. 9.

T _B1 and T _B2 can be computed by piecewise calculations. As can be seen in Fig. 9, the temporal overlap between B1 and B2 depends on the difference T _A2 − T _A1 between the durations of the stages A1 and A2, and on the SOA. Obviously, in the interval SOA + T _A2 − T _A1, the processing rate for B1 equals r _max. The work done in this interval is (SOA + T _A2 − T _A1) · r _max. The remaining work [W _B1 − (SOA + T _A2 − T _A1) · r _max] is subsequently completed with a rate of r _B1. Thus, taken together, T _B1 can be computed as

$$ T_{{{\text{B}}1}} = {\text{SOA}} + T_{{{\text{A}}2}} - T_{{{\text{A}}1}} + {{\left[ {W_{{{\text{B}}1}} - \left( {{\text{SOA}} + T_{{{\text{A}}2 }} - T_{{{\text{A}}1}} } \right) \cdot r_{ \max } } \right]} \mathord{\left/ {\vphantom {{\left[ {W_{{{\text{B}}1}} - \left( {{\text{SOA}} + T_{{{\text{A}}2 }} - T_{{{\text{A}}1}} } \right) \cdot r_{ \max } } \right]} {r_{{{\text{B}}1}} }}} \right. \kern-\nulldelimiterspace} {r_{{{\text{B}}1}} }}. $$

(1)

In graphical terms, T _B1 is reflected by the length of area B1 (see Fig. 9). An important prediction can be derived by Eq. 1. Because the work for B1 is fixed, the duration (length) of this stage depends on the processing rate (height). RT1 is thus predicted to increase with a decreasing rate. That is, the more capacity is shared with B2, the more costs are produced for RT1. The duration of B2 is calculated analogously to that of B1:

$$ \begin{aligned} T_{{{\text{B}}2}} & = {{\left[ {W_{{{\text{B}}1}} - \left( {{\text{SOA}} + T_{{{\text{A}}2}} - T_{{{\text{A}}1}} } \right) \cdot r_{ \max } } \right]} \mathord{\left/ {\vphantom {{\left[ {W_{{{\text{B}}1}} - \left( {{\text{SOA}} + T_{{{\text{A}}2}} - T_{{{\text{A}}1}} } \right) \cdot r_{ \max } } \right]} {r_{{{\text{B}}1}} }}} \right. \kern-\nulldelimiterspace} {r_{{{\text{B}}1}} }} + {{\left( {W_{{{\text{B}}2}} - \left[ {W_{{{\text{B}}1}} - \left( {{\text{SOA}} + T_{{{\text{A}}2}} - T_{{{\text{A}}1}} } \right) \cdot r_{ \max } } \right]/r_{{{\text{B}}1}} \cdot r_{{{\text{B}}2}} } \right)} \mathord{\left/ {\vphantom {{\left( {W_{{{\text{B}}2}} - \left[ {W_{{{\text{B}}1}} - \left( {{\text{SOA}} + T_{{{\text{A}}2}} - T_{{{\text{A}}1}} } \right) \cdot r_{ \max } } \right]/r_{{{\text{B}}1}} \cdot r_{{{\text{B}}2}} } \right)} {r_{ \max } }}} \right. \kern-\nulldelimiterspace} {r_{ \max } }} \\ & = T_{{{\text{A}}1}} - T_{{{\text{A}}2}} - {\text{SOA}} + W_{{{\text{B}}1}} /r_{ \max } + W_{{{\text{B}}2}} /r_{ \max } . \\ \end{aligned} $$

(2)

If we consider Eq. 2, then it is obvious that the dependence of T _B2 on the SOA explains the PRP effect. Apart from that, T _B2 depends on the work required for the processing of Task 1 and Task 2. However, T _B2 does not depend on the relative rates (see Fig. 9). This property of the model implies that T _B2 and, thus, also RT2 is not affected by the degree of capacity sharing. In other words, sharing capacity with Task 2 does not produce any benefit on RT2, but it produces costs on RT1. Therefore, according to the CCS model, a strict serial processing strategy would be optimal. Furthermore, RT1 is predicted to decrease with an increasing SOA. Moreover, this effect should be the stronger, the smaller r _B1. Only if r _B1 equals r _max, then no influence of SOA on the performance for Task 1 should be observed.

Appendix 2

Fitting the CCS model to the data

The model was fitted to 48 mean data points from the Experiments 1A and 1B. Twelve of these points were from the Fixed-flanker condition: six points for RT1, and six points for RT2. The six points for each response type represent the three instructions and the two congruency conditions (the small first-color effect was ignored). The other 36 data points were taken from the Altering-flanker condition. The data pattern was the same as for the Fixed-flanker condition. However, there was one pattern for each of the three First-part intervals. Furthermore, only those trials were included in which the congruency type of the First-part flanker was the same as that of the Second-part flanker.

The use of the same judgment type for both tasks might be problematic in some respect. For fitting the data to a formal model, however, this condition is favorable, because it needs only a relatively small number of free parameters for its description. This is a crucial prerequisite in view of our restricted number of data points. We started with a formal version of the standard CCS model (see “Appendix 1”), and considered congruent trials as the standard situation for this version. We then extended the model to also account for the performance on incongruent trials, where we considered only the Fixed-flanker trials and those Altering-flanker trials, where the congruency type did not change between the First-part flankers and the Second-part flankers.

Because the same judgment types were used for both tasks in Experiment 1A and 1B, we made the reasonable assumption that the processing of Task 1 required the same work as that of Task 2. It follows that T _A1 = T _A2, and T _C1 = T _C2. Furthermore, as there was no variation in the duration of perceptual processing or in the execution of responses, W _A and W _C were considered as constant for all conditions. Apart from that, we assumed that capacity was shared in such a way that T _B1 was always less or equal to T _B2. In other words, less capacity was allocated to S2 than to S1 during the processing of Task 1.

The deficit of the standard CCS model with respect to the present experiments is that it cannot account for congruency effects. Therefore, we added the following assumptions to the model: first, also the crosstalk between the tasks consumes part of the central capacity. However, we supposed that for congruent stimuli the costs are outweighed by the positive response priming. Thus, the congruent situation was considered as functionally equivalent to the standard situation. For incongruent stimuli, though, the situation is different. They produce negative response priming, so that there is no compensation for the capacity reduction. Thus, for incongruent stimuli, we had to implement a reduction of central capacity. This was modeled by multiplying the processing rates r _B1 and r _B2 by a common reduction parameter z, which could vary between 0.5 and 1.

To show this in more detail, let us first consider the Fixed-flanker condition. Because we had a simultaneous onset of the stimuli in this case, the equations are relatively simple (the small First-color effect is ignored). Altogether, with our assumptions, Eq. 1 is modified in Eq. 3:

$$ T_{{{\text{B}}1}} = [1/(r_{{{\text{B}}1}} \cdot z)] \cdot W_{{{\text{B}}1}} . $$

(3)

A preliminary attempt to fit the model to the data confirmed our suspicion that the reduction of capacity was not constant but depended on the degree of capacity sharing. With a constant z, no satisfactory results were obtained for RT2. Presumably, this reflects the mechanism that an increased sharing leads to an increased crosstalk, which, in turn, consumes more central capacity. To take this into account, and to limit the number of extra parameters, z was defined as a linear function of r _B1, i.e.: z = d · r _B1/r _max + e, where d and e are free parameters.

The duration of stage B2 is determined by the length of the central overlap interval between the tasks, [1/(r _B1 · z)] · W _B1], and by the time needed for the remaining work (W _B2 − [1 / (r _B1 · z)] · W _B1 · r _B2 · z) · r _max. Taken together, we have in Eq. 4:

$$ T_{{{\text{B}}2}} = \left[ {1/\left( {r_{{{\text{B}}1}} \cdot z} \right)} \right] \cdot W_{{{\text{B}}1}} + \left( {W_{{{\text{B}}2}} - \left[ {1/\left( {r_{{{\text{B}}1}} \cdot z} \right)} \right] \cdot W_{{{\text{B}}1}} \cdot r_{{{\text{B}}2}} \cdot z} \right)/r_{ \max } . $$

(4)

Simplification leads to

$$ T_{{{\text{B}}2}} = W_{{{\text{B}}1}} \cdot \left( {1/r_{{{\text{B}}1}} } \right)\left( {1/z - 1} \right) + W_{{{\text{B}}1}} /r_{ \max } + W_{{{\text{B}}2}} /r_{ \max } . $$

As can be seen, if z = 1, as is assumed for congruent stimuli, we have the same results for T _B1 and T _B2 as in Eqs. 1 and 2. If capacity is reduced, however, response times are increased. Because z is also part of the equation for T _B2, RT2 increases with the degree of capacity sharing, in contrast to Eq. 2.

These basic equations also apply to the Altering-flanker condition. However, because the results are relatively complicated for the condition where the congruency type changed from the First-part to the Second-part flankers, we included only the trials in which First-part and Second-part flankers were of the same congruency type. For these trials, the performance was always rather similar to that in the corresponding Fixed-flanker conditions. We merely had to take the First-part interval variation into account. Our results show that the change of flanker identity on these trials produced some costs, which increased with increased duration of the First-part interval. Therefore, we modeled the Altering-flanker data in the same way as the Fixed-flanker data, except that the term ‘g · INT’ (INT = interval), was added to the equations. This term increases the time for the central stages in Eq. 5, as compared to Eqs. 3 and 4 for the Fixed-flanker trials, by some duration that is proportional to the First-part interval:

$$ \begin{aligned} T_{{{\text{B}}1}} & = \left[ {1/\left( {r_{{{\text{B}}1}} \cdot z} \right)} \right] \cdot W_{{{\text{B}}1}} + g \cdot {\text{INT}}. \\ T_{{{\text{B}}2}} & = W_{{{\text{B}}1}} \cdot \left( {1/r_{{{\text{B}}1}} } \right)\left( {1/z - 1} \right) + W_{{{\text{B}}1}} \cdot \left( {\text{s/c}} \right) + W_{{{\text{B}}2}} \cdot \left( {\text{s/c}} \right) + g \cdot {\text{INT}}. \\ \end{aligned} $$

(5)

The work for the central stages, W _B1 and W _B2, was deliberately set to 500 c, respectively. Because the pre- and post-central stages were assumed to be identical for both tasks and constant for all conditions, they were modeled by a single additive constant. Moreover, because the predicted RTs had also to be scaled in order to be in the same range as our data, we used a single linear transformation to obtain the estimated RTs, i.e., RT1 = a · T _B1 + b and RT2 = a · T _B2 + b.

Altogether, we had 6 relevant parameters for fitting the 48 data points: three values for r _B1 corresponding to the three instruction conditions, two linear parameters d and e for computing the capacity reduction z, and one parameter g for the first-part interval effect. The model was fitted by a routine (SIMPLEX) that estimated the parameter values by minimizing the sum of squared errors.

As estimation for r _B1 the procedure revealed the values 0.872, 0.747, and 0.666 for the serial, neutral, and parallel condition, respectively. The values of the parameters d and e to determine the reduction parameter z were 0.803 and 0.0729, respectively. The obtained value for parameter g was 0.633. Finally, the scaling parameters a and b were 0.684 and 226. As can be seen in Fig. 9, with these parameters the model fits the data very well. This is also reflected by the corresponding R² of 0.999.