Psychometric functions for one subject in the difficult (A) and easy (B) versions of the contrast decrement detection task. Probe detection probability increases steadily with probe duration (note the log scale on the x axis). Although the functions are steeper and shifted to the left in the easy version compared with the difficult version, set size (the number of cued elements) significantly affects performance in both cases.

Reaction times (A) and error rates (B) for contrast decrement detection in the control experiment, as a function of set size (number of precued elements), for the two different difficulty levels (n = 8). Whereas error rates are unaffected by task difficulty, reaction times strongly differ depending on the task. The search slopes (average RT increase for each set size increase) are 38 ms per item and 1 ms per item, respectively, for the difficult and easy versions of the task, suggesting that the task difficulty alters the underlying attentional requirements.

Experimental design. Four disks filled with high-contrast dynamic random noise were constantly shown to the subjects. At the beginning of each block, a cue display indicated which of the disks were likely to contain the probe for the following trials (set size 1–4). The probe was a contrast decrement of variable duration, occurring with 50% probability in one of the cued elements. At set size 1, “full attention” could be allocated to the probe, whereas at set sizes 2, 3, and 4, attention had to be “divided” between multiple targets. In 10% of trials (including probe-present trials), a catch stimulus (similar to the probe) was shown at an uncued location. By definition, we assume that it received “minimal attention.” (Set size 4 trials had no uncued location and therefore could not serve as catch trials). At the end of each trial, subjects reported whether the probe was presented among the cued locations, and (using a separate key) whether a catch stimulus had been detected at an uncued location. Probe and catch stimulus duration varied independently between 50 and 1,000 ms. The strength of the contrast decrement could either be 50% (“easy” version) or 30% (“difficult” version).

Model prediction errors for one subject (A and B, same subject as in Fig. 3) and averaged across eight subjects (C and D), in the difficult (A and C) and the easy (B and D) versions of the contrast decrement detection task. The prediction error is the r.m.s. distance between observed and predicted data in the divided attention conditions (i.e., set sizes 2, 3, and 4). This error is computed for each possible value of the model parameter (sampling period for the sample-always and the sample-when-divided models, lower horizontal axis; division cost for the “parallel” model, upper horizontal axis). The mean prediction error (±SEM, shaded area) across set sizes is plotted for each model. The optimal model is the one yielding the lowest prediction error (horizontal arrows on the left vertical axis), which in A is the sample-always model and in B is the parallel model. The optimal parameter values for each model are indicated by the vertical arrows. (C) Average prediction errors across eight subjects in the difficult version of the task. The shaded area represents SEM across subjects. The sample-always model was the optimal for all subjects tested, with an optimal “sampling period” of 140 ms on average (range 100–190 ms). (D) Average prediction errors across eight subjects in the easy version of the task. The optimal model in this case is the parallel model, with an optimal “division cost” of 17% on average (range 9–25%). (E and F) Prediction error of each model at its optimal parameter value (average across eight subjects). (E) For a difficult contrast decrement detection task, the model that best reflected human performance was a sampling model, in which periodic sampling occurred even during undivided attention (sample-always model). (F) For an easier version of the same task, the parallel strategy yielded the lowest prediction error. The classic idea of a switching spotlight (sample-when-divided model) did not closely reflect our observers' strategy in either situation.

Computational principle. The detection probability of a probe event (e.g., brief contrast decrement) increases with its duration. Using the psychometric functions obtained for full attention (when the probe location is known) and minimal attention (when the probe occurs at an unexpected location), one can derive predicted functions for divided attention (when the probe occurs at one of multiple possible target locations). Depending on the strategy assumed for attentional allocation (“parallel,” “sample-when-divided,” or “sample-always”), the predicted functions will differ. We can thus distinguish among these three alternative models of divided attention. (A) Within a parallel model, probe information for divided attention accrues at a rate intermediate between that obtained for full and for minimal attention; performance only depends on performance with full and minimal attention at the same probe duration. (B) Within a sampling model, performance for divided attention (shown here for one example trial with a set size of 2) originates from a series of periods during which information accrues either at the rate of full attention (periods marked 1 and 3 in this example) or at the rate of minimal attention (period 2). Only the onset portion of the full attention psychometric curve, with values below the postulated sampling period, is used to generate these predictions (because attention will not rest at a given location for longer than the postulated sampling period). Note that the illustration is only meant to exemplify what happens on a single hypothetical trial; this explains the accelerating shape of the information accrual function for this illustrative trial. In the general case, because the onset time of each attentional sample on any given trial is unknown, we integrate the depicted calculation over all possible onset times (see SI Appendix); this integration will result in regular nonaccelerating psychometric functions (data not shown). (C) In the “sample-always” model, information accrues in discrete epochs, even when a single location is cued. The “set size 1” psychometric function therefore reflects a collection of such epochs and not the true full attention function. The full attention curve can be recovered from the set size 1 curve, however, by inverting the previously described calculation (see SI Appendix). Performance with divided attention is then derived as before (B) by combining periods of full attention and minimal attention.