Other than exhaustive search, self-terminating search, which incorporates an additional stopping rule in which the participant scans the arrows one by one until the majority threshold is reached (e.g., 2 arrows pointing in the same direction within a set size of 3 arrows), is clearly the second most straightforward algorithm. Consistent with this algorithm, reveals that RT increases with the computational load in two cases: (a) across the three congruent conditions, in which all arrow(s) in each set point in the same direction, as indicated by the dashed line; and (b) within the two conditions of set size 3 and within the three conditions of set size 5, as indicated by the solid lines. However, the opposite prediction for the 2
1 and 5
0 conditions based on self-terminating search stands as evidence against the possibility that subjects adopted this algorithm.
The fact that human visual attention can be directed towards more than one item simultaneously allows for the possibility of a grouping search algorithm, in which participants first select a sample of arrows with a size equal to the majority threshold and then process the sample. This is similar to perceptual grouping 
. If all arrows in the sample happen to point in the same direction (congruent), then a response can be quickly generated. If not, a re-sampling kicks in until a congruent sample is found and a response is generated. We estimate the computational load for the grouping search as a logarithmic function of the product of the grouping size and the expected number of groups that need to be sampled in order to obtain one that is congruent. It is clear that RT increases monotonically as a function of the computational load and that this relationship is well approximated by a linear function (). The linear relationship between RTs and the computational load based on the grouping search strategy may suggest a tree-like structure representing the arrows to be sampled and a dichotomizing test.
These results support the idea that RT is determined not only by the amount and content of the input but also by the algorithms of mental operations that people adopt in the face of information uncertainty. Situations in the real world are often more complex than laboratory choice-RT tasks and require more voluntary control. The majority function task is interesting in that it requires greater voluntary control of computation than tasks used for testing the conflict effect (e.g., 
), although it also uses conflicting information to manipulate information uncertainty. With this task we highlight the role of voluntary control during information processing and provide a more general framework to account for the conflict effect. For example, in a variation of the flanker task 
in which people were asked to detect the direction of the target arrow and ignore the distracters, we observed a typical conflict effect–the RT difference between the incongruent and congruent conditions–of about 50–150 ms. This can be accounted for by the computational load framework because the computational load for the congruent condition is 1 bit for the two alternative responses, whereas it is less than or equal to 2 bit for the incongruent condition because of incongruent flankers.
In addition, performance in the majority function task cannot be fully predicted by a conflict effect account. For example, comparing two conflicting conditions in which the distribution of arrows are 2
1 and 4
1 in set sizes 3 and 5 respectively, the RT of the latter condition is significantly longer (1121 vs. 1261, t(23)
4.55, p<0.001) , which is opposite of what the conflict effect account predicts, since the non-target to target ratio is larger in the former. Similar to other categorization tasks 
, the goal of computation is to identify the majority based on the input. Because any arrow could potentially belong to the majority subset if the set size is equal to or greater than 3, more than one arrow needs to be processed, either scanned one-by-one or randomly sampled and grouped as we tested above. However, the degree of uncertainty caused by conflicting information predicts RT within a given set size if the grouping search is adopted. For example, for the set size 5, the most uncertain condition with the distribution of arrows of 3
2 requires 10 grouping attempts to be made on average before a solution is reached based on the grouping search algorithm. This may explain why its RT is much longer (1615 ms) compared to another less uncertain but also conflicting condition with the distribution of arrows of 4
1 (1261 ms), which requires only 2.5 attempts on average to obtain a congruent sample.
The majority function task reported in this paper has features of conflict, grouping, and input variation that are often elements of many separate tasks in the literature. The methods to compute the computational load in this task may be used to account for discrepancy between findings of previous studies on conflict effect using different tasks. This majority function task is similar to the visual motion task used in studies of perceptual decision making (e.g., 
), but here we examine and model decision making on a system level by considering the algorithms potentially adopted by the brain to process discrete information. We cannot exclude other possible factors that might contribute to the current results such as the Gestalt effect based on the holistic perception of all congruent arrows, information reduction 
, or perceptual grouping 
. This may account for overall faster responses and the relatively flat slope for the congruent conditions. Although certain common mechanisms might be involved in voluntary control, the underlying algorithms will vary and be task specific in different situations depending on different computational goals 
. For example, under the high input information condition, which is beyond the grouping capacity limit (e.g., more than 5 arrows), other algorithms, such as those suggested for perceptual decision making regarding motion coherence, might be adopted by humans to find the majority.
We argue that voluntary control is implemented by algorithms of mental operations, which are in turn implemented by brain networks. This study demonstrates that it is important and plausible to analyze the underlying algorithms for voluntary control by examining the relationship between the amount and content of input and RT. RT is a basic and central measure of mental operations in almost all cognitive tasks 
. Early studies based on information theory 
have found that choice RT is determined by the amount of information in bits that has to be processed to generate a correct response (e.g., 
), though the causality in this relationship has been challenged 
. Some elegant models for the central mechanisms of choice RT have been proposed, and changes in RT as a function of information processing have been studied in the context of perceptual decision making (e.g., the sequential-sampling models, for a review, see ref. 
), mental addition (subtraction) 
, visual search 
, and categorization 
. In this study, we explicitly considered the underlying algorithms for voluntary control of information processing.