A new measure of group decision-making efficiency

Hsieh, Cheng-Ju; Fifić, Mario; Yang, Cheng-Ta

doi:10.1186/s41235-020-00244-3

Original article
Open access
Published: 17 September 2020

A new measure of group decision-making efficiency

Cognitive Research: Principles and Implications volume 5, Article number: 45 (2020) Cite this article

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Abstract

It has widely been accepted that aggregating group-level decisions is superior to individual decisions. As compared to individuals, groups tend to show a decision advantage in their response accuracy. However, there has been a lack of research exploring whether group decisions are more efficient than individual decisions with a faster information-processing speed. To investigate the relationship between accuracy and response time (RT) in group decision-making, we applied systems’ factorial technology, developed by Townsend and Nozawa (Journal of Mathematical Psychology 39, 321–359, 1995) and regarded as a theory-driven methodology, to study the information-processing properties. More specifically, we measured the workload capacity C_AND(t), which only considers the correct responses, and the assessment function of capacity A_AND(t), which considers the speed-accuracy trade-off, to make a strong inference about the system-level processing efficiency. A two-interval, forced-choice oddball detection task, where participants had to detect which interval contains an odd target, was conducted in Experiment 1. Then, in Experiment 2, a yes/no Gabor detection task was adopted, where participants had to detect the presence of a Gabor patch. Our results replicated previous findings using the accuracy-based measure: Group detection sensitivity was better than the detection sensitivity of the best individual, especially when the two individuals had similar detection sensitivities. On the other hand, both workload capacity measures, C_AND(t) and A_AND(t), showed evidence of supercapacity processing, thus suggesting a collective benefit. The ordered relationship between accuracy-based and RT-based collective benefit was limited to the A_AND(t) of the correct and fast responses, which may help uncover the processing mechanism behind collective benefits. Our results suggested that A_AND(t), which combines both accuracy and RT into inferences, can be regarded as a novel and diagnostic tool for studying the group decision-making process.

Significance

Previous studies have shown the so-called collective benefit. That is, performance is more accurate when participants work as a group in which they can communicate with each other verbally or non-verbally, and with an exchange of decision evidence or internal estimate of confidence. However, it is still unclear whether group decisions are more efficient than individual decisions since a tradeoff may exist between speed and accuracy. In other words, increasing the number of group members may increase the group’s response accuracy, but at the same time would slow down the processing speed. The aim of the study was to learn about the relationship between accuracy and response-time (RT) measures in group decision-making. To sum up, our results replicated the previous findings that showed the collective benefit for accuracy: the group’s detection sensitivity was higher than the best individual’s detection sensitivity only when group members’ detection sensitivities were similar. The measures of processing speed, the workload capacity measures, revealed that group decision-making was of supercapacity processing. In addition, our results suggested that the assessment function of workload capacity, which combines both accuracy and RT into inferences, can be regarded as a novel and diagnostic tool to study the group decision-making process. The current study is not only a replication of the previous studies, but also highlights the importance of combined accuracy and RT measures in the inference of the group decision-making process.

Introduction

An old saying goes, “Two heads are better than one.” Combining group members’ opinions to make a coherent decision is usually regarded as a better means of decision-making than having an individual make a decision alone (Clemen, 1989). Many important real-world decisions are collective decisions, such as juries rendering verdicts or a team of radiologists reading X-rays. The situation in which group decisions are considered to be superior to individual decisions is termed “wisdom of crowds”^{Footnote 1} (Surowiecki, 2004).

In the literature, the primary focus has been on learning about the mechanism that underlies group decisions. Previous studies have investigated the properties of cooperation between two or more participants in perceptual decision tasks (Bahrami, Olsen, Latham, Roepstorff, Rees, & Frith, 2010; Bahrami, Olsen, Bang, Roepstorff, Rees, & Frith, 2012a; Sorkin & Dai, 1994; Sorkin, Hays, & West, 2001; Sorkin, West, & Robinson, 1998) and found that a group exhibits a decision advantage—a so-called “collective benefit”—over an individual decision-maker. There are several proposed possible explanations for the existence of the collective benefit.

First, it is possible that the collective benefit results from a reduced workload, as group members strategically split information among themselves. The fact that each member can focus attention on a subset of information (which means the group does not have to focus its attention on the entirety of information) could lead to an increase in collective processing efficiency. However, group decision accuracy may be limited by individual ability because the group must rely on the capabilities of each member. This possibility has been challenged and ruled out by Barr and Gold (2014). Barr and Gold (2014) manipulated the group size (one to four members) and the quantity of information (partial or full) that each member received. Their results showed that groups viewing the entirety of information significantly outperformed groups whose members viewed limited portions of information and suggested that strategically splitting information does not necessarily lead to a collective benefit.

Second, the collective benefit could be due to the statistical facilitation effect (Green & Swets, 1966; Lorge & Solomon, 1955; Sorkin & Dai, 1994; Sorkin et al., 2001; Swets, Shipley, McKey, & Green, 1959). In stochastic modeling, adding more independent random variables to a parallel processing system can lead to a faster and more accurate task completion. The statistical facilitation that is achieved through the redundancy gain, has been used by engineers to decrease failure rate of their machines. Analogously, one can demonstrate a similar effect in human group decision-making, by increasing the number of independent decision-makers who work in parallel. The overall group’s achievement will be better than that of any individual member working alone. Interestingly, the collective benefit, that is due to the statistical facilitation and the redundancy gain, is merely an outcome of the statistical improvement – that is, the group benefit is not achieved by the group members’ interaction. Such a statistical facilitation effect is conditioned on the use of the so-called first-termination rule,^{Footnote 2} which means that the system would wait for the fastest and correctly responding unit to complete and would then use it to make the final decision while ignoring the unfinished or incorrectly responding units.

Third, the collective benefit could be a result of the integration of evidence collected by each group member via social interaction (Bahrami et al., 2010; Lorenz, Rauhut, Schweitzer, & Helbing, 2011). This explanation is different from the first two in that it assumes the collective sum of knowledge occurs not only as a simple sum of individual knowledge, but as novel knowledge created through a series of social interactions. As a result, the performance of a group is better than that of the best observer or exceeds the expectations of individual members working in isolation (Collins & Guetzkow, 1964; Davis, 1969, 1973). Consistent with the coactive model^{Footnote 3} (Houpt & Townsend, 2011; Schwarz, 1989, 1994; Townsend & Nozawa, 1995), the collective benefit may have occurred because the individual contribution is weighted and integrated into a single information channel following a “weighted-and-sum” principle of information integration.

Recently, an increasing number of studies has challenged the idea that group decisions would always outperform individual decisions. For example, Fific and Gigerenzer (2014) suggested that adding more decision-makers does not necessarily enhance the group performance. The best individual may match the collective decision accuracy or even outperform the group, especially when free riders exist. In addition, a single expert, under certain conditions, can outperform the group (e.g., Gordon, 1924; Graham, 1996; Winkler & Poses, 1993). The debate over the potential negative effect of collaboration on decision-making has intensified work that explores the conditions under which the collective benefit/cost may arise.

A potential factor that may influence the collective effect was performance similarity between group members. In a two-interval forced-choice oddball detection task in which participants had to decide which interval contained an odd target, Bahrami et al. (2010) investigated whether participants can utilize their partner’s confidence rating to improve group decision sensitivity.^{Footnote 4} The results showed that only in a consistent group, in which the two group members had similar detection sensitivity, was the group decision superior to individual decisions. Specifically, the authors used S_min and S_max to represent the detection sensitivity of the worse and better individual, respectively, and only when S_min/S_max ≥ 0.4 did the group show the collective benefit, i.e., S_dyad/S_max > 1 (S_dyad denotes group detection sensitivity). By contrast, in an inconsistent group (i. e. , S_min/S_max < 0.4), the group decision was worse than the decision made by its better group member. Bahrami et al. (2010) further suggested that the results supported the weighted confidence-sharing model (WCS), which assumes that individuals can take advantage of the confidence information, i.e., an internal estimate of the probability of being correct; the final decision is made based on the weighting function of the group members’ confidence. The WCS model can be considered a variant of the coactive models.

Another important factor in understanding the collective is the tradeoff between accuracy and speed in group decision-making (see Heitz, 2014 for a review). The collective effect can be measured by both response accuracy and RT. However, when used in the same task, the two measures can have an inverse relationship. That is, increasing the number of group members may increase the group’s response accuracy, but, at the same time, could create longer RTs; for example, as the group size increases, group members require more time to communicate with each other to reach a consensus. Thus, it is reasonable to speculate that collaboration can increase response accuracy but slow down the decision RT. The above-mentioned studies focused mainly on the effect of collaboration on accuracy measures (e.g., Bahrami et al., 2010, 2012a, b) while neglecting its effect on the measure of processing speed.

Two other studies utilized RT measures (e.g., RT distribution analysis) to assess the collective effect (e.g., Brennan & Enns, 2015; Yamani, Neider, Kramer, & McCarley, 2017). Brennan and Enns (2015) tested the violation of the race-model inequality to infer the collective benefit. In general, the race model assumes that two processing units are racing to reach a certain decision criterion (Miller, 1982). The two units work independently, and the faster unit, which reaches the decision criteria first, will determine the response outcome. In our domain of interest, the units are defined as the individual decision-makers. The race-model inequality assumes that two group members work independently and in parallel (simultaneously). In a nutshell, a violation of the race-model inequality would suggest that the two decision-makers did not work independently of each other and that at some point, prior to making a final decision, they interacted with each other. In terms of modeling processing systems, this situation is defined by coactive processing. Using a visual enumeration task in which participants were required to count the number of targets (0/1/2) presented against the distractors, Brennan and Enns (2015) demonstrated that the observed RT data violated the race-model inequality, thereby supporting the notion that the two decision-makers did not work independently and collaboration would facilitate decision RTs.

Using another RT measure, i.e., workload capacity, Yamani et al. (2017) examined how collaboration affects individual processing efficiency in terms of the information-processing speed. Workload capacity is a measure of the change in processing efficiency (speed) at the individual subject level when the system’s workload (i.e., the number of decision-makers) increases, as proposed by the framework of Systems Factorial Technology (SFT, Little, Altieri, Fific, & Yang, 2017; Townsend & Nozawa, 1995). According to SFT, increasing the number of processing units (i.e., the system’s workload) can have three different effects on the processing speed of an individual processing unit. In the case of limited-capacity processing, the speed of processing per processing unit slows down when more units operate at the same time. In the case of unlimited-capacity processing, the speed of processing per processing unit remains unchanged when more processing units are added. In the case of supercapacity processing, the speed of processing per processing unit speeds up by the addition of more decision units. This could be the result of a facilitatory interaction between decision units. It is notable that an unlimited-capacity system implies that the efficiency of an individual unit remains unchanged, whereas the system overall can be performing better compared with the individual unit, due to stochastic considerations. In the context of group decision-making, the limited capacity indicates some form of inhibition between individual decision-makers when they work as a group. In the case of unlimited capacity, the addition of more group members does not affect individual efficiency. In the case of supercapacity, efficiency improves as a result of the group members’ facilitatory interaction. Yamani et al. (2017) study adopted the shared-gaze technique which allows participants to see where their partner is looking in order to study whether collaboration can benefit scan in teams. Results showed supercapacity processing when both group members were required to find and respond to a target; by contrast, limited capacity was found when the faster searcher found and responded to the target. The supercapacity results suggested that, by holding fixation on the target, the faster searcher can cue their partner to the target location, which, in turn, boosts the processing speed of the slower searcher. The limited-capacity results suggested that shared gaze offered no benefits but slowed down the processing for the faster searcher.

To summarize, collective benefit/cost can be measured by either response accuracy or RTs. These two measures play complementary roles in understanding the mechanism underlying group decisions. However, to our knowledge, there is no prior study that combines accuracy and RT measures to infer the dynamic process of group decision-making. This raises several related questions. Are the inferences from the two measures consistent enough to draw similar conclusions? Is it possible to use a single performance index to quantify the collective effect by considering the two measures simultaneously? In the present study, we integrated, within one study, the two approaches by applying SFT (Little et al., 2017; Townsend & Nozawa, 1995). Before we go into the details about the present study, we first briefly introduce the theory and methodology of Systems Factorial Technology (SFT).

Systems Factorial Technology

SFT (Little et al., 2017; Townsend & Nozawa, 1995) is a useful tool for analyzing and diagnosing the dynamic decision-making process. A wide range of fields in cognitive research have utilized SFT, such as visual search (Zehetleitner, Krummenacher, & Müller, 2009), memory search (Townsend & Fific, 2004; Van Zandt & Townsend, 1993), face perception (Ingvalson & Wenger, 2005; Yang, Altieri, & Little, 2018), classification (Fific, Nosofsky, & Townsend, 2008), change detection (Yang, 2011; Yang, Chang, & Wu, 2013; Yang, Hsu, Huang, & Yeh, 2011), cued detection (Yang, Little, & Hsu, 2014; Yang, Wang, Chang, Yu, & Little, 2019), word processing (Houpt, Sussman, Townsend, & Newman, 2015; Houpt, Townsend, & Donkin, 2014), audiovisual processing (Altieri & Yang, 2016; Yang et al., 2018; Yang, Yu, & Chang, 2016), and group decision-making (Yamani et al., 2017). According to SFT, two important information-processing properties of group decision-making can be uncovered, such as a group’s organization during task participation (i.e., how two individuals work together to achieve a group decision) and workload capacity (i.e., individual decision efficiency varies as a function of the number of decision-makers).

In this paper, we used the workload capacity measure to quantify group decision-making efficacy. The workload capacity measures can be used to indicate the amount and type of the potential collective benefit. Here, we introduced two types of capacity measures. First is the AND capacity^{Footnote 5} (C_AND(t)), a standard measure of workload capacity, developed by Townsend and colleagues (e.g., Townsend & Nozawa, 1995), which considers only the RT data of correct responses. The AND capacity is analyzed by comparing the group processing efficiency to a baseline predicted from the UCIP model (i.e., unlimited-capacity, independent, parallel model), which assumes that all group members work independently and in parallel. The workload capacity is formalized as a ratio of the cumulative reverse hazard functions, K(t) = ln F(t) where F(t) = P (RT ≤ t) (Chechile, 2003, 2011; Townsend & Eidels, 2011; Townsend & Wenger, 2004), and is expressed as:

$$ {C}_{AND}(t)=\frac{K_1(t)+{K}_2(t)}{K_{12}(t)} $$

(1)

for t > 0, where K₁, K₂, and K₁₂ represent the cumulative reverse hazard function of the two non-collaborative conditions, in which participants perform the task independently, and the collaborative condition, in which participants work together with social interaction (here, the non-verbal communication), respectively. The interpretation of C_AND(t) > 1 implies that group performance is better than the prediction from the UCIP model—that is, the system engages in supercapacity processing. When C_AND(t) = 1, it suggests an unlimited-capacity processing system, implying that individual decision performance is unchanged when the number of group members increases. When C_AND(t) < 1, it suggests a limited-capacity processing system, implying that social interaction may, in fact, even slow down individual decision time.

Second, we introduced a new measure, assessment function of workload capacity A_AND(t), to analyze group decision efficiency. To our knowledge, A_AND(t) has not been used to study the group decision-making process. Similar to C_AND(t), A_AND(t) has the advantage of inferring the dynamic processing efficiency as a function of RT (Donkin et al., 2014; Townsend & Altieri, 2012). Better than C_AND(t), A_AND(t) combines both accuracy and RT data into the analysis, such that we can analyze the decision efficiency of four response conditions: (a) correct and fast, (b) correct and slow, (c) incorrect and fast, and (d) incorrect and slow. The inferences of A_AND(t) were similar to C_AND(t); the details of the data analysis and inferences will be introduced in the “Data analysis” section. Please see Table 1 for the SFT-related theoretical glossary. More details can also be found in Townsend and Altieri (2012).

Table 1 Systems Factorial Technology (SFT)-related theoretical glossary

Full size table

The present study

In the present study, we conducted two psychophysics experiments by collecting both accuracy and RT data to test whether collaboration through the exchange of confidence information can promote group decision efficiency. In the first experiment, we extended the study by Bahrami et al. (2010) by adopting the two-interval forced-choice oddball detection task. In the second experiment, we used a yes/no Gabor detection task. Both accuracy-based and time-based measures were computed to infer the collective effect. First, the accuracy-based collective effect was computed by comparing the dyad’s sensitivity with the maximum individual sensitivity (i.e., S_dyad/S_max). Of particular interest was testing whether this effect would change as a function of relative detection sensitivity between the two group members (i.e., S_min/S_max). Second, the time-based collective effect was inferred from two workload capacity measures (C_AND(t)) and A_AND(t))) as introduced in the previous section.

Our first goal of the study is to replicate the effect of sensitivity similarity on joint decisions. We expect that only when group members have similar detection sensitivities, the collective benefit would be observed; that is, group detection sensitivity will be higher than the detection sensitivity of the best observer. Our second goal is to evaluate how RT and accuracy measures are consistent enough to draw similar conclusions about the collective effect. We expect to observe significant correlations between the time-based and accuracy-based measures of collective effect. Our last goal is to establish the assessment function of workload capacity as a standard measure of group decision efficiency. Considering the speed-accuracy trade-off effect, A_AND(t), which combines both accuracy and RT into the analysis, can be regarded as a better index for quantifying the group decision advantage.

Experiment 1

In Experiment 1, a two-interval forced-choice oddball detection task was adopted. The relative detection sensitivity between the group and the best observer was computed to infer the accuracy-based collective effect. Additionally, the workload capacity of group decision-making was assessed as a time-based measure of the collective effect. If both accuracy-based and time-based measures can reflect the collective benefit, we should expect that the accuracy-based collective effect (i.e., S_dyad/S_max) is greater than 1 and that time-based measures reveal supercapacity results.

Method

Participants

Fourteen (eight male and six female; age: 21.3 ± 2.05 years) undergraduate students at National Cheng Kung University volunteered to participate in this experiment and were randomly divided into seven pairs. All of the participants were right-handed. Before the experiment, participants signed an informed consent form. The ethics approval for the study was obtained from the Human Research Ethics Committee of National Cheng Kung University, and the experiment was conducted in accordance with the approved guidelines and regulations. The participants were either compensated in the amount of NTD 140 per hour or received class credit for their participation.

Equipment

A desktop computer with a 3.20 G-Hz Intel Core i7–8700 Processor, Intel UHD Graphics 630, and 8 GB of RAM controlled the display and recorded the manual responses. Stimuli were presented on a 19-in. CRT monitor with a refresh rate of 75 Hz and a display resolution of 1024 x 768 pixels. The viewing distance was kept at 60 cm and a chin-rest was used to prevent any head movements. The experiment was programmed using Psychtoolbox (http://psychtoolbox.org/) from MATLAB (Mathworks Inc., Natick, MA, USA).

Design, stimuli, and procedure

Figure 1 shows the flowchart of the task design and Fig. 2 shows an illustration of the trial procedure for each condition.^{Footnote 6} All the participants participated in three cooperation conditions: the individual condition, the non-collaborative condition, and the collaborative condition. Participants first performed the individual condition. Then they were randomly paired to form a dyad.^{Footnote 7} As a dyad, each participant faced their screen and was positioned next to their partner at a distance of 55 cm. They could only see their partner’s responding hand. Therefore, we believe that other forms of communication (e.g., body language) are not available. The order of the collaborative and non-collaborative conditions was counterbalanced across dyads (see Fig. 1).

The response mode was slightly different across the three cooperation conditions. In the individual condition, participants were required to complete the task alone and to provide a response by clicking the left or right button of the mouse to indicate which interval contained the odd target. After they had made their decision, they were asked to rate their confidence in their judgment using a Likert-type scale from 1 (“very doubtful”) to 5 (“absolutely sure”). The confidence information was used for the collaborative condition in the next stage.

In the non-collaborative condition, two participants working as a dyad performed the task together but without any communication. One participant delivered a response with a mouse click while the other participant responded with a keyboard press. During each trial, only one of the participants was required to respond and the person who was required to respond was dependent on the color of the question mark, with the color green indicating that the participant with the keyboard should respond and the color red indicating that the participant with the mouse should respond. The color (green/red) was randomly selected with equal probability. See Fig. 2 for an example, because the question mark is colored in red, the participant with the mouse should respond and the participant with the keyboard does not need to respond. After a response was made, a feedback display was shown to indicate the correctness of the present decision (middle) and the correctness of the participants’ individual decision about the same trial tested in the individual condition (top and bottom colored in red and green, respectively).

In the collaborative condition, the procedure was the same as that in the non-collaborative condition except that each participant’s confidence information was presented on a horizontal line to indicate the participants’ confidence in the judgment regarding the same trial that was tested in the individual condition. Participants could only communicate by exchanging their confidence information and no other forms of communication (e.g., verbal communication) were allowable. Specifically, the confidence information was represented by two colored marks (i.e., red and green marks representing the confidence rating made by the two participants, respectively). The farthest-left side represented the fact that the participants were very confident that the target was presented at Interval 1, while the farthest-right side represented the fact that the participants were very confident that the target was presented at Interval 2. This display of the participants’ confidence allowed them to communicate with each other and non-verbally exchange their confidence ratings.

Each trial started with a fixation cross displayed for a random duration of between 500 and 1000 ms. Afterward, two consecutive stimulus displays were presented for 85 ms with a 1000-ms blank interval in between the two displays. Each display contained six Gabor patches, which were placed in an imaginary circle with a radius of 232.7 pixels (8.0°). All the Gabor patches were placed equidistant from each other. Each Gabor patch was vertically oriented (standard deviation of the Gaussian envelope: 0.45°; spatial frequency: 1.049 cycle/°). In one of the two displays, there were an odd target and five distractors; in the other display, there were six identical distractors. The contrast of the distractor was 10%. The contrast of the target was 1.5, 3.5, 7.0, or 15.0% higher than the distractor (i.e., the contrast level of the target was 11.5, 13.5, 17.0, or 25.0%) and was randomly selected from one of the four contrast levels. The background was colored in gray with a luminance of 36.34 cd/m². The participant’s task was to make a two-interval forced-choice (2IFC) to indicate which interval contained the odd target, as accurately and quickly as possible after the question mark was presented. The question mark was presented until the participants delivered a response or 2 s had elapsed.

Each cooperation condition involved two sessions in order to collect enough data points. In each session, participants first performed a practice block of 64 trials and then 10 blocks of formal trials. Each block consisted of 2 (target was presented at Interval 1 or 2) × 4 (contrast levels of the target) × 8 (trials per combination).

Data analysis

The practice trials were excluded from the analysis. Correct RTs within a range of the 2.5% quartile and the 97.5% quartile were extracted for further analysis to exclude the outliers. To compute the collective effect, we compared the data of the collaborative condition (i.e., two participants working together with an exchange of the partner’s confidence) to the data of the non-collaborative condition (i.e., two participants working independently without any communication). In doing that, we were able to conclude that the collective benefit stems from the non-verbal communication rather than from the social facilitation effect. It is notable that, in the following, the term “non-collaborative-individual” represents the individual performance extracted from the non-collaborative condition rather than from the individual condition. The reason why we did not directly compare the data of the collaborative condition to the data of the individual condition is that a large processing difference existed between the two conditions. That is, in the collaborative and non-collaborative conditions, participants were required to first process the color of the question mark such that they would know who was going to respond in the present trial, whereas there was no need to process the color of the question mark in the individual condition. If the processing was less efficient in the collaborative condition than it was in the individual condition, we did not know whether the results were due to collective cost or whether making a group decision would require an additional process.

First, we conducted a mixed-design analysis of variance (ANOVA) to test the differences in accuracy and mean correct RTs, with the social condition (better observer, worse observer, collaboration) serving as a between-subject factor and with contrast level (11.5, 13.5, 17.0, or 25.0%) serving as a within-subject factor. It is notable that the data of the worse observer and better observer was extracted from the data of the non-collaborative condition and that the collaboration data was extracted from the data of the collaborative condition. We did not incorporate the data of the individual condition into analysis, as mentioned above; however, the data would provide necessary information for the collaborative condition.

The participants’ perceptual sensitivity was derived from response accuracy. To quantify perceptual sensitivity, the maximum slope of the psychometric function of contrast sensitivity was estimated for individuals and dyads, respectively (Bahrami et al., 2010, 2012a, b). We fitted the data with a cumulative Gaussian function to estimate the maximum slope of the psychometric function. A larger slope indicates higher sensitivity. The cumulative Gaussian function with two parameters, i.e., bias (b) and variance (σ²), was fitted by a procedure of maximum likelihood estimation via the MATLAB Palamedes toolbox (http://www.palamedestoolbox.org/) (Mathworks Inc., Natick, MA, USA). The psychometric curve, denoted as P(∆c) where ∆c is the contrast difference between the target and distractor, can be expressed as:

$$ P\left(\Delta c\right)=H\left(\frac{\Delta c+b}{\sigma}\right) $$

(2)

where H(z) is the cumulative Gaussian function:

$$ H(z)\equiv {\int}_{-\infty}^z\frac{dt}{{\left(2\pi \right)}^{\frac{1}{2}}}{e}^{\frac{-{t}^2}{2}} $$

(3)

Here, P(∆c) corresponds to the probability of responding that the odd target was presented during the second interval. By definition, the variance σ² is related to the maximum slope of the psychometric curve, denoted as s, which can be expressed as:

$$ s=\frac{1}{{\left(2\pi {\sigma}^2\right)}^{\frac{1}{2}}} $$

(4)

We defined “collective effect” as the ratio of the dyad’s slope (S_dyad) to that of the better observer (i.e., the individual with a larger slope, S_max). A collective effect larger than 1 indicates that the dyad managed to gain an advantage over its better individual, suggesting collective benefit. Values below 1 indicate that collaboration was counterproductive and that the dyad did worse than its better individual—namely, collective cost. The effect of relative detection sensitivity on the collective benefit can be revealed by plotting the collective effect against the relative sensitivity between the worse and the better observer (S_min/S_max).

Moreover, we adopted SFT to calculate the workload capacity of group decision-making (Little et al., 2017; Townsend & Nozawa, 1995). We used the SFT R package (Houpt, Blaha, McIntire, Havig, & Townsend, 2014) to compute workload capacity C_AND(t) and the assessment function of workload capacity A_AND(t) (please see the introduction for more details). Notably, A_AND(t) incorporates both RT and accuracy data into the analysis. The probabilities of responses can be classified into four categories: (a) the probability that a correct response is made by time t (correct and fast decision), (b) the probability that an incorrect response is made by time t (incorrect and fast decision), (c) the probability that a correct response will be made but it has not happened by time t (correct and slow decision), and (d) the probability that an incorrect response will be made but it has not happened by time t (incorrect and slow decision).

The influence of additional decision-makers was measured for each of the four response types by comparing the collaborative performance with the predicted UCIP baseline derived from the non-collaborative condition. This interpretation is more nuanced than the standard capacity coefficient. In contrast to the standard capacity coefficient C_AND(t), when interpreting the A_AND(t) function, one must consider the type of response being made. The following examples may help readers understand how we make inferences using A_AND(t). The interpretation of A_AND(t) for correct and fast responses bears the closest resemblance to the standard capacity coefficient. When A_AND(t) = 1 for correct and fast responses, the implication is that the observed responses made before time t are as probable as expected by the UCIP baseline model. A correct and fast A_AND(t) > 1 means that participants make more correct responses by time t than expected and, thus, exhibit a form of supercapacity. Similarly, correct and fast A_AND(t) < 1 implies that fewer correct responses are made by time t than expected by the UCIP model (i.e., capacity is limited). The interpretation differs for the other types of responses. For example, for the incorrect and slow responses, A_AND(t) > 1 indicates that more incorrect responses are made after time t than is expected by the UCIP model, which implies a form of limited capacity.

Finally, a quantitative comparison was made to observe the relationship between the accuracy-based measure and time-based measure. We first employed functional principal components analysis (fPCA) with varimax rotation to decompose C_AND(t) and A_AND(t) into several principal components (Burns, Houpt, Townsend, & Endres, 2013). The SFT R package was adopted (Houpt, Blaha, et al., 2014). fPCA is a structural extension of standard PCA (Ramsay & Silverman, 2005) and can be used to describe the entire functions using a small number of scalar values (i.e., the loading of the principal component) (Burns et al., 2013). The approach enabled us to describe which part of the function-level property is crucial for distinguishing collective effect as a function of RT. Then, we tested the relationship between the factor scores and the accuracy-based collective effect measured by sensitivity such that the relationship between the accuracy-based measure and the component functions can be uncovered.

In order to let the readers follow our SFT analysis steps to replicate our results (e.g., results of C_AND(t), A_AND(t), and fPCA analysis), please see the Supplementary material (S.1). We upload our script and data into Github. Files can be downloaded at https://github.com/hanekaze/A-new-measure-of-group-decision-making-efficiency.

Results

Table 2 presents the mean correct RTs and accuracy for all the combinations of the social condition and contrast level.

Table 2 Mean correct response time (RT) and accuracy for all the combinations of social condition and contrast level

Full size table

ANOVA

For RT, the results showed a significant main effect of contrast level [F (3, 54) = 63.46, p < 0.001, $ {\eta}_p^2 $ = 0.78]. Post hoc comparison showed that the mean RT of the contrast level of 25.0% was the fastest and that the mean RT of the contrast level of 17.0% was faster than that of the contrast level of 13.5% and that of the contrast level of 11.5% (ps < .01 for all comparisons). The difference between the contrast levels of 13.5% and 11.5% was not significant. There was a main effect of social condition [F (2, 18) = 4.005, p = 0.04, $ {\eta}_p^2 $ = 0.31]. Post hoc comparison showed that the mean RT of the better observer was faster than that of the worse observer (ps < .05). However, there were no significant differences between the performance of the collaboration and the better observer or worse observer. The two-way interaction did not reach the significance level.

For accuracy, the results showed that a significant main effect of contrast level [F (3, 54) = 560.53, p < 0.001, $ {\eta}_p^2 $ = 0.97]. Post hoc comparison showed that accuracy increased as the contrast level increased and that the differences between every two contrast levels were all significant (ps < .01 for all comparisons). Moreover, there was a significant main effect of social condition [F (2, 18) = 3.936, p = 0.04, $ {\eta}_p^2 $ = 0.30]. Post hoc comparison showed that the accuracy of the worse observer was lower than that of the better observer (ps < .05). However, there were no observable, significant differences between collaboration and the better observer or worse observer. The two-way interaction did not reach the significance level.

Detection sensitivity

Figure 3 plots the relationship between the relative sensitivity between the worse observer and the better observer and the accuracy-based collective effect (i.e., S_dyad/S_max). Although we observed a slight trend of positive correlation, as found in the Bahrami et al. (2010) study, this positive correlation did not reach the significance level (R² = 0.16, slope = 0.58, p = 0.37). The non-significant result may have occurred because of the restriction of range—namely, in the present experiment, the relative detection sensitivities of all dyads were above 0.6.

Capacity coefficient^{Footnote 8}

Figure 4 shows the plot of the capacity coefficient function for each dyad. The capacity functions are plotted according to the level of the accuracy-based collective effect (i.e., S_dyad/S_max) to reveal the relationship between the capacity level and the accuracy-based collective effect. The brightness level represents the level of the accuracy-based collective effect. Our visual inspection indicated that all dyads were of supercapacity with all C_AND(t) greater than 1 for all times t. However, we did not find a robust relationship between the capacity values and the level of the accuracy-based collective effect.

Assessment function

Figure 5 shows the A_AND(t) for all four response types; the functions were plotted according to the level of the accuracy-based collective effect. The assessment functions for each response type can be summarized as follows:

1.
For the correct and fast responses, values of A_AND(t) were consistently greater than 1, suggesting that correct collaborative responses were faster and more frequent than expected—that is, supercapacity processing.
2.
For the correct and slow responses, values of A_AND(t) were greater than 1 at the faster RTs and were then less than 1 at the slower RTs. This indicates that correct and slow responses made after time t were more probable than expected at faster RTs but less probable than expected at slower RTs.
3.
For the incorrect and fast responses, the results showed that most dyads made more incorrect responses by time t at the faster RTs; however, incorrect and fast responses were less probable than expected at the slower RTs.
4.
For the incorrect and slow responses, values of A_AND(t) were less than 1 for all times t, indicating that fewer incorrect and slow responses were observed than the expectation from the UCIP model.

To sum up, the results of A_AND(t) indicate that the collaborative performance was processed more efficiently than the predicted baseline model, suggesting supercapacity processing. However, similar to the results of C_AND(t), we did not find a strong relationship between A_AND(t) and the collective effect indicated by relative detection sensitivity. Thus, the current results did not provide evidence in support of the relationship between time-based measures and accuracy-based measures.

fPCA and quantitative comparison

We employed fPCA with the varimax rotation to decompose C_AND(t) and A_AND(t) into several component functions and plotted the factor score of each component against the accuracy-based collective effect to investigate the relationship between the component function and the accuracy-based collective effect. However, we failed to find a robust relationship between the component functions and the accuracy-based collective effect—that is, all the correlations did not reach the significant level (ps > .05). Due to the non-significance, we do not present the results in the main text. Please refer to the Supplementary material (S.2) for detailed descriptions of the fPCA results.

Discussion

In Experiment 1, we conducted a two-interval forced-choice oddball detection task as employed in the study by Bahrami et al. (2010, 2012a, b) but with several modifications on the test procedure. We measured both accuracy and RT, from which we estimated the accuracy-based and time-based collective effects, respectively. Our accuracy results were consistent with the Bahrami et al. (2010) findings. That is, when two individuals had similar detection sensitivity (S_min/S_max > 0.8), we observed a collective benefit with S_dyad/S_max larger than 1. Note that the sensitivity measure did not reveal a collective benefit in all dyads. In four of seven dyads, S_dyad/S_max was larger than 1, while in the other three, the effect was less than or equal to 1.

In addition, the collective benefit was inferred from the RT data in three ways. First, at the mean RT level, we found no significant difference between the mean RT of the collaboration condition and that of the worse or better observer, suggesting a lack of collective benefits at the mean RT level. Second, the observed C_AND(t) values which were greater than 1 for all times t, suggested supercapacity for all dyads—namely, collective benefits were consistently observed for all pairs. Third, when RT and accuracy were combined, the results of A_AND(t) again supported supercapacity processing. In particular, the results of A_AND(t) with respect to correct and fast responses showed that dyads made correct and faster responses more frequently than expected, suggesting supercapacity processing. On the other hand, the correct and slower responses were more probable than expected at the faster RTs, also implying supercapacity processing. For the two types of incorrect responses, the results of A_AND(t) suggested that dyads were more efficient since fewer incorrect responses were made.

It is interesting to observe that, despite the various applied measures that demonstrated the collective effect, the time-based and accuracy-based measures did not show a robust correlation between the two. Here, we came up with two likely explanations for the non-significant correlational results. First, we consider a likely case of the range of restriction. Participants in the present study had a similar detection sensitivity that restricted the full range of possible sensitivities in the general population, which could lead to non-significant sample correlations. To ameliorate a possible restricted range issue in Experiment 2, we introduced a stimulus noise; an extra background white noise was added on top of the visual display of one of the participants, thus heightening the relative differences in detection sensitivity between the two participants. Second, we suspect that there is another reason why the procedure of Experiment 1 was not sensitive enough to measure the time-based collective benefit. In the current setting, participants’ RT was recorded from the onset of a question mark. However, the participants may have already made their decisions while the two intervals were presented; therefore, the RT may reflect only the motor execution time rather than the information accumulation and decision time. To ameliorate this issue and correctly measure the decision time in Experiment 2, we replaced the 2IFC oddball detection task with a yes/no Gabor detection task. With a better estimation of the decision time, we expect to find a robust relationship between the time-based and accuracy-based measures.

Experiment 2

In Experiment 2, a yes/no Gabor detection task was conducted to enable a better estimation of the information accumulation time and decision time. In addition, we manipulated the transparency of the noise mask that was superimposed on the target stimulus, to manipulate the detection difficulty and thereby estimate the psychometric function. The experiment could be separated into Experiments 2a and 2b depending on whether an additional background noise was introduced to one of the participant’s displays. In Experiment 2a, dyads were tested without any additional background noise such that participants might have had similar detection sensitivity. In Experiment 2b, one of the participant’s displays was covered by additional background noise such that participants might have had different detection sensitivities. This manipulation could solve the problem of the restriction of range raised by Experiment 1 and allowed us to observe the effect of the relative detection sensitivity on collaborative performance. Following the Bahrami et al. (2010) study, we expected that relative detection sensitivity would reduce the collective benefit, which could be observed by both RT- and accuracy-based measures.