Runninghead:HIDDENPROFILEPARADIGM 1Hidden-Profile Paradigm, a Tautology or an Uninteresting Argument at Best: Toward Complex

and Dynamic Models of Group Information Processing

Introduction

Despite the prevalent belief that two heads are better than one, small group researchers

have found a considerable amount of empirical evidence showing that groups frequently fail to

outperform individuals in decision making. For example, since firstly formulized by Janis (1972),

groupthink has been successfully replicated and developed by many follow-up studies (e.g.,

Esser, 1998; McCauley, 1989; Schafer & Crichlow, 1996) and now became a classic theory that

accounts for the failure of group decision-making. Also, group polarization studies constitute

another stream of research on the failure of groups in decision making (e.g., Moscovici &

Zavalloni, 1969), which has been extended into political settings (e.g., Sunstein, 2002; 2004).

Although much empirical research has presented for the last several decades that groups

are necessarily better than individuals in decision making, there has been no unanimous

agreement upon the group mechanism that leads to the failures. For example, Janis (1972)

pointed out a higher level of group cohesiveness as a main factor that groups tend to reduce the

possibility of comprehensive examination of all potential alternatives by preventing

contradictory views from being expressed. However, many other empirical studies suggested that

the impacts of group cohesiveness on group performance, specifically decision quality is

mediated or moderated by many other intervening variables (for overview, see Esser, 1998) .

As an alternative theoretical explanation for the failure of groups in decision making,

Stasser and Titus (1985; 1987) suggested an insightful model, called information sampling

model. In their conception, group decision-making is regarded as or at least involves the process

by which the information unevenly distributed among individuals is collected and aggregated so

Runninghead:HIDDENPROFILEPARADIGM 2that the group can establish a pool of information as the basis of its decision-making. Specifically,

based on their simple and intuitive formula, they argued that decision-making groups tend to

focus their discussions on information that their members already know (i.e., shared

information) rather than what individual members uniquely possess (i.e., unshared

information) (for overviews, see Stasser & Titus, 2003; Wittenbaum, Hollingshead, & Botero,

2004, etc.).

Furthermore, they attributed the failure of groups in decision making to the group

tendency to exclude the information that group members uniquely possess (what they called

overlooked gems: Stasser & Titus, 2003, p. 304) from the information pool. Such argument

indeed shifted the focus of group decision-making research from the characteristics of groups

and their members to the processes and mechanisms of group decision making, more specifically,

information pooling: for example, comparison between the choosing-the-best and rank-order

procedures or between computer-mediated and face-to-face discussions (Hollingshead, 1996). In

short, Stasser and Tituss (1985; 1987) simple idea established the foundation of a new research

area, called hidden-profile paradigm.

For the last 25 years since Stasser and Titus proposed the information sampling model

and developed it into the hidden-profile paradigm, many empirical studies have been conducted

and provided consistent findings to some degree. A recent meta-analysis (Li, Yuan, McLeod, &

Xia, 2010) found that (a) groups mentioned more common information than unique information;

(b) hidden profile groups were seven times less likely to find the correct answer than

information-all-shared groups; (c) whereas mentioning unique information indeed leads to better

decision quality, discussion focus did not affect group decision quality; and (d) face-to-face

groups did not differ from CMC groups in group decision quality. Despite the consistency of the

Runninghead:HIDDENPROFILEPARADIGM 3previous findings, a close logical examination of those findings reveals that many of them are no

more than tautological statements or uninteresting arguments at best. Moreover, as Reimer,

Reimer, and Czienskowski (2010) pointed out in their meta-analysis, a fundamental question of

the hidden-profile research still remains as unanswered: How do groups manage to attenuate the

group discussion bias toward shared information, and thereby How do groups establish a more

comprehensive information pool? In other words, a large part of the questions concerning group

information processing and its mechanisms have not been fully addressed.

From this perspective, the present study attempts to point out the limitations of the

previous studies on hidden-profiles due to both theoretically and methodologically problematic

approaches, and to suggest a new approach appropriate for dealing with the complex and

dynamic nature of group processes. For these purposes, a set of mathematical analyses using

differential equations were carried out. Based on the findings of the analyses, general behaviors

of decision making groups are discussed in the relation with the previous research. Finally, the

study suggests the directions for further research.

Problems with the Information Sampling Model

To answer the question of why groups often fail to outperform individuals in decision

making, Stasser and Titus (1987) proposed a simple mathematical formula that describes the

probability that unshared information that individual members uniquely possess is mentioned

during group discussion as a function of group size (i.e., the number of participants). Specifically,

provided that the probability that any participant mentions his or her own unique information is

equal to p, and that each of n participants has an independent and identical chance to mention his

or her own unique information (i.e., p = pi = pj for all i and j; the iid assumption), the probability

Runninghead:HIDDENPROFILEPARADIGM 4that the group as a whole does not mention the unique information (that is, the probability of the

group mentioning common information only, P(Gshared)) can be written as following:

1 1 (1)Where n is the number of participants and p is the probability that an individual participant

mentions his/her own unique information. Since p is a probability, it is a real number between 0

and 1. Therefore, as n increases, (1 p)n approaches 0, while P(Gshared) approaches 1 (Figure 1):

that is,

lim 1

As shown in Figure 1, a group with a large number of participants tends to discuss shared

information much more than unshared information. Even when the probability that unshared

information is mentioned by an individual, p, is exactly equal to the probability that shared

information is mentioned, (1 p), and/or even when the group size is quite small (e.g., a 2-

person group), the group is more likely to discuss the shared information than the unshared

information.

[Insert Figure 1 about here]

Such discussion bias favoring shared information is easily expected to result in the

dominance of the information common among group members over the unique information. As

this continues, the unique information will be squeezed out of the group discussion, and the

group will reach a conclusion without a critical review of the unique information that might be

important for making a better decision. Stasser and Titus (1985; 1987) argue that the failure in

group decision-making is mainly due to the incompleteness of the information pool

overwhelmed by the shared information. Based on this, it is easy to infer that a group would a

worse decision or not a best decision at best without well-designed group discussion rules that

Runninghead:HIDDENPROFILEPARADIGM 5prevent the dominance of shared information and promote the exchanging of unshared

information.

With no doubt, Stasser and Tituss equation above precisely captures the probability of

the dominance of shared information during group discussions, but only in terms of frequencies.

To be sure, in a group with a large number of members the shared information is more likely

(more precisely speaking, more frequently) to be discussed than the unshared information.

However, this does not necessarily mean that the amount of the shared information has been

accumulated more than that of the unshared information. By definition, shared information is

what group members already possess in common. Therefore, no matter how many times or how

frequently it is mentioned, the total amount of shared information can never be accumulated,

simply because mentioning another piece of shared information is nothing other than just

repeating what has been previously mentioned (recall the concept of information redundancy of

Shannon, 1948). In contrast, no matter how infrequently it is mentioned, mentioning a piece of

the unshared information will linearly increase its total amount, because any piece of the

unshared information is completely new and not redundant at all to what has been previously

mentioned. Thus, if a group makes a decision relying not upon simply the frequencies by which

different types of information are mentioned but upon the amount of the information that has

been accumulated so far, it is not impossible for a group with a large number of members to

reach a conclusion that the unshared information designates.

If the total amount of information does matter in group decision making, it might not be

surprising that a group can choose the alternative that unshared information (or less dominant

information in terms of frequency) supports. If it is the case, such possibility will lead us to

suspect whether or not any particular group information processing mechanisms actually change

Runninghead:HIDDENPROFILEPARADIGM 6the direction of group decisions at all, because the change of the decisional direction may happen

by chance. This is one of the most important reasons that previous research on hidden-profiles

should be reexamined.

Impacts of sharing unique information on group decisions

In empirical research on hidden-profiles, more generally, studies on group information

sampling, experimental groups are typically asked to reach a joint decision by choosing one out

of a set of choice alternatives. Such tasks include choosing a company for a hypothetical

financial investment, selecting a suspect in a murder mystery, selecting an individual for student

body president, or developing a diagnosis for a medical case (for overviews, see Stasser &

Birchmeier, 2003). In order for the groups to make a choice, they are given multiple sets of

information items, each of which supports one choice alternative than any others. For example,

in frequently cited experiment research conducted by Hollingshead (1996), a number of 3-person

groups were asked to choose one out of three companies for a hypothetical financial investment,

and they were provided with three different sets of information items, each of which supports

each company. However, the information was not randomly distributed among the three

members within each group; but rather, the sharedness of the information among the members

was sophisticatedly manipulated in a systematic way. In particular, the information items that

support one company, say A, (i.e., the correct choice) were given to single members separately

(i.e., unshared information), while those that support the other companies, say B and C, (i.e., the

incorrect choices) were given to all group members in common (i.e., shared information). Then,

the researcher examined whether one discussion rule that apparently facilitates to share the

unique information (i.e., the rank-order procedure) would lead to the correct decision (i.e.,

company A) than the other rule that might inhibit exhaustive consideration of all information

Runninghead:HIDDENPROFILEPARADIGM 7items, specifically unshared ones (i.e., the choose-the-best procedure). Finally, the researcher

concluded that the groups following the rank-order rule were more likely to choose the correct

alternative than the others, because the rank-order procedure did facilitate sharing the unique

information items. In short, when unique information is shared with all the group members, the

group will make a right decision. Although the effect sizes of sharing unique information on

group performance vary, previous research on hidden-profile tasks generally confirmed this

finding (Lu, et al., 2010). Then, what is the hidden problem of this finding?

To illustrate the problem embedded in hidden-profile research, let us suppose a typical

and general experimental setting for hidden-profile tasks used in previous studies. Groups are

given a decision-making task that requires choosing one out of multiple alternatives (say,

alternative A, B, C, ). In any cases, it is reasonable to assume that which alternative is better

than any others is completely unknown, at least to the participants. If the best choice is already

known in advance, the decision-making task itself is unnecessary. That is, a group of people do

not have any practical reasons to waste their valuable resources, including time, in order to

choose the best one already known.

Further suppose that multiple sets of information are given, each of which supports one

alternative exclusively (say, information , , , ). Then, the question is whether or not it is possible for the participants (either as individuals or as a whole group) to determine the

significance of a particular information item in their decision-making: that is, in Stasser and

Tituss terms, Is this information item an overlooked gem or not? The logical answer to this

question is No. No matter how it looks significant during group discussion, the significance of

an information item completely depends on the correctness of the alternative that the item

supports, which is, in turn, completely unknown. For instance, an information item is

Runninghead:HIDDENPROFILEPARADIGM 8significant, only if alternative A that the item supports turns out to be the correct one; otherwise,

the item is not significant at all. However, since the correctness of alternative A is undermined in advance, so is the significance of the information item . In short, the significance of a given information item is determined not by itself but by the correctness of the corresponding

alternative, which is completely unknown. Logically speaking, thus, picking up and examining

an information item during group discussion is a blind procedure to the group members.

Let us go back to the typical findings of hidden-profile research when unique

information is shared with all the group members, the group will make a right decision. As

described above, the significant information items (known to researchers but unknown to

participants) were given to individual members separately (i.e., unshared information). Then, the

typical findings can be rephrased as When unshared information is significant, and when the

unshared significant information is eventually shared through a certain discussion process, the

group will make a correct choice. Obviously, this statement is a tautology or an uninteresting

statement at best.

Suppose that a researcher manipulates the sharedness of information in the opposite way

to the previous research: that is, he or she distributes significant information to all the group

members (i.e., shared information), whereas distributes insignificant information to individual

members separately (i.e., unshared information). And he or she asks the groups to follow a

certain discussion rule that facilitates sharing the unshared information. Two possible results can

be conceived. First, if the groups make the wrong choice, this proves that the general finding of

hidden-profile research is tautological, because the hypothetical experiment confirms that the

quality of group decision depends on the significance of the unshared information. Second, the

groups still make a correct choice, this proves that neither the group discussion rules nor the

Runninghead:HIDDENPROFILEPARADIGM 9manipulation of the sharedness of the information has no impacts on the quality of group

decision. Therefore, the hidden-profile paradigm is a tautology or an uninteresting argument at

best.

Nonetheless, such tautological research design can be also useful when it is used to trace

in what way a particular information item or a set of information items is disseminated,

especially when the information is unshared prior to group discussion. It is not very interesting to

merely address whether a group following a specific rule will make a correct decision, because it

totally depends on the manipulation of the sharedness of information. Rather, it might be more

interesting to examine how the relative proportion (not the relative frequency but the amount) of

a particular type of information fluctuates over time through complex and dynamic group

processes.

Complex and Dynamic Models of Group Information Processing

Group processes, either those by which groups generate new ideas or those by which they

make joint decisions, are usually considered as to be complex and dynamic. Therefore, these two

words are ones of the most frequently appear in the small group literature. However, few studies

in the research area, if any, have attempted to conceptualize or to operationalize the notion of

complexity and dynamics of group processes, unlike in other fields: e.g., Kolmogorov

complexity in computer science (Kolmogorov, 1963). The lack of clear or at least agreeable

definitions of complexity and dynamics results in inconsistent findings and more seriously

prevents us from deepening our understanding of group processes, while it remains as so-called a

black box. Sometimes, the complexity and dynamics of group processes are mentioned in

order to justify the poor explanatory power of the research models or the inconsistent findings

even within single studies. I do not believe that it is possible to investigate something complex

Runninghead:HIDDENPROFILEPARADIGM 10and dynamic without clear definitions of them. This is not a mere personal belief but a general

principle of scientific research that is articulated in any social scientific research textbooks.

This is, I suppose, in large part because of the a-century-long convention in psychological

research, in general, in social scientific research. Unlike other sciences in which complex and

dynamic phenomena are handled very well, social science has relied mainly upon correlational

statistical analytic methods, which assume methodological individualism (i.e., the iid assumption)

and linear relationships among variables (e.g., ANOVA family, various regression analysis).

Although it is undeniable that the conventional statistical methods in social science have led to

tremendous advance in the development of human knowledge for the last century, they are not

suitable for the examination of complex and dynamic phenomena in behavioral and social

sciences (for the detailed discussion, see Brown, 2007, Chapter 1). In brief, conventional

statistical models that require programmable formulae in advance of any empirical tests, such as

t-distribution for t-test, are not appropriate to complex and dynamic phenomena, those whose

statistical behaviors are hard to specify in advance.

Pointing out the limitations of the conventional statistical methods, many scholars have

suggested new analytical tools, such as differential and difference equations, as alternatives for

examining complex phenomena (e.g., Coleman, 1964; Brown, 1995; Rapoport, 1983; Simon,

1957, etc.). Ironically, however, most small group researchers, those who (over)emphasize the

importance of sharing unshared information for successful group performance, have ignored the

unfamiliar approaches and exploited only familiar analytical tools: i.e., conventional statistical

methods. As many small group theories predict, attachment only to conventional methods will be

detrimental to the development of the small group research area.

Runninghead:HIDDENPROFILEPARADIGM 11

From this perspective, the present study proposes a new model of group decision-making

employing the new analytical approach, specifically differential equation techniques, as part of

effort to introduce unshared information into the small group research area as a decision-

making group with expectation that it will contribute to deepening our understanding of complex

and dynamic group processes.

A General Setting of Hidden-Profile Experiment

In order to build a mathematical model of group information processing, this study adopts

the manipulation of the sharedness of information used in Hollingsheads (1996) experiment as

the simplest prototype to be extended into a general setting of hidden-profile experiment. In her

experiment, she assigned every three participants into a group (i.e., 3-person groups) and asked

either to choose one out of three companies (i.e., the choose-the-best procedure) or to rank all the

three companies (i.e., the rank-order procedure) for a hypothetical financial investment. A total

of six information items were given to each group members within a group: two of which favor

one company, say A; the rest of which support the other two companies, say B and C,

respectively. The first two information items were given to individual members uniquely (i.e.,

unshared information); while the rest were given to the group members in common (i.e., shared

information). The manipulation of the sharedness of the information described here is

summarized in Figure 2.

[Insert Figure 2 about here]

It should be noted that while the total number of information items is equal to 18 (= 3

persons * 6 items), the total amount of information is equal to 10 (= 2 * 3 unique information

items + 4 * 1 common information) because the redundant information items are not counted as

additional information. Provided that a group makes a decision based on the amount of

Runninghead:HIDDENPROFILEPARADIGM 12information rather than the number of information items available, when all the information

items become available through group discussion (i.e., 6 items favoring company A > 4 items

favoring either company B or C) the group will choose company A (assumed to be the correct

choice); but when only the common information items become available (i.e., no item favoring

company A < 4 items favoring either company B or C) as the information sampling model

predicts, the group will choose either company B or C (assumed to be the wrong choices). In the

case in which group members make their own decisions prior to group discussion, each member

is more likely to choose either company B or C than A, since the amount of information favoring

B or C is greater than A (i.e., 2 item favoring company A < 4 items favoring either company B or

C).

Let us extend this prototypical setting into a general one. The group size needs not to be

limited to three; but rather we can extend it to n-person cases. Similarly, the number of

information items given to individual members is not necessarily limited to 6, either; but can be

extended to K. At the individual member level, the number of information items supporting one

alternative (i.e., company A), which belong to each group member individually, p, should be less

than that of information items supporting the other alternatives (i.e., companies B and C), which

is common to all the members, q. So that when decisions are made individually, individual

members are more likely to choose either B or C than A: that is, p < q. Accordingly, the sum of

both types of information items that an individual member hold should be equal to K (= p + q).

At the group level, the total number of information items is equal to Kn; whereas the total

amount of information is equal to p + nq. In order for the group to choose company A with all

information available, the amount of information supporting company A should be greater than

that supporting either company B or C: i.e., p < nq. To avoid notational confusion, hereafter,

Runninghead:HIDDENPROFILEPARADIGM 13denote the total amount of information items supporting company A as X instead of p; and that of

information items supporting either company B or C as Y instead of nq: hence, X < Y. The

general setting described here is diagramed as Figure 3.

[Insert Figure 3 about here]

Finally, it is necessary to consider the scale of time unit. Unlike the previous empirical

studies in which time is typically measured in minutes, in this study time is measured in a

theoretical manner: that is, a unit of time is defined as the amount of time that should be spent for

processing a single information item (i.e., mentioning and discussing it). Such a theoretical

measure is not impossible, if every information item is perfectly manipulated as having an equal

length and an equal level of readability and difficulty or if the variance of the time spent for

processing the items is statistically controlled.

The general and theoretical setting of hidden-profile experiment developed in this section

has several advantages over conventional experimental designs. First, it enables to overcome the

limitations of laboratory settings usually reported at the end of research articles: for example,

limited group sizes (typically 3 to 5), limited sample sizes (no more than 200), and sampling bias

(e.g., college students). Second, it removes any possible measurement errors and allows to

examine the pure effects of variables/manipulation of interest. As mentioned earlier, for example,

the theoretical measure of time allows us to completely remove any possible noises related to

empirical time measurement and to look at the pure effects of time on group decision-making

processes. Third, the general and theoretical setting makes it possible to manipulate key

(independent) variables that are impossible to in real world. For example, the following section

will set four different rules of the sequence of information items to process: (1) independent, (2)

self-reinforcing, (3) negative, and (4) positive mutual influences among different types of

Runninghead:HIDDENPROFILEPARADIGM 14information. Furthermore, in order to test the sensitivity of group processes to initial conditions,

the impacts of all possible initial settings will be tested.

Model 1: Independent model1

The first model, labeled independent model, assumes the simplest situation, in which

every group member has an equal chance to provide one out of K information items at a time,

every single item is to be selected and discussed with an equal probability, and moreover the

chance to select one type of information item is independent of what was discussed previously.

Now, let x denote the amount of the information supporting alternative A at a time point

of t. The increase rate of x at t can be expressed in a differential equation as:

1 (2)

where p is the probability of the information items supporting A to be selected, X is the total

amount of information supporting A, and thus, 1 x/X is the proportion of the information items

that have been not discussed until t. The solution of the differential equation is

(3)

where the initial value of x at t = 0 is equal to 0: that is, none of the information items supporting

A has been discussed prior to group discussion.

As shown in both equations (2) and (3), x monotonously increases over time: i.e., dx/dt >

0 for all t. However, the increase rate itself decreases over time: i.e., d2x/d2t < 0 for all t. This is

because additional items are likely to be redundant with the items that have been already

discussed. Also, x tend to approach to its upper bound X over time: i.e., lim . 1Sofar,thegeneralandtheoreticalsettinghasassumeda3alternativesituation(i.e.,companiesA,B,andC).Intherestofthepaper,itwillbereducedintoatwoalternativesituationbymergingthelasttwoalternativesintoasingleone(i.e.,alternativeAvs.B).Thereasonistwofold:first,a3alternativesituationinvolvesaseriesofcomplicatedmathematicalcalculationofpartialdifferentialequationsthatmightexceedthepagelimit;second,theresultsofanalysesofa3alternativesituationishardtodisplayina2dimensionalplane.

Runninghead:HIDDENPROFILEPARADIGM 15

In a similar way, the growth curve of the other type of information supporting B, say y,

can be derived from its differential equation and solution:

1 (3)

(4)

where q is the probability of information items supporting B to be selected, Y is the total amount

of information supporting B, and thus, 1 y/Y is the proportion of the information items that

have been not discussed until t.

The growth curve for y shows the similar shape to that of x. However, the increasing rate

of y is much smaller than that of x at early stage, while it is much larger than that of x at late

stage, since p > q and X > Y. In fact, it is easily shown that y eventually exceeds x, say at a time

point of t* by the intermediate value theorem. This implies that where any information items are

freely exchanged by any members (i.e., equal chance to be mentioned), the shared information is

dominant over the unshared information at early stage of group discussion (i.e., x > y at 0 < t <

t*), but the unshared information eventually becomes dominant (i.e., x < y at t > t*).

Figure 4 presents the time series plot of a numerical example in which X = 10, Y = 12, p

= .6, and q = .4. As shown in the figure, both curves increase sharply at early stage, but the

increasing rates slow down, and eventually approach to their upper bounds. The curve for x

increases faster than the curve for y at early stage; but the cumulative amount of information y

exceeds that of x at t = 46. That is, from the beginning up to t = 46, x is dominant over y. After

the time point, however, y becomes dominant.

[Insert Figure 4 about here]

Runninghead:HIDDENPROFILEPARADIGM 16

From the findings of the simplest model, we can draw two important conclusions. First,

no matter what size of a decision-making group, the amount of unshared/unique information

eventually exceeds that of shared/common information specifically after t* because of

information redundancy, which is opposite to what the information sampling theory predicts.

Second, whether a group relies upon the shared or the unshared information is determined by the

amount of time given for group discussion. However, in laboratory experiments, different groups

following different discussion rules are given the equal amount of time. For example, in

Hollingsheads (1996) study, four different groups (the choose-the-best vs. the rank-order

procedure * face-to-face vs. CMC groups) were all given 20 minutes, but it was shown that the

rank-order and FtF groups were more likely than the others to rely upon unshared information.

Thus, it might be seen that time had no significant impacts on group processes. However, this

previous finding can be interpreted to indicate that a unit amount of information could be

processes in a shorter time under the condition of the rank-order rule and FtF than the others. For

example, the one-to-one comparison between alternative companies may facilitate processing

contrasting information items, and the nonverbal cues used in FtF communication may also

reduce the information processing time. Therefore, it can be said that those groups spent more

amount of the theoretical time than the others, even though they spent the same amount of actual

time as the others.

The independent model allows further examination of infinitely many different cases in

which initial conditions are differently set up. For example, what if one type of information was

already dominant over the other even before group discussion begins (i.e., x(t) > y(t) 0 or y(t) > x(t) 0). Such case can be also interpreted as the situation in which power relations among group members are not evenly distributed. That is, a unit of information provided by a powerful

Runninghead:HIDDENPROFILEPARADIGM 17member will be weighted more than that by a powerless member. For this, phase diagram

techniques were used and summarized in Figure 5.

The solid curve from the origin of the coordinate (i.e., x(0) = y(0) = 0: no preexisting

information dominant) to the equilibrium point represents the trajectory of the relative

proportions of the amounts of two types of information x and y. When the curve is below the

straight line y = x, x is dominant over y, vice versa. By setting the initial condition differently

(i.e., x(0) y(0) 0), it is possible to investigate the impacts of the initial conditions on the direction of trajectories and the area through which they pass. As shown in Figure 5, however,

different initial conditions do not generate any complex behaviors during group discussion,

compared to complex phenomena found in natural worlds: e.g., the Lorenz attractors and the

competitive Lotka-Volterra model. In fact, all initial points are directed toward the equilibrium

point at the top of the plane. Moreover, even when the complexity of group process rules

increases, the impacts of different initial conditions remains less complex (discussed later). In

turn, this implies that group processes may not be complex enough to justify the poor and/or

inconsistent findings of previous research.

[Insert Figure 5 about here]

Model 2: Self-reinforcing model

The independent growth model examined previously does not appear to be complex or

dynamic enough to reflect real-world group processes. Therefore, the second model, self-

reinforcing model, is suggested to capture self-reinforcing positive feedback such that makes the

model more realistic as well as more complex and dynamic. As its name itself articulates, the

self-reinforcing positive feedback refers to as a situation in which the more one type of

information was mentioned previously, the more likely the same type of information is to be

Runninghead:HIDDENPROFILEPARADIGM 18mentioned, vice versa. This assumption reflects the violation of the independence assumption

between the chances of information items being mentioned at different time points. This implies

that conventional statistical methods are no longer suitable to test the self-reinforcing model.

This model can be expressed in differential equations as following:

1 (4)

1 (5)

and their solutions are:

(6)

(7)

Figure 6 presents the time series plot of a numerical example of the model in which X =

10, Y = 12, p = .6, and q = .4: the same values as the previous example. While the increase rates

of the two curves in the independent model monotonously decrease (Figure 4), those in the self-

reinforcing model fluctuate over time (Figure 6). At early stage of group discussion, both types

of information increase slowly because neither of them have been mentioned enough. However,

both curves increase sharply around the inflection points (i.e., the point at which the second-

derivative equals 0). Compared to the independent model, the amounts of both types of

information grow much more slowly on average. For example, x in the independent model

almost reaches its upper bound around t = 100, while that in the self-reinforcing model reaches

around t = 150. This is because the self-reinforcing process encourages the same type of

information to be selected in the following rounds, and thereby increases the information

Runninghead:HIDDENPROFILEPARADIGM 19redundancy, too. Although the two curves eventually intersect with each other, that is, y exceeds

x, it also takes much longer until the time point (t* = 151; cf. t* = 46 in the previous model).

[Insert Figure 6 about here]

The self-reinforcing model seems more complex than the previous one due to the

fluctuating increase rates of both curves. However, a phase diagram analysis reveals almost the

same pattern as the independent growth model (Figure 7). Even though the trajectory that begins

from the origin (the solid curve in Figure 7) stays longer the area below the y = x line and passes

through the deep right-bottom corner, it eventually pass through the y = x line and end up at the

equilibrium point. The impacts of different initial conditions shows similar patters to the

previous model, but the initial points around the center of the plane tend to change their phases at

faster rates than those around the peripheral.

Therefore, the two conclusions drawn previously are still held even when additional

complexity and dynamics are added to group processes: (1) the amount of unshared/unique

information eventually exceeds that of shared/common information because of information

redundancy; (2) whether a group relies upon the shared or the unshared information is

determined by the amount of time given for group discussion.

Model 3.1: Interdependent model with negative mutual influences

The previous two models the independent model and the self-reinforcing model do

not take into account the possibilities of interdependent relationships between different types of

information. Two possible scenarios can be considered: one is that the increase in one type of

information suppresses that in the other (i.e., negative mutual influences); the other is that the

increase in one type of information leads to the increases in the other type (i.e., positive mutual

Runninghead:HIDDENPROFILEPARADIGM 20influences). The first possible interdependent models can be expressed in a system of two

differential equations with two unknowns as following2:

1

1

(10)

Figure 8 presents the time series plots of numerical examples of the two models in which

X = 10, Y = 12, p = .6, and q = .4: the same values as the previous examples. As shown in Figure

8, the negatively interdependent growth model shows almost an identical pattern with the

independent growth model. The phase diagram also looks similar to both the independent growth

model and the self-reinforcing model with slight differences (Figure 9)

[Insert Figure 8 and 9 about here]

The two conclusions drawn previously are still held even when eve more complexity and

dynamics are added to group processes, i.e., the interdependence between the two different types

of information: (1) the amount of unshared/unique information eventually exceeds that of

shared/common information because of information redundancy; (2), whether a group relies

upon the shared or the unshared information is determined by the amount of time given for group

discussion.

Model 3.2: Interdependent model with positive mutual influences (the ping-pong model)

The final model of this study also involves the possibility of interdependent relationships

between different types of information but in the opposing way to the previous model: i.e.,

positive mutual influences. In other words, the model assumes that the increase in one type of

information leads to the increases in the other type. This mechanism seems like playing a ping-

2Duetothecomplexityofthesystemsofdifferentialquestions,theexplicitsolutionsforneitherofsystemscanbefound.Forthisreason,numericalintegrationmethodswereused.

Runninghead:HIDDENPROFILEPARADIGM 21pong game: the increase in one type of information causes the increase in the other type, and it

will once again increases the first type of information. The model can be formulized as a system

of two differential equations with two unknowns as following:

1

1

(11)

Figure 10 presents the time series plots of numerical examples of the two models in

which X = 10, Y = 12, p = .6, and q = .4: the same values as the previous examples. Unlike the

three previous models the independent model, the self-reinforcing model, and the negatively

interdependent growth model, this final model shows the most striking patterns: both curves run

parallel to each other until they begin to converge to their asymptotes.

The phase diagram shows such distinctive patters more clearly (Figure 11). The trajectory

that begins from the origin goes along with the y = x line, although it is slightly above the

straight line, which means that both types of information increases at an almost equal rate for

regardless of the proportions, relative amounts, or the upper bound of different types of

information. In addition, the initial points around the straight lines appear to be quite static (i.e.,

the short velocity arrows).

The behavioral patterns of group processes shown in the positively interdependent

models can be summarized as: (1) unshared/unique information is as likely to be mentioned as

shared/common information, which is contrasting to both the prediction of the information

sampling model and the previous conclusion of this study; and (2) whether a group tends to rely

upon the shared or the unshared information is relatively independent of the amount of time

spent for group discussion, which is also contrasting to the previous conclusions. In short, the

Runninghead:HIDDENPROFILEPARADIGM 22positively interdependent model allows groups to build their information pools in which different

types of information increase at almost an equal rate. However, it is noted that the model does

not guarantee better decisions, simply because the significance of the unshared information is

unknown.

Conclusion

This study takes a close look at the major findings of hidden-profile research and the

information sampling model (Stasser & Titus, 1985; 1985) from which hidden-profile paradigm

has been originated. As a result, a set of critical limitations of hidden-profiles has been found.

First, the information sampling model fails to pay attention to the information redundancy and

therefore the amount of information, mainly focusing on the frequencies of information items

mentioned during group discussion. Therefore, it predicts that groups tend to prefer the

information that all group members share in common (i.e., shared information) over the

information that individual members possess individually (i.e., unshared information). According

to the probability model that they proposed (Equation 1), it is true that the share information will

be mentioned more frequently than the unshared information. However, because of the

informational redundancy, the shared information cannot be accumulated even when it is

repeatedly mentioned. Therefore, the unshared information is not necessarily squeezed out of

group information processing. Furthermore, the possibility of the survival of the unshared

information without any intentional or pre-designed discussion rules implies that certain group

discussion mechanisms, which were previously considered as critical for the survival of the

unshared information, might not have significant impacts on the survival of the unshared

information or the quality of group decision.

Runninghead:HIDDENPROFILEPARADIGM 23

Second, the present study found that the key findings of the hidden-profile research are

tautological statement. Logically speaking, the significance of a given piece of information,

either shared or unshared, is completely unknown. Nevertheless, the previous research assumes

that the unshared information is significant and critical for group decisions (overlooked gems),

and examined whether or not sharing the unshared information would lead better decisions.

However, such tautological research design might be very useful, if it is used for tracing certain

types of information (e.g., shared or unshared; correct or incorrect, etc.) during group discussion

processes. However, most studies failed to examine the group processes (Reimer, et al., 2010).

This study pointed out the lack of appropriate analytical tools as the main reason that the

important questions have long been unanswered. Using different equations as alternative

analytical tools (Brown, 2007), the present study suggested four different models of group

information processing: (1) the independent model, (2) the self-reinforcing model, (3) the

negatively interdependent model, and (4) the positively interdependent model, each of which

represents group discussion rules with different levels of complexity and dynamics, and detected

the behavioral patterns of competitive dominance of different types of information within a

group. The results from the first three models showed that the unshared information is not

necessarily squeezed out of group information processing but rather become dominant over the

shared information at late stage. Also, the results showed that whether groups rely upon the

shared or the unshared information is determined by the amount of time spent for group

discussion. On the other hand, the forth model, the positively interdependent model, suggested

the most striking patterns. When two different types of information encourage the likelihood of

each other to be mentioned, the unshared information is exchanged as much as the shared

information, regardless of the amount of time spent.

Runninghead:HIDDENPROFILEPARADIGM 24

In this study, one of the simplest forms of differential equations was used as the

unshared information to the group of researchers to examine the emerging patterns of group

information processing and to suggest a new direction of hidden-profile paradigm and small

group research, in general. However, as discussed above, the unshared information is not always

significant than the shared information. Therefore, the suggested direction should be developed

into a set of hypotheses against which empirical evidence can be tested.

Runninghead:HIDDENPROFILEPARADIGM 25References

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Runninghead:HIDDENPROFILEPARADIGM 27

Figures

Figure 1. A numerical example of the information sampling model with p = .5

Figure 2. A typical research design for hidden-profile task

Runninghead:HIDDENPROFILEPARADIGM 28

Figure 3. The general and theoretical setting of hidden-profile experiments

Runninghead:HIDDENPROFILEPARADIGM 29

Figure 4. Growth curves of the independent model, where X = 10, Y = 12, p = .6, and q = .4

Figure 5. A phase diagram of the independent model, where X = 10, Y = 12, p = .6, and q = .4

Runninghead:HIDDENPROFILEPARADIGM 30

Figure 6. Growth curves of the self-reinforcing model, where X = 10, Y = 12, p = .6, and q = .4

Figure 7. A phase diagram of the self-reinforcing model, where X = 10, Y = 12, p = .6, and q = .4

Runninghead:HIDDENPROFILEPARADIGM 31

Figure 8. Growth curves of the negatively interdependent model, where X = 10, Y = 12, p = .6,

and q = .4

Figure 9. A phase diagram of the negatively interdependent model, where X = 10, Y = 12, p = .6,

and q = .4

Runninghead:HIDDENPROFILEPARADIGM 32

Figure 10. Growth curves of the positively interdependent model, where X = 10, Y = 12, p = .6,

and q = .4

Figure 11. A phase diagram of the positively interdependent model, where X = 10, Y = 12, p = .6,

and q = .4