Review on Energy Resilience

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International Journal of Production Economics

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Are distrust relationships beneficial for group performance? The influence of the scope of distrust on the emergence of collective intelligence Giovanni F. Massaria, Ilaria Giannoccaroa,∗, Giuseppe Carbonea,b a Department of Mechanics, Mathematics, and Management, Politecnico di Bari, viale Japigia 182, 70126 Bari, Italy b Center for Nonlinear Science, University of North Texas, P. O. Box 311427, Denton, TX 76203-1427, USA

A R T I C L E I N F O

Keywords: Collective intelligence Group decision-making Scope of distrust Social relationships Simulation

A B S T R A C T

Collective intelligence is defined as the collective ability of human groups in solving different tasks. It explains why some teams perform better than others by exploiting the power of social relationships, so motivating re- search on which features of social relationships can improve it. We contribute to this line of research by ana- lyzing the effect of distrust relationships, in which individuals involved tend to make antagonistic decisions, on collective intelligence. Borrowing from previous studies that recognize consensus seeking among self-interested individuals as a critical process for the emergence of collective intelligence, we investigate the relationship between scope of distrust (i.e., the extent to which distrust relationships are spread in the group) and group performance (measured as efficacy to solve a decision making problem), in different conditions of strength and density of social relationships. To do this, we employ a simulation model coming from statistical physics, where collective dynamics is governed by a continuous-time Markov process. Results show that scope of distrust can be beneficial or not for group performance, depending on the value of the strength and the density of social in- teractions. When the strength (density) of social relationships is too low, any scope of distrust is detrimental for group performance, while when the strength (density) of social relationships is moderately high, low scope of distrust can be useful to improve group performance. Theoretical and managerial implications of these findings are finally discussed.

1. Introduction

Collective intelligence (CI) is a powerful concept proposed in the literature to explain why some groups perform better than others in a variety of different tasks (Woolley et al., 2010). It is a form of dis- tributed intelligence, which arises from the collaboration and compe- tition of many individuals (Levy, 1997). Similarly to swarms of birds, schools of fishes, and colonies of ants, just to name a few of most popular natural systems that exhibit a similar property (known as swarm intelligence), human groups are able to reach higher perfor- mance than single individuals, by exploiting the power of social re- lationships (Pentland, 2007; Bonabeau, 2009; Krause et al., 2009; Woolley et al., 2010).

Improving group performance by increasing CI is an important re- search issue to address. In this regard, it is fundamental to understand the processes that let CI emerge in human groups and to investigate the features of social relationships that, influencing these processes, lead to superior group performance. Previous studies have argued that CI emerges from both the interaction and the combination of bottom-up

and top-down processes (Woolley et al., 2015). In particular, bottom-up processes involve features characterizing the individuals and enhancing their collaboration (such as cognitive diversity and social perceptive- ness), while top-down processes include aspects affecting the co- ordination of the social relationships, such as group structures, norms and routines. Despite the importance of the topic, research is still at its infancy (Schut, 2010; Woolley and Fuchs, 2011) and further in- vestigation is needed.

In this paper we focus on the process of consensus seeking among group members and identify the features of social relationships that, affecting this process, influence CI.

Despite literature has investigated the relationship between con- sensus and group performance (Janis, 1982; Priem, 1990; Jehn, 1997; Esser, 1998; Jehn and Mannix, 2001; Hinds and Bailey, 2003), it is only recently that this process has been related to CI. Recent studies on group decision-making show that CI results from an adaptive process where individuals make decisions exploring the problem space, driven by two competing forces: 1) the search for solutions with higher per- formance and 2) consensus seeking. The strength of social relationships

https://doi.org/10.1016/j.ijpe.2018.12.005 Received 13 April 2018; Received in revised form 20 September 2018; Accepted 7 December 2018

∗ Corresponding author. E-mail addresses: [email protected] (G.F. Massari), [email protected] (I. Giannoccaro), [email protected] (G. Carbone).

International Journal of Production Economics 208 (2019) 343–355

Available online 08 December 2018 0925-5273/ © 2018 Elsevier B.V. All rights reserved.

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controls the extent to which the interacting members are pushed to be in agreement. A critical value of strength of social relationships is found, at which the group suddenly moves from low to high consensus state, characterized by the highest group performance (De Vincenzo et al., 2017). Neither too low or too high levels of consensus is shown to maximize group performance (Carbone and Giannoccaro, 2015).

In these studies social relationships are consider conduits of social influence pushing individuals to seek consensus. However, this is rea- sonable under the implicit assumption that the social relationships are not distrustful. Distrust is defined as negative expectations of in- dividuals regarding the “conduct” of the interacting ones, in terms of what they say, do, and how make decisions (Lewicki et al., 1998; Lewicki and Wiethoff, 2000). Therefore, rather than to increase con- sensus, distrust relationships push individuals involved to be in dis- agreement. Since this affects the process of consensus seeking, distrust relationships should have a role in the emergence of CI.

Therefore, in this paper we extend the models above to investigate the effect of distrust relationships on group performance. In particular, since an optimal level of consensus is required to maximize group performance and since distrust relationships reduce the level of con- sensus, we argue that a moderate number of distrust relationships (i.e. scope of distrust) can be beneficial in specific conditions. These con- ditions are characterized by two variables influencing the pressure to- wards consensus: (i) the strength and (ii) density of social relationships. We argue that when a group is characterized by high strength and density of social relationships, a given number of distrust relationships, contrasting consensus reaching, could help the system achieve the right level of consensus enabling the emergence of CI (i.e., a high performing state). Studies on minority dissent and beneficial effect of the “ad- vocate's devil” procedure on group performance support this argu- mentation (Schulz-Hardt et al., 2002). Therefore, we investigate the moderating role of strength and density of social relationships on the relation between scope of distrust and group performance.

To do this, we adopt a simulation approach based on a technique coming from statistical physics, where the effect of social relationships is modelled by means of the Ising model of interacting spins. The Ising model has been largely applied in social science to model the influence of social processes (Bordogna and Albano, 2007a, 2007b; Wei-Xing and Sornette, 2007; Stauffer, 2007; Sornette, 2014; Oh and Jeon, 2007; Giannoccaro and Carbone, 2017). Here, we refer to its recent applica- tion to model collective decision-making and to simulate group per- formance in complex environments (Carbone and Giannoccaro, 2015; De Vincenzo et al., 2017; Giannoccaro et al., 2018). The reason for employing the Ising-Glauber dynamics in a group context is justified by social influence theory. This theory argues that individuals make changes to their feelings, behaviors, and decisions, as a result of the interaction with the others (Di Maggio and Powell, 1983). Therefore, this model applies very well to groups, where individual's decision is affected by the opinion of neighbors or interacting members.

The paper is organized as follows. First, we briefly review the recent literature on collective intelligence. Then, we develop our theory con- cerning the effect of scope of distrust on the emergence of CI. Successively, we describe the model we adopt to conduct the simulation analysis. We end with a discussion of results and conclusions.

2. Collective intelligence

Collective intelligence is not at all a new concept. It is related to the swarm intelligence, i.e. the collective behavior of social insects (e.g., beehives, ant colonies, swarms of birds), which despite the simplicity of each single agent, are collectively able to do intelligent things (Bonabeau et al., 1999; Bonabeau and Meyer, 2001; Krause et al., 2009). It is also linked with the wisdom of crowds, i.e. the ability of crowds to make decisions better than the average of single individuals (Surowiecki, 2005; Lorenz et al., 2011).

Nevertheless, it is a new concept with reference to human groups.

Woolley et al. (2010) conduct an experimental study with human groups ranged in size from two to five and working on multiple tasks, including creative brainstorming problems, puzzles involving verbal or mathematical reasoning, negotiation tasks, and moral-reasoning pro- blems. By carrying out a factor analysis of the groups score, Woolley and colleagues find that a single dominant factor explains 43% of the variance in performance. As for individuals, they called this factor “the collective intelligent factor g” and defined CI as the ability of human groups to perform well on a variety of tasks. We refer to this con- ceptualization in our study.

In the referring literature, the processes letting CI emerges and the features influencing it have been analyzed. Two main processes are distinguished: 1) bottom-up and 2) top-down ones. Bottom-up processes refer to improved collaboration among group members. They are mainly influenced by group features such as social sensitivity (Woolley et al., 2010) and cognitive diversity (Kozhevnikov et al., 2014; Aggarwal and Woolley, 2013). As to social sensitivity, in their experi- mental study Woolley et al. (2010) demonstrate that groups with higher proportion of people able to judge others' emotions perform better than other groups. As to cognitive diversity, Woolley et al. (2015) find that an inverted U-shaped relationship exists between cognitive diversity and CI.

In combination with bottom-up processes, CI is the result of top- down processes. These mainly concern the coordination of social in- teractions and are affected by group structures, norms, and routines. One of the most important aspect in this regard is that groups where people communicate and participate more equally exhibit higher CI (Woolley et al., 2010) in both face-to-face and on-line groups (Engel et al., 2014; Kim et al., 2015; Woolley et al., 2010). Furthermore, groups ruled by incentive systems wherein agents are rewarded for expressing accurate minority opinions show to produce stable, near- optimal CI (Mann and Helbing, 2017).

In addition to these processes and features, recent modeling studies have shown that CI is related to the process of consensus seeking among interacting group members and investigated the effect of the strength of social relationships on group performance. A critical value of strength of social relationships is found, at which group performance is opti- mized. This critical threshold corresponds to a phase transition from low to high level of consensus, which assures the effective exploration of problem space (De Vincenzo et al., 2017).

When the strength of social relationship is too low, the group is not able to reach an adequate level of consensus, individuals behave as single units, explore independently part of the performance space, and make decisions often conflicting among each other. As a consequence, performance suffers. For too high strength of social relationships, the individuals in the group are prematurely and strongly forced to be in agreement. Consensus limits effective exploration while fosters con- formity. This determines negative performance.

We are interested to more in depth investigate this process leading to the emergence of CI by introducing the role of distrust relationships.

3. Theory

3.1. Conceptualization of distrust in decision-making groups

Distrust is an important dimension of interpersonal relationships (Rempe et al., 1985). Distrust has been traditionally conceptualized with reference to trust, reflecting the end of a continuum. The latter is a multidimensional construct embracing diverse dimensions, such as vulnerability, benevolence, cooperation, non-opportunism, positive expectation, dependence, and goodwill (Seppänen et al., 2007). A large body of literature investigates trust at the interpersonal, organizational, and inter-organizational levels (for reviews, see Dirks and Ferrin, 2002; Seppänen et al., 2007; Schoorman et al., 2007). Overall, trust is defined as a party's confident positive expectations regarding intentions, mo- tives and behavior of another party (Mayer at al. 1995; Das and Teng,

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2001; Inkpen and Currall, 2004). Following Lewicki et al. (1998), with reference to interpersonal level, trust is defined as the individual's confident positive expectations regarding the conduct of another, where conduct includes not only what the other says and does, but also how he/she makes decisions. According to the traditional conceptualization, distrust, reflecting the end of a continuum, is defined as low trust. It follows that distrust concerns low positive expectations regarding the conduct of another, and, in particular, his/her actions and decisions.

More recently, distrust has gained an independent meaning. It is viewed as a separate and distinct construct by trust (Hardin, 2004; Lewicki et al., 1998; Sitkin and Roth, 1993; Vlaar et al., 2007, Lewicki et al., 2006). According to this recent perspective, distrust concerns a pervasive negative lens through which the counterpart is perceived, and a negative expectation regarding the behavior or intentions of another (Dimoka, 2010; Krämer et al., 1994, 1996; Lewicki et al., 1998; Sitkin and Roth, 1993). Following this consideration, distrust is defined as confident negative expectations regarding another's intentions, beha- vior, actions and decisions. As such, distrust relationships are associated with caution, defensiveness, and vigilance (Lewicki et al., 1998). In this paper we refer to this recent conceptualization and we are interested to analyze the effect of distrust relationships on the efficacy of collective decision-making. In particular, we define the scope of distrust as the extent to which distrust relationships are spread in the group. It is operationalized by the ratio between the number of distrust relation- ships and the total number of social relationships in the group.

3.2. Relationship between distrust and level of consensus in decision-making groups

In addition to caution and defensiveness, distrust relationships are found to be related to lack of cooperation (Cho, 2006; Vlaar et al., 2007), negative social influence (Blau, 1964; Sheppard and Tuchinsky, 1996), avoidance of interaction (Bies and Tripp, 1996), unwillingness to share views and preferences (Bijlsma-Frankema, 2004; March and Olsen, 1975), reduced information sharing (Gillespie and Dietz, 2009), and intergroup conflicts (Fiol et al., 2009; Tomlinson and Lewicki, 2007). In particular, once such negative expectations are created, the conflict tends to rise in scope and intensity so as to become often in- tractable (Lewicki and Wiethoff, 2000).

Distrust relationships, rather than pushing group members to reach an agreed solution, involve dissent and disagreement. Because distrust regards negative expectations concerning the others' conduct and, in particular, how well they make decisions, individuals involved in dis- trust relationships tend to make antagonistic decisions. This in turn makes more difficult reaching consensus in group. Therefore, as the number of distrust relationships rises, the level of consensus within the group diminishes. Coherently, within social-psychological and socio- logical literature, while trust is considered an ingredient for social order, distrust is associated with disorder and emergence.

3.3. Relationship between distrust, level of consensus, and collective intelligence

As argued above, social relationships induce group members to share information, adapt their actions, and converge toward common understanding and agreed decisions (Kelman, 1958). CI emerges when a right level of consensus within the group is reached. When the level of consensus within a group is too low, individuals propose alternative solutions and assess these solutions only on the basis of their knowl- edge. In doing this, they incur in decision-making biases (Bonabeau, 2009), which negatively influence the exploration of the solutions, and lead them to identify low-performing solutions. Furthermore, in- dividuals making independent decisions tend to be conflicting one with each other. When conflicts remain unresolved, group performance suffers (De Dreu and Weingart, 2003). On the contrary, when the level of consensus within the group is too high, individuals are strongly in

agreement. A strong pressure towards consensus creates conformity and limit creativity with negative consequence on the exploration of the solution space (Janis, 1982; Esser, 1998). The group comes up with a limited number of alternatives, because prefers to converge on an agreed solution rather than explore new solutions. The efficacy of the decision making process is thus undermined.

The level of consensus reached within a group depends on two features of the network of social relationships, i.e. the strength and the density. The strength of social relationships concerns the intensity of the link and, in particular, the extent to which the individuals involved in the relationship influence one with other (Marsden and Campbell, 1984). The density refers to the number of social relationships within a group in percentage to the total number of possible social relationships involving group members. Both variables positively affect social influ- ence within the group and lead the latter to reach a given level of consensus.

When the strength and the density of social interaction are too high, the pressure towards consensus may be too strong, so that the level of consensus reached within the group may be too high, leading to di- minished group performance. In this situation, distrust relationships, which involve dissent and contrast consensus seeking, can play a ben- eficial effect on group performance. By forcing disagreement within the group, distrust reduces the level of consensus, introduces emergence in the system, and let the group broadly explore the space of solutions. Until the right level of consensus is not reached, increasing the number of distrust relationships is beneficial for group performance. However, above a certain threshold, a further increase of the scope of distrust is no longer beneficial, since a strongly decreased level of consensus makes the members of the group to almost independently explore the solution space, leading to unresolved conflicts with negative con- sequences on performance. Based on the above, we argue that when the strength and the density of social relationships are too high, the scope of distrust first increases and then decreases group performance. Conversely, when the strength and the density of social relationships are too low, the presence of distrust relationships are always detri- mental. They decrease consensus and move away the system from reaching the right level of consensus with detrimental effect on group performance.

4. Model

We consider a group of M individuals collectively solving a complex decision making problem. The decision-making process is modelled referring to the model first developed by Carbone and Giannoccaro (2015) and then by De Vincenzo et al. (2017). In these models, the group is conceived as engaged in solving a combinatorial decision- making problem, consisting in identifying the combination of multiple and interdependent decisions d = (d1, d2, …,dN), yielding to the highest payoff for the group P(d).

The problem space is generated by means of the NK fitness land- scape (Kauffman and Levin, 1987, Kauffman, 1993), where N stands for multiple binary decisions and K (<N) is the average number of inter- dependencies among the decisions. K corresponds to the extent to which decisions are synergistic with the respect to their impact on system performance (Lazer and Friedman, 2007). This means that the impact of any decision to the system performance is contingent on the value of multiple interacting decisions (Rivkin, 2001). N captures the reality that problems are characterized by multiple decisions; K cap- tures the complexity of the problem (Rivkin and Siggelkow, 2003). The higher K, the more complex the landscape.

This problem space (referred to as fitness landscape) consists of 2N

possible combinations of choices on decisions, each with a fitness payoff associated. Specifically, the NK fitness landscape is generated by following a stochastic procedure, which permits to assign the payoff, P (d), to each combination of choices on decisions d=(d1,d2, …,dN). The payoff value, P(d), is computed as follows (De Vincenzo et al., 2017):

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d dP V N V V( ) ¯ [ ( ) ¯ ]= + (1)

where,

d d

V C

N ( )

( )j N

j1= = (2)

and V̄ is the statistical average of V d( ). Cj is the contribution that the decision j leads to the total system payoff. The latter is drawn at random from a uniform distribution [0,1]. Notice that, as effect of the inter- dependencies among decisions (K), Cj depends not only on how the single decision j is resolved but also on the choice on its interdependent decisions. For details about the landscape generation see Carbone and Giannoccaro (2015) and De Vincenzo et al. (2017).

The NK fitness landscape methodology was chosen for multiple reasons. It is consistent with a long tradition of creating simple yet insightful models of complex adaptive systems referring to single or- ganizations (Rivkin, 2000, 2001; Rivkin and Siggelkow, 2003; Siggelkow and Levinthal, 2003; Siggelkow and Rivkin, 2005; Ethiraj and Levinthal, 2004), groups of individuals (Barkoczi and Galesic, 2016; Carbone and Giannoccaro, 2015; De Vincenzo et al., 2017; Giannoccaro et al., 2017; Giannoccaro et al., 2018; De Vincenzo et al., 2018) and supply chains (Giannoccaro, 2011; Capaldo and Giannoccaro, 2015a, 2015b; Giannoccaro, 2015; Giannoccaro et al., 2018). Compared to other simulation models (such as systems dynamics and agent-based modeling), it is well-suited to model problem com- plexity arising from the interdependence among decisions (K) in a controlled manner (Davis et al., 2007). Furthermore, this approach helps in capturing the emergent nature of group decision-making, be- cause the group is conceived as a group of agents collectively per- forming an adaptive walk on the performance landscape in search of the highest peak (Giannoccaro et al., 2017).

4.1. The drivers of individual decision-making process

Any individual k in the group formulates his/her own opinion ( , , , )k k k kN1 2= … concerning the preferred combination of choices

on the decisions. In doing so, the individual is driven by two forces: 1) the improvement of the personal payoff (perceived payoff), which de- pends on the level of knowledge of the individual and 2) the social influence exerted by means of the social relationships.

To define the perceived fitness of the individual Vk, we first in- troduce the level of knowledge of the individual. This is modelled by specifying which contributions to the system payoff the individual knows. For example, in the case of a group developing a new product made up of a mechanical engineer and a marketing expert, it is likely

that the mechanical engineer knows the contributions to the system fitness of the decision concerning the materials to use or the production cycle to adopt, but could not know the contributions associated with the decision on the marketing campaign. Vice versa is true for the mar- keting expert.

Being D the matrix whose element Dkj takes the value of 1 when the single agent k knows the contribution Cj(σ) to the total fitness and 0 when he/she does not know it. To generate D, we define the probability p that the single agent knows the contribution Cj(σ) to the total fitness. Then, we set that Dkj takes the value of 1 with probability p and 0 with probability 1- p. The perceived fitness of the agent k is so defined (Carbone and Giannoccaro, 2015):

V D C

D ( )

( ) k k

j N

kj j k

j N

kj

1

1

= =

= (3)

Note that in this way the perceived fitness is the average of con- tributions of the decisions known by the individual k.

We consider that group members are involved in a network of social relationships during the decision-making process. Since group members make all the decisions, different social relationships can occur for any decision with a specific pattern. Therefore, we modelled the network of social relationships by means of a multiplex network made up by N layers corresponding to the N decisions dj. On each layer the nodes are the individuals and the links are the social relationships occurring among the group members concerning that specific decision. This net- work is coded by a N-block diagonal adjacency matrix A (see Fig. 1 as an example).

Based on the social influence theory, we argue that an individual involved in social relationships adapts his/her behavior, beliefs, mental models to the behaviors, beliefs or mental models of the interacting members (Kelman, 1958; Leenders and Fearon, 1997). Social relation- ships are in fact conduits of opinion formation and stimulate the con- vergence towards a common understanding of a situation and shared mental models among individuals (Liu et al., 2012). Therefore, the in- dividual k, as a consequence of the social relationship with l, will tend to modify his/her opinion to be in agreement with l.

However, distrust may characterize social relationships in the group. Distrust relationships, rather than inducing group members to be in agreement, involve dissent. As explained in Section 3.2, since the individual has negative expectations concerning the conduct and, in particular, the goodness of the decisions made by the interacting members, he/she prefers to make antagonistic decisions. This implies that the individual k while interacting with l will tend to modify his/her opinions attempting to reach a disagreement with l.

Fig. 1. Multiplex network of the social relationships within the group and attendant adiacency matrix.

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To model these dynamics, an Ising-like approach is employed (Bordogna and Albano, 2007a). We refer to the model by Carbone and Giannoccaro (2015) and, for any given decision layer, we defined the energy level associated with individual k, Ek, as follows:

E A J ,k j l

kl kl k j

l j=

(4)

where k j is the current opinion of individual k on the given decision j

and l j is the current opinion on the same decision of the interacting

individual l. Jkl is the element of a weight matrix J, which models the strength of social relationships in the network. It is also employed to model distrust relationships.

Note that the individual k formulates his/her opinion to minimize the energy level. This implies that if Jkl is positive, the effect of social relationship is to push the individual k to make his/her opinion to be in agreement with the interacting individual l. On the contrary, if Jkl is negative, the social relationship has the opposite effect, i.e. the in- dividual k is induced to make opinions in disagreement with l. Therefore, the social relationship with Jkl <0 is distrustful.

The scope of distrust is operationalized by introducing increasing number of distrust relationships and defined as the percentage of dis- trust relationships on the total number of social relationships in the group.

4.2. The model of the collective decision making process

The group decision making dynamics is modelled by means of a continuous-time Markov chain, whose transition rates are defined so as to capture the two drivers of individual behavior in groups, i.e. the optimization of the perceived payoff and the consensus reaching.

The state of the whole system is:

s s s ss ( , , ..., , ..., ) ( , , ... , , , ... , ..., , , ... ).l M N N N M M MN1 2 11 12 1 21 22 2 1 2= =×

Let be P (s, t) the probability that, at time t, the state vector takes the value s out of 2M×N possible states. The time evolution of the probability P (s, t) satisfies the following master equation:

s s s s s s s

dP t dt

w P t w P t ( , )

( ) ( , ) ( ) ( , )k k k k k k k l

= + (5)

where s s s s s( , , , , , )k k M N1 2= … … × and s s s s s( , , , , , )k k M N1 2= … … × . Eq. (5) represents a Markov time-continuous chain where the

transition rate (i.e., the probability per unit time that the opinion sk flips to − sk while the others remain temporarily fixed) is given by (De Vincenzo et al., 2017):

s sw s J A s exp V s s( ) 1 2

[1 tanh( ) ] { [ ( , ) ] }k k k l

kl kl l k k= × (6)

In Eq. (6) β' is referred to as the level of confidence the members have about their perceived fitness, is the mean degree connectivity of the network of social relationship among the agents on each decision layer, and V s s( , )k k is the change in the perceived payoff if the in- dividual k modifies his/her opinion from sk to sk.

Note that the transition rate is the product of two terms: 1) the Weidlich exponential rate (Weidlich, 1991), exp V s s{ [ ( , ) ] },k k which models the improvement of perceived payoff and 2) an Ising-Glauber

term (Glauber, 1963), s J A s[1 tanh( ) ],k l kl kl l 1 2

which models the process of social influence. For this reason, our model differs from a classical straightforward Ising model, which includes only the effect of the social interactions among spins (i.e. the social interactions).

We employ the Gillespie algorithm to generate the stochastic pro- cess given by equations (5) and (6). This algorithm is summarized in Appendix A.

It is noteworthy that using this approach the effect of social re- lationships on the system dynamics is not imposed but it is emergent and self-organized. This means that a distrust relationship does not imply that the individuals involved are forced to make opposite choices, but that they will be pushed to behave in this way. The actual choice will be the result of both the influences, i.e. the maximization of the perceived payoff and the social influence resulting from the entire complex network of social relationships.

4.3. Group decision-making performance

Group performance measuring the efficacy of the group to find the best choice configuration is computed at the end of simulation. To do so, the group choice configuration given the choice configuration of all group members is defined. Different rules may be employed at this aim such as majority, best member, and random member (Hastie and Kameda, 2005; Sorkin et al., 2001; Sorkin et al., 1998). We selected the majority rule because it is consistent with our theory concerning con- sensus reaching inside the group. It is also proved to perform better than the best and random member rules in different situations (Hastie and Kameda, 2005; Barkoczi and Galesic, 2016).

Thus, given the set of opinions ( , , ,j j M j

1 2 … ) that the agents have about the decision j at time step t, we set the group choice on the de- cision j as follows:

d M j Nsgn , 1,2, .j k

k j1= = …

(7)

If M is even and in the case of a parity condition, dj is uniformly chosen at random between the two possible values + 1, −1.

In particular, the group performance is calculated in terms of effi- cacy of group decision making by normalizing V respect to the max- imum fitness value on the landscape (Vmax). A value of 1 means that the group was able to identify the optimal solution.

We also compute the level of consensus among the agents in the group as follows:

M N 1

j

N

kh

M

k j

h j

2 1 1

= < > = = (8)

Note that 0 1< < . The higher , the higher the level of consensus. 1= means that all M members agree on all N decisions. Table 1 summarizes the operationalizations of the main variables.

5. Simulations analysis and results

We simulate a group with M=21 solving a combinatorial decision- making problem defined by a NK fitness landscape with N=15 and K=1, 3, 5, 7. We set β’=3 and β=1. The network of social

Table 1 Variables and operationalizations.

Variable Operationalization

Network of social relationships Multiplex network modelled by matrix A Density of the network of social relationships Number of social relationships on the total number of possible social relationships Strength of the social relationship between l and k | Jlk | where the symbol | | stands for the absolute value Distrust relationship between l and k Jlk < 0 Scope of distrust Number of distrust relationships among the group members on the total number of social relationships Group performance V/Vmax where the vector of group decisions is computed by applying the majority rule

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relationships is generated according to Erdős and Rényi (1960)'s random graph. All social relationships are assumed to have the same strength intensity |Jlk|=J > 0 for any l and k.

The simulation is carried out by adopting the Gillespie algorithm for a simulation period of 500000 time steps and 100 replications. The simulation results consist in the efficacy of the group in solving the decision making problem (V/Vmax) and in the level of consensus reached at the end of simulation, both averaged across replications.

5.1. Baseline model results

We first simulate group performance in absence of distrust re- lationships. In particular, we simulate groups characterized by four values of strength of social relationships (0.5, 1, 2, 4). These values are chosen to be lower (0.5), slightly higher (2), and higher (4), than the threshold optimal value (1) leading to the emergence of CI. The threshold value is given by De Vincenzo et al. (2017). Similarly, we simulate groups with three levels of density of social relationships in the group (0.1, 0.3, and 0.7).

Results concerning the efficacy of group and the level of consensus are presented in Table 2 for any level of strength of social relationships (J), density of social relationships (DENS), and level of interdependence among decisions (K).

They show that as the strength of social relationships (J) increases, the level of consensus grows, the value of interdependence (K) and density of social relationships (DENS) fixed. For example for K=1 and DENS=0.1, the level of consensus grows from 0.2086 to 0.6885, as J increases from 0.5 to 4. Similarly, for K=3 and DENS=0.3, the level of consensus ranges from 0.2982 to 0.9788, when J moves for 0.5 to 4. Similarly, we also note that as the density of social relationship rises, the level of consensus increases, K and J fixed. For example, in the case of K=3 and J=1, as the density increases from 0.1 to 0.3 and to 0.7, the level of consensus moves from 0.5230 to 0.7389, and to 0.8003, respectively.

These findings, running as expected, are a test of the internal va- lidity of our simulation model.

Results also show that when the level of consensus is too low, group performance is quite low. Increasing the level of consensus, the group performance rises. However, when the level of consensus becomes too high, group performance diminishes. For example, let consider the case of K=3. For J=0.5 and DENS=0.1, the level of consensus is 0.2525 and group performance is 0.8250. Increasing J to 1 and density to 0.3, the level of consensus becomes 0.7389 and group performance reaches 0.9993. However, for J=4 and DENS=0.7, the level of consensus becomes 0.9925 and group performance decreases to 0.6435.

5.2. Results for increasing scope of control

We simulate group performance in different scenarios characterized by increasing values of scope of distrust. We considered ten values of scope of distrust from Z=0.05 to Z=0.5. To model the scope of dis- trust, on a generic decision layer, we draw at random, with probability Z, certain distrust relationships. This result is then replicated on all the decisions layers. Therefore, using this method, the agents involved in distrust relationships are the same on all the decision layers and the scope of distrust is about Z.

Similarly to the baseline model, we simulate groups characterized by four values of strength of social relationships (0.5, 1, 2, 4) and three levels of density (0.1, 0.3, and 0.7), solving problems with K=1,3,5,7. Thus, the total plan of experiments is made by 528 cases, including the baseline. Results are shown in Table 3.

We analyze the relationship between scope of distrust and group performance. Results confirm our theoretical argumentations. When the strength of social relationships is lower than (J=0.5) or equal to the critical threshold (J=1), the effect of scope of distrust is always det- rimental for group performance, independently of K and density values. For higher values of strength of social relationships (J=2, 4), group performance first increases and then decreases, as the scope of distrust rises. For example, for K=3 and DENS=0.3, in the case of J=1, moving from Z=0 to Z=0.5, performance reduces from 0.9993 to 0.6741, while in the case of J=2 first increases from 0.9213 to 0.9855 as Z rises from 0 to 0.2, and then decreases to 0.7302 when Z=0.5. For J=4, group performance increases from 0.7130 to 0.9363 moving from Z=0 to Z=0.25, then diminishes to 0.7051 when Z=0.5.

Note that, as the strength of social relationships increases, the highest performance is achieved for higher value of scope of distrust, compared to previous cases. On average, for J=2, a minimal scope of distrust (Z=0.05) is optimal for group performance; for J=4, the optimal value of scope of distrust is higher (Z=0.25).

Fig. 2a (2b) shows group performance (level of consensus) as a function of the scope of distrust (Z) for the four J values, averaged across density and K values. These figures clearly show the two trends: 1) the decreasing relationship between group performance and scope of distrust for low values of J and 2) the inverted-U shape between group performance and the scope of distrust for high values of J.

When the strength of social relationships is too low (e.g. J=0.5) and the level of consensus reached within the group is low, any scope of distrust is detrimental for group performance. Introducing a distrust relationship decreases the level of consensus, so impeding the in- dividuals to collectively explore the landscape in search of better so- lutions. Each individual independently explores the landscape with a detrimental effect on group performance.

As the strength of social relationship increases, the level of

Table 2 Results of the baseline model (Z=0)*.

DENS Group performance Level of consensus

J=0.5 J=1 J=2 J=4 J=0.5 J=1 J=2 J=4

K=1 0.1 0.8555 0.9096 0.9309 0.8414 0.2086 0.3776 0.6215 0.6885 0.3 0.8568 0.9911 0.9859 0.7353 0.2123 0.6718 0.9179 0.9759 0.7 0.8542 0.9954 0.9896 0.7121 0.2208 0.7168 0.9553 0.9919

K=3 0.1 0.8250 0.9951 0.9565 0.8444 0.2525 0.5230 0.6484 0.6864 0.3 0.8213 0.9993 0.9213 0.7130 0.2982 0.7389 0.9308 0.9788 0.7 0.8308 0.9976 0.9259 0.6938 0.3086 0.8003 0.9651 0.9936

K=5 0.1 0.6700 0.9302 0.8944 0.7959 0.1813 0.5057 0.6435 0.6674 0.3 0.6237 0.9249 0.8538 0.7088 0.1891 0.7435 0.9366 0.9814 0.7 0.6327 0.9223 0.8579 0.6435 0.1887 0.8127 0.9741 0.9925

K=7 0.1 0.5345 0.9491 0.8956 0.8080 0.1475 0.5025 0.6248 0.6273 0.3 0.4621 0.9276 0.8606 0.7193 0.1369 0.7267 0.9393 0.9835 0.7 0.4981 0.9292 0.8533 0.6795 0.1432 0.8259 0.9743 0.9932

*standard deviation given in Appendix B.

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consensus reached within the group also grows and group performance improves. However, a too high the level of consensus (e.g., J=4) be- comes detrimental for group performance, because prematurely hinders the exploration of the landscape looking for configurations with higher fitness. In such a case, since a distrust relationship decreases the level of consensus reached in the group, it improves the exploration of the landscape thus increasing the chance to find high performing config- urations. However, when the scope of distrust rises too much, the level of consensus becomes low and performance diminishes. Since the higher the strength of social relationships, the higher the level of con- sensus and the lower the group performance are, the optimal number of distrust relationships grows as the strength of social relationships rises.

We now analyze the influence of the density of social relationships

on the relationship between scope of distrust and group performance. In Table 4, for each value of the density and scope of distrust, the averaged group performance and the averaged level of consensus reached within the group are shown. They are computed averaging results across the scenarios characterized by the four values of strength of social re- lationships (J) and the three values of interdependence (K). Findings show that the density of social relationships affects the relationship between scope of distrust and group performance. For low density (0.1), group performance diminishes as the scope of distrust rises. For medium and high density values (0.3 and 0.7), an inverted-U trend is achieved. In particular, when density is 0.3, a scope of distrust of 0.2 is optimal for group performance; for density equal to 0.7, a value Z=0.3 assures the highest performance. This result is explained by the

Table 3 Results of simulations*.

Z=0 Z=0.05 Z=0.1 Z=0.15 Z=0.2 Z=0.25 Z=0.3 Z=0.35 Z=0.4 Z=0.45 Z=0.5

J=0.5 DENS=0.1 K=1 0.8555 0.8554 0.8480 0.8498 0.8318 0.8332 0.8270 0.8201 0.8092 0.8124 0.8038

K=3 0.8250 0.8334 0.8028 0.7559 0.7707 0.7279 0.7395 0.6965 0.6940 0.6767 0.6996 K=5 0.6700 0.6502 0.6352 0.6163 0.5326 0.5511 0.5382 0.5272 0.5117 0.5013 0.5071 K=7 0.5345 0.5475 0.5171 0.4836 0.4811 0.4644 0.4206 0.4317 0.4412 0.3941 0.3976

DENS=0.3 K=1 0.8568 0.8612 0.8487 0.8348 0.8372 0.8258 0.8265 0.8184 0.8133 0.8056 0.7860 K=3 0.8213 0.8105 0.7979 0.7769 0.7551 0.7351 0.7004 0.6758 0.6975 0.6591 0.6356 K=5 0.6237 0.6223 0.5709 0.5652 0.5722 0.5355 0.5371 0.5137 0.5430 0.5190 0.4800 K=7 0.4621 0.4823 0.4817 0.4424 0.4706 0.4707 0.4003 0.3786 0.4182 0.4004 0.4008

DENS=0.7 K=1 0.8542 0.8514 0.8478 0.8450 0.8396 0.8272 0.8339 0.8227 0.8123 0.8098 0.8173 K=3 0.8308 0.8158 0.7784 0.7547 0.7624 0.7410 0.6869 0.6939 0.6592 0.6581 0.6447 K=5 0.6327 0.5972 0.5659 0.5493 0.5433 0.5119 0.5316 0.5082 0.5133 0.5092 0.4906 K=7 0.4981 0.4718 0.4356 0.4180 0.4380 0.4281 0.4216 0.4071 0.4130 0.3838 0.4277

Average 0.7054 0.6999 0.6775 0.6577 0.6529 0.6376 0.6220 0.6078 0.6105 0.5941 0.5909 J=1 DENS=0.1 K=1 0.9096 0.9043 0.8819 0.8831 0.8686 0.8630 0.8241 0.8204 0.8291 0.8317 0.8157

K=3 0.9951 0.9746 0.9275 0.8921 0.8622 0.8219 0.8153 0.7861 0.7512 0.7470 0.7075 K=5 0.9302 0.9116 0.8584 0.8090 0.7902 0.7221 0.7176 0.6684 0.6232 0.5803 0.5625 K=7 0.9491 0.9158 0.8849 0.8297 0.7643 0.7183 0.6936 0.6116 0.6568 0.5716 0.5700

DENS=0.3 K=1 0.9911 0.9871 0.9809 0.9744 0.9709 0.9578 0.9428 0.9272 0.9256 0.9138 0.9121 K=3 0.9993 0.9939 0.9663 0.9339 0.8959 0.8279 0.8091 0.7623 0.7459 0.6990 0.6741 K=5 0.9249 0.9103 0.8951 0.8453 0.7713 0.7069 0.5876 0.5952 0.5635 0.5136 0.4768 K=7 0.9276 0.9293 0.9129 0.8433 0.7215 0.6332 0.5319 0.4688 0.4239 0.4043 0.4377

DENS=0.7 K=1 0.9954 0.9897 0.9786 0.9774 0.9699 0.9548 0.9547 0.9270 0.9305 0.8898 0.9049 K=3 0.9976 0.9916 0.9812 0.9213 0.8875 0.8092 0.8004 0.7483 0.7038 0.6621 0.6666 K=5 0.9223 0.9174 0.8958 0.8342 0.7551 0.6399 0.5745 0.5480 0.5123 0.4888 0.4841 K=7 0.9292 0.9409 0.9162 0.7683 0.6089 0.4627 0.5079 0.4317 0.3757 0.4146 0.3983

Average 0.9559 0.9472 0.9233 0.8760 0.8222 0.7598 0.7299 0.6913 0.6701 0.6431 0.6342 J=2 DENS=0.1 K=1 0.9309 0.9222 0.8978 0.8997 0.8702 0.8653 0.8442 0.8359 0.8267 0.8289 0.8114

K=3 0.9565 0.9644 0.9034 0.9258 0.8907 0.8682 0.8077 0.8004 0.7424 0.7472 0.7170 K=5 0.8944 0.8974 0.8780 0.8375 0.8082 0.7502 0.7435 0.6780 0.6977 0.6995 0.6155 K=7 0.8956 0.8738 0.8441 0.7875 0.7800 0.7501 0.6934 0.6727 0.6384 0.6366 0.5704

DENS=0.3 K=1 0.9859 0.9990 0.9966 0.9970 0.9907 0.9852 0.9706 0.9581 0.9404 0.9107 0.8983 K=3 0.9213 0.9640 0.9657 0.9826 0.9855 0.9713 0.9246 0.8797 0.7996 0.7590 0.7302 K=5 0.8538 0.8682 0.8975 0.9043 0.9205 0.9016 0.8446 0.7987 0.7334 0.6158 0.6500 K=7 0.8606 0.8774 0.8906 0.9101 0.9047 0.9072 0.8677 0.8461 0.6960 0.6188 0.5692

DENS=0.7 K=1 0.9896 0.9957 0.9995 0.9992 0.9955 0.9908 0.9827 0.9612 0.9411 0.9222 0.8803 K=3 0.9259 0.9507 0.9687 0.9709 0.9907 0.9876 0.9692 0.9228 0.8277 0.7296 0.6818 K=5 0.8579 0.8742 0.8777 0.9026 0.9184 0.9218 0.8623 0.7904 0.6875 0.5888 0.5534 K=7 0.8533 0.8895 0.8964 0.9029 0.9173 0.9277 0.9198 0.7521 0.6218 0.5320 0.4634

Average 0.9105 0.9230 0.9180 0.9183 0.9144 0.9023 0.8692 0.8247 0.7627 0.7158 0.6784 J=4 DENS=0.1 K=1 0.8414 0.8538 0.8465 0.8316 0.8284 0.7960 0.7938 0.7991 0.7690 0.7490 0.7587

K=3 0.8444 0.8584 0.8266 0.8482 0.8332 0.7721 0.7629 0.7279 0.7597 0.7320 0.7160 K=5 0.7959 0.7873 0.7735 0.7309 0.7402 0.7184 0.6823 0.6250 0.6289 0.6184 0.5659 K=7 0.8080 0.7760 0.7930 0.7412 0.7095 0.6732 0.6602 0.6267 0.5867 0.5808 0.5443

DENS=0.3 K=1 0.7353 0.8147 0.8864 0.9479 0.9632 0.9902 0.9675 0.9352 0.9224 0.8600 0.8620 K=3 0.7130 0.7860 0.8407 0.8925 0.9322 0.9363 0.8989 0.8812 0.8585 0.7733 0.7051 K=5 0.7088 0.7510 0.8002 0.8248 0.8497 0.8682 0.8503 0.8485 0.8053 0.7384 0.6905 K=7 0.7193 0.7755 0.7971 0.8306 0.8544 0.8588 0.8236 0.8012 0.7610 0.7587 0.7015

DENS=0.7 K=1 0.7121 0.7476 0.8557 0.9135 0.9735 0.9945 0.9917 0.9831 0.9663 0.9251 0.8978 K=3 0.6938 0.7405 0.8065 0.8640 0.8923 0.9365 0.9531 0.9412 0.9001 0.8342 0.7270 K=5 0.6435 0.7036 0.7430 0.7841 0.8280 0.8812 0.9036 0.8753 0.8385 0.7608 0.6505 K=7 0.6795 0.7388 0.7842 0.8152 0.8392 0.8744 0.8982 0.9102 0.7934 0.7460 0.6757

Average 0.7412 0.7778 0.8128 0.8354 0.8537 0.8583 0.8488 0.8296 0.7992 0.7564 0.7079

*standard deviation given in Appendix B.

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influence of the density of social relationships on the process of con- sensus reaching within the group. When the density of social relation- ships is too low, there are few social links among the group members and the level of consensus reached in the group is in turn quite low, because individuals interacting less are not so prone to change their opinions to be in agreement with others. A low level of consensus en- tails that individuals in the group make independent decisions on the basis of their personal knowledge and perspective, often resulting in conflicting positions that will remain unresolved. Introducing distrust relationships in this condition is detrimental. Distrust relationships negatively influence the level of consensus, making the system per- forming even worse.

When the density of social relationships is quite high (0.3, 0.7), social influence takes place, individuals are engaged in intensive dia- logue and interactions, so that the group is able to reach higher con- sensus. In such a case, distrust relationships are beneficial because, reducing consensus, make the system better explore the landscape without converging too soon in an agreed suboptimal solution. However, when the scope of distrust becomes too high, the level of consensus decreases, impeding the emergence of CI, and performance reduces (see Fig. 3).

5.3. Validation

In order to validate our model, we perform a sensitivity analysis to test the robustness of the results to the value of parameter β’ (the level of confidence of individual of their knowledge). In particular, we per- form additional simulations for β'=10. Results confirm the same

trends observed above, concerning the moderating effect played by the strength of social relationships and the density of social relationships on the relation between group performance and scope of distrust. In Appendix B we reported the results of regression analyses made on all the simulation data, statistically confirming the moderating effect played by the strength of relationships (Model 1) and density of re- lationships (Model 2) on the relation between scope of control and group performance. In fact, as to the effect of the strength of social relationships, results statistically confirm that the effect of Z is linear and negative when J=0.5 and J=1 (, whereas when J=2 and J=4 the linear effect of Z becomes positive and the quadratic effect of Z is negative. Similarly, as to the effect of the density of social relationships, results statistically confirm that when DENS=0.1 the effect of Z is significant and negative, while when DENS=0.3 and DENS=0.7 the linear effect of Z is positive and the quadratic effect of Z is negative.

6. Discussion and conclusions

How can the emergence of collective intelligence be fostered to increase performance of decision-making groups? Our paper answers to this enduring question investigating the effect of the scope of distrust on group performance.

Our main finding is that the relationship between scope of distrust and group performance depends on the strength and the density of social relationships. In particular, depending on the values of these variables, two trends emerge: 1) the scope of distrust negatively affects group performance and 2) the relationship between scope of distrust and group performance follows an inverted-U shape. The negative trend

Fig. 2. Group performance (a) and Level of consensus (b) for different strength of social relationships.

Table 4 Averaged performance for different density of social relationships.

Z=0 Z=0.05 Z=0.1 Z=0.15 Z=0.2 Z=0.25 Z=0.3 Z=0.35 Z=0.4 Z=0.45 Z=0.5

Averaged group performance DENS=0.1 0.8523 0.8454 0.8199 0.7951 0.7726 0.7435 0.7227 0.6955 0.6854 0.6692 0.6477 DENS=0.3 0.8190 0.8395 0.8456 0.8441 0.8372 0.8195 0.7802 0.7555 0.7280 0.6844 0.6631 DENS=0.7 0.8135 0.8260 0.8332 0.8263 0.8225 0.8056 0.7995 0.7640 0.7185 0.6784 0.6478 Averaged level of consensus DENS=0.1 0.4942 0.3968 0.2965 0.2450 0.1946 0.1531 0.1265 0.1073 0.0904 0.0774 0.0681 DENS=0.3 0.7101 0.6687 0.6089 0.5448 0.4467 0.3492 0.2532 0.1876 0.1413 0.1060 0.0885 DENS=0.7 0.7411 0.7114 0.6722 0.6137 0.5437 0.4672 0.3646 0.2594 0.1703 0.1200 0.0908

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is found for low values of strength (density) of social relationships, while the inverted-U shape is achieved when the strength (density) of social relationships is high. This implies that for high strength (density) of social relationships a moderate scope of distrust is beneficial for group performance. In fact, in absence of distrust relationships, high values of the strength (density) of social interactions drive the group towards a too high level of consensus, which hampers a broad ex- ploration of the landscape, thus leading to inadequate group perfor- mance. In these conditions we showed that distrust relationships, by reducing the level of consensus, have a beneficial effect on exploration and leads to higher group performance. However, when the number of distrust relationships rises too much, performance diminishes, because the level of consensus becomes too low and individuals behave as in- dependent agents exploring each limited portion of the landscape. When the strength (density) of social relationships is low and, thus, the level of consensus reached within the group is low, distrust relation- ships reduce more the level of consensus and have a detrimental effect on group performance.

Our study makes multiple contributions to the literature. From a theoretical point of view, we enrich CI research by highlighting a new process leading to the emergence of CI, i.e. consensus seeking, while previous studies analyzed the processes of collaboration and co- ordination of social relationships (Woolley et al., 2015). We showed that an adequate pressure towards consensus seeking is required to permit individuals to sufficiently explore the landscape in search of high performing solutions but simultaneously to foster the agreement among the group members on a common solution, so avoiding conflicts. This finding sheds light on the mechanisms used by decision making groups to overcome individual's biases when solving complex problems (Bonabeau, 2009).

We also contribute to literature concerning the drivers of CI. While previous studies mainly focus on individual and group features (e.g., social identity, cognitive diversity, group incentives) affecting colla- boration and coordination (Woolley et al., 2015), we analyzed a vari- able characterizing the nature of social relationships in the group, i.e. the scope of distrust. We were able to clearly define when distrust re- lationships are beneficial for group performance and at which scope. These findings also complement previous research on consensus-per- formance relationship and the role of minority dissent on the efficacy of group decision-making. In particular, we contribute to explain contra- dictory results concerning the effect of consensus on group performance

(Dess and Origer, 1987; Priem, 1990; Jehn and Mannix, 2001; Hinds and Bailey, 2003), emphasizing the role of two moderators character- izing the network of social relationships (i.e., the strength of social relationships and the density of social relationships), while previous studies focused on group attributes, such as homogeneity (Priem, 1990), social identity (Hinds and Mortensen, 2005), geographical dis- tribution (Hinds and Bailey, 2003), and contextual variables, such as problem complexity (De Dreu and Weingart, 2003), environmental dynamism (Priem, 1990; Homburg et al., 1999), and firm strategy (Homburg, Krohmer et al., 1999). We also confirm that minority dissent provoked by distrust relationships is beneficial (De Dreu and West, 2001; De Dreu, 2002), because can increase the ability of decision- making groups to find optimal solution to the problem, but it should be introduced at a moderate extent. In particular, we add in which con- ditions (high strength and high density of social relationships) dissent should be instigated within group to enhance its performance, for ex- ample by assigning controversial roles to the group members resorting to the so-called ‘devil's advocacy’ procedure (Herbert and Estes, 1977; Janis, 1982; Schwenk, 1984). According to this practice, someone role- plays a position critiquing the decisions favored by the other in- dividuals to increase diversity and improve the quality of group deci- sions. This corresponds to intentionally introduce distrust relationships within the group, which, reducing the high level of consensus, may foster the coordinated exploration of solution space.

Furthermore, our results may inform managers on how to design web-based platforms exploiting CI (Bonabeau, 2009). Referring to the study by Malone et al. (2010), who classified the genome of CI plat- forms, our study refers to the “How” question and “Group decision” and “Individual decisions” genes. As to the group decision, our study shows that consensus is useful at a moderately and not high extent. As to the “Individual decisions”, we find that the strength and the density of social connections have an important role and should be moderate. Thus, our findings suggest to limit social connections among users and to control the strength of social relationships among them. A too high strength, due for example to multiple, long-term, or friendship inter- actions, could be risky for CI because could determine a too high level of consensus. Therefore, random connections with limited amount should be preferred when designing such a CI platform.

Finally, our study presents an advance in modeling group decision- making using the Ising approach. We defined the transition rate by adding to the classical Ising-Glauber term (modeling the social

Fig. 3. Group performance (a) and Level of consensus (b) for different density of social relationships.

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influence), the Weidlich exponential rate, which takes into account the search of high performing solutions. This means that the individuals involved in social relationships are not forced to make the antagonistic or similar decisions depending on the nature of social relationships, but that they will be driven to behave in these ways. The actual choice of the individual will be in fact the result of both the tendency to max- imize the perceived performance and the social influence resulting from the entire complex network of social relationships. This makes the model more complex and more realistic compared to previous studies

that neglect performance and include only social influence (Galam, 1997; Galam and Zucker, 2000; Holme and Jo, 2016).

This paper has some limitations. We consider that distrust involving two individuals occur on all the decisions they are taking. The network of social relationships is assumed to be a random one, while interac- tions especially in social systems may follow different patterns, such as small-world or scale-free ones. Replicating the study for groups showing these types of pattern can be useful to extend the boundary conditions of our theory.

Appendix A

Here, we report a brief explanation of the Gillespie algorithm, used to solve equations (5) and (6). It consists in the following steps:

1) Choose at random the initial state s of the system 2) Calculate all the transition rates s sw l n N M( ), 1, ,l l = … = 3) Calculate the total rate s sw w ( )l lT l= 4) Normalize all the transition rates as w s s w( )/l l l T= and builds the cumulative distribution F ( )l from the probability mass function l. 5) Calculate the time t to the next opinion flip by drawing from an exponential distribution with mean w1/ T, i.e. chooses a real number r0 1 from a uniform distribution and set t w rlog ( )T 1=

6) Identify the l-th opinion that flips from sl to sl, by drawing from a discrete distribution with probability w s w( ) /l l T= , i.e. draw a real random number s0 1 from a uniform distribution and chooses l so that F s F( ) ( )l l1 < < .

7) Update the state vector and returns to step 2 or quit.

Appendix B

Table B1 Standard Deviations of the results of the baseline model (Z=0).

Group performance Level of consensus

DENS J=0.5 J=1 J=2 J=4 J=0.5 J=1 J=2 J=4

K=1 0.1 0.0627 0.0363 0.0148 0.0690 0.0500 0.0546 0.0581 0.0506 0.3 0.0626 0.0195 0.0518 0.1334 0.0508 0.0644 0.0295 0.0169 0.7 0.0615 0.0160 0.0444 0.1559 0.0518 0.0616 0.0229 0.0073

K=3 0.1 0.0942 0.0195 0.0517 0.1024 0.0677 0.0614 0.0523 0.0519 0.3 0.1065 0.0059 0.0800 0.1311 0.0714 0.0686 0.0346 0.0162 0.7 0.1036 0.0127 0.0763 0.1265 0.0739 0.0536 0.0199 0.0069

K=5 0.1 0.1586 0.0503 0.0649 0.0841 0.0599 0.0747 0.0572 0.0491 0.3 0.1739 0.0304 0.0696 0.1296 0.0660 0.0687 0.0354 0.0144 0.7 0.1484 0.0323 0.0745 0.1505 0.0637 0.0670 0.0193 0.0070

K=7 0.1 0.2316 0.0490 0.0585 0.0884 0.0541 0.0803 0.0643 0.0781 0.3 0.2156 0.0498 0.0748 0.1109 0.0437 0.0870 0.0365 0.0149 0.7 0.2178 0.0444 0.0787 0.1459 0.0500 0.0792 0.0193 0.0072

Table B2 Standard deviations of the results of simulations.

Z=0 Z=0.05 Z=0.1 Z=0.15 Z=0.2 Z=0.25 Z=0.3 Z=0.35 Z=0.4 Z=0.45 Z=0.5

J=0.5 DENS = 0.1 K=1 0.0627 0.0633 0.0719 0.0679 0.0721 0.0659 0.0785 0.0799 0.0780 0.0732 0.0830

K=3 0.0942 0.1086 0.1138 0.1533 0.1447 0.1725 0.1399 0.1748 0.1550 0.1449 0.1507 K=5 0.1586 0.1496 0.1698 0.1389 0.1722 0.1718 0.1847 0.1627 0.1856 0.1767 0.1668 K=7 0.2316 0.2281 0.2215 0.2175 0.2349 0.2186 0.2023 0.2024 0.2015 0.1799 0.2049

DENS = 0.3 K=1 0.0626 0.0626 0.0689 0.0626 0.0744 0.0820 0.0732 0.0759 0.0791 0.0828 0.0848 K=3 0.1065 0.1017 0.1270 0.1292 0.1464 0.1275 0.1624 0.1365 0.1501 0.1471 0.1609 K=5 0.1739 0.1607 0.1546 0.1590 0.1678 0.1458 0.1549 0.1684 0.1717 0.1681 0.1931 K=7 0.2156 0.2033 0.2042 0.1935 0.1809 0.2021 0.1930 0.1930 0.1762 0.1898 0.1735

DENS = 0.7 K=1 0.0615 0.0621 0.0690 0.0687 0.0753 0.0745 0.0721 0.0661 0.0767 0.0726 0.0842 K=3 0.1036 0.1058 0.1226 0.1455 0.1351 0.1491 0.1513 0.1439 0.1538 0.1783 0.1563 K=5 0.1484 0.1410 0.1601 0.1658 0.1824 0.1710 0.1552 0.1732 0.1573 0.1664 0.1686 K=7 0.2178 0.2230 0.2118 0.1902 0.2073 0.2014 0.1681 0.2123 0.1833 0.1945 0.1925

J=1 DENS = 0.1 K=1 0.0363 0.0508 0.0635 0.0623 0.0679 0.0744 0.0802 0.0810 0.0820 0.0824 0.0769

K=3 0.0195 0.0779 0.1104 0.1596 0.1563 0.1795 0.1932 0.1890 0.1891 0.1951 0.2113 K=5 0.0503 0.0966 0.1514 0.1920 0.2096 0.2314 0.2247 0.2453 0.2503 0.2600 0.2106 K=7 0.0490 0.1137 0.2065 0.2471 0.2542 0.2929 0.2438 0.2973 0.2808 0.3229 0.2871

DENS = 0.3 K=1 0.0195 0.0272 0.0283 0.0334 0.0379 0.0483 0.0616 0.0725 0.0801 0.0892 0.0730 K=3 0.0059 0.0254 0.0703 0.1027 0.1056 0.1180 0.1250 0.1536 0.1462 0.1727 0.1511

(continued on next page)

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Table B2 (continued)

Z=0 Z=0.05 Z=0.1 Z=0.15 Z=0.2 Z=0.25 Z=0.3 Z=0.35 Z=0.4 Z=0.45 Z=0.5

K=5 0.0304 0.0548 0.0916 0.1207 0.1544 0.1658 0.2091 0.1888 0.1690 0.1570 0.1718 K=7 0.0498 0.0668 0.0984 0.2012 0.2336 0.2506 0.2284 0.2385 0.1674 0.2063 0.1920

DENS = 0.7 K=1 0.0160 0.0217 0.0286 0.0308 0.0403 0.0506 0.0488 0.0813 0.0776 0.1002 0.0871 K=3 0.0127 0.0241 0.0473 0.0831 0.0972 0.1104 0.1149 0.1423 0.1562 0.1580 0.1502 K=5 0.0323 0.0441 0.0647 0.1159 0.1383 0.1452 0.1659 0.1634 0.1741 0.1803 0.1717 K=7 0.0444 0.0444 0.1107 0.2385 0.2531 0.1959 0.2160 0.1977 0.1737 0.1883 0.2078

J=2 DENS = 0.1 K=1 0.0148 0.0444 0.0598 0.0535 0.0737 0.0785 0.0780 0.0881 0.0874 0.0756 0.0914

K=3 0.0517 0.0534 0.1413 0.1251 0.1664 0.1798 0.2133 0.1915 0.2085 0.2163 0.2021 K=5 0.0649 0.0702 0.1140 0.1896 0.1942 0.2199 0.2107 0.2421 0.2413 0.2201 0.2354 K=7 0.0585 0.1250 0.1487 0.2077 0.2290 0.2542 0.2495 0.2470 0.2821 0.2290 0.2510

DENS = 0.3 K=1 0.0518 0.0058 0.0111 0.0111 0.0229 0.0273 0.0471 0.0510 0.0760 0.0969 0.0921 K=3 0.0800 0.0550 0.0551 0.0383 0.0357 0.0595 0.1300 0.1652 0.1844 0.1872 0.2104 K=5 0.0696 0.0673 0.0581 0.0478 0.0465 0.1076 0.1684 0.2081 0.2286 0.2180 0.2282 K=7 0.0748 0.0643 0.0576 0.0541 0.1035 0.1362 0.1744 0.2297 0.2683 0.3121 0.2806

DENS = 0.7 K=1 0.0444 0.0287 0.0037 0.0045 0.0144 0.0195 0.0304 0.0556 0.0669 0.0810 0.0996 K=3 0.0763 0.0652 0.0522 0.0460 0.0307 0.0352 0.0602 0.0908 0.1426 0.1672 0.1564 K=5 0.0745 0.0661 0.0590 0.0494 0.0463 0.0466 0.1326 0.1923 0.2077 0.1765 0.1851 K=7 0.0787 0.0575 0.0514 0.0519 0.0477 0.0737 0.1261 0.2550 0.2629 0.2614 0.2194

J=4 DENS = 0.1 K=1 0.0690 0.0621 0.0736 0.0837 0.0874 0.1015 0.0928 0.0760 0.0973 0.1081 0.0946

K=3 0.1024 0.1059 0.1198 0.1177 0.1382 0.1553 0.1604 0.1809 0.1736 0.1777 0.1747 K=5 0.0841 0.1221 0.1524 0.1886 0.1627 0.2061 0.1841 0.2007 0.2197 0.1879 0.2162 K=7 0.0884 0.1367 0.1387 0.2275 0.1997 0.2136 0.2284 0.2412 0.2725 0.2486 0.2555

DENS = 0.3 K=1 0.1334 0.1273 0.1170 0.0864 0.0774 0.0296 0.0722 0.0866 0.1072 0.1508 0.1474 K=3 0.1311 0.1034 0.1076 0.0978 0.0859 0.0808 0.1451 0.1525 0.1613 0.2319 0.2379 K=5 0.1296 0.1270 0.1038 0.0993 0.1019 0.1085 0.1402 0.1499 0.1851 0.2390 0.2172 K=7 0.1109 0.0987 0.0997 0.0905 0.0786 0.1027 0.1682 0.2061 0.2227 0.2211 0.2478

DENS = 0.7 K=1 0.1559 0.1556 0.1155 0.1089 0.0624 0.0304 0.0280 0.0328 0.0478 0.1051 0.1103 K=3 0.1265 0.1011 0.1071 0.0927 0.0839 0.0707 0.0703 0.0983 0.1462 0.1794 0.2235 K=5 0.1505 0.1335 0.0913 0.1080 0.0843 0.0706 0.0701 0.1169 0.1840 0.2374 0.2562 K=7 0.1459 0.1180 0.1013 0.0863 0.0777 0.0633 0.0748 0.0967 0.2196 0.2457 0.2730

Table B3 Results of the regression analyses with group performance as dependent variable.

Model 1 Model 2

J=0.5 J=1 J=2 J=4 DENS=0.1 DENS=0.3 DENS=0.7

Constant 0.7509** 0.9972** 1.0091** 0.8800** 0.9849** 0.8914** 0.8397** 0.0172 0.0201 0.0167 0.0184 0.0140 0.0202 0.0230

β' 0.0429** 0.0189** 0.0057** 0.0077** 0.0122** 0.0218** 0.0224** 0.0014 0.0017 0.0012 0.0014 0.0012 0.0015 0.0017

K −0.0370** −0.0297** −0.0261** −0.0256** −0.0353** −0.0293** −0.0243** 0.0022 0.0026 0.0019 0.0021 0.0018 0.0023 0.0027

Z −0.2130** −0.4749** 0.2380* 0.6540** −0.3803** 0.2870** 0.4044** 0.0314 0.0368 0.1020 0.1126 0.0256 0.1231 0.1406

Zˆ2 −1.2122** −1.5604** −1.1319 −1.2637 0.1965 0.2169 0.2372 0.2709

Model fit F 411.7000** 142.1300** 105.0800** 60.8900** 237.2111** 115.7694** 77.8215** R-squared 0.8261 0.6212 0.6187 0.4847 0.6716 0.5716 0.4729 Adj R-squared 0.8241 0.6168 0.6129 0.4767 0.6688 0.5667 0.4668

(**p < 0.001; *p < 0.05.).

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