Boosting Cooperation Between Agents in Diverse Groups: A Dynamical Model of Prosocial Behavior, Free-Riding  and Coercive Solutions

 

Nazaria Solferino^1*
Serena  Fiona Taurino²
M. Elisabetta Tessitore³

1,2,3Ecoomics Department, University of Rome Tor Vergata" Via Columbia 2, 00133 Rome,  Italy.	Abstract Cooperation is usually stronger towards in-group members, because giving an up-right signal about themselves implies higher possibilities of reciprocity among members with the same social identity. We examine the case where collaboration between two groups is a mandatory condition to achieve success in a particular project, but in the first one, the social identity is quite strong. We show that the existence of a small share of prosocial players in the first group can create a sort of "imitation effect" so that each new member puts more effort in cooperating with the outsiders.  On the other side, to avoid free-riding effort should be conditional to the other's commitment. This way to boost cooperation is usually more efficient than a coercive strategy in the presence of significant sized majorities or feelings of resentments. Our analysis suggests that it is appropriate, under some circumstances, to stimulate a multicultural paradigm devoted to value and manage diversity through an acculturation process emphasizing adaptation, interdependence, and mutual appreciation of different cultures.
Licensed: This work is licensed under a Creative Commons Attribution 4.0 License.
Keywords: Cooperation Dynamical analysis Groups Identity. JEL Classification: C61; C71; D71.
(* Corresponding Author)

Funding: This study received no specific financial support.

Competing Interests: The authors declare that they have no competing interests.

1. Introduction

It is commonly believed that people act more prosocially towards members of their group than with those outside. There is a broad range of interdisciplinary literature (Everett, Faber, & Crockett, 2015) and bibliography therein cited), interested in explaining such in-group bias. Everett et al. (2015) explore in particular to what extent this behavior is driven by preferences for the welfare of the one own's group or from the beliefs of future reciprocal action by their own, as well as the outside group. In this context, a key concept is that of social identity (Tajfel, 1974) according to which  a social group is defined as the collection of individuals who perceive to be a member of the same category sharing common objective or subjective criteria (nationality, gender, jobs, etc.). Bidirectional actions define the social identity, i.e. cooperation, which implies the payment of a cost regarding the efforts devoted to realizing a common outcome which will benefit all members, namely themselves and the other group members (Dovidio, Piliavin, Schroeder, & Penner, 2006).

The whole outcome of this cooperation is strongly related to the agents' effort and therefore to the potential team members showing to work synergistically (Alchian & Demsetz, 1972). The individual benefits, instead, can be perceived as immediate or related to reputational concerns. In the literature on the reputation-based cooperation (see Everett et al. (2015)) as well as their bibliography), the most relevant input to push cooperative actions are the beliefs that the others will reciprocate now or later. In this context, it is important to give a signal to be perceived as a trustworthy person.

Milinski, Semmann, and Krambeck (2002) show that with the goal of building a good reputation people cooperate more when they have to contribute to the public good formation, and the Tragedy of Commons never applies. This cooperation is usually stronger towards in-group members, as giving a good signal about themselves implies higher possibilities to be reciprocated from those who share the same social identity (Everett et al., 2015).

Apparently this in-group bias can generate negative externalities for the outside group  as the preferential in group treatments are usually done at the expenses of the outsiders,  also concerning reduced cooperation. According to Sheremeta (2015) in order to succeed people tend to cooperate more with the others in the same group but, as the effort is not costless, they may tend to free ride with the members of the same group and become more competitive towards the outside groups. The author argues that the main factors affecting  these results are related to group size, sharing rule, interactions context and social impact function, heterogeneity of players. As pointed out by Everett et al. (2015) it is unclear whether the cause of this in group bias is to be found in preferences related the welfare of   in-group members or in beliefs about the behavior of in group and out group members.

In this work, by analyzing cooperation between and within groups, we focus on the issue of reputational concerns and indirect reciprocity (Alexander, 1987; Nowak & Sigmund, 1998, 2005). Naturally, indirect reciprocity happens when individuals decide to make cooperative and helpful actions towards others in a strategic way to build up a positive personal reputation because they will need help, at some point, from the others. As Nietzsche will put it, it is the selfishness of the generous.

Harris, Herrmann, Kontoleon, and Newton (2015) show that when in group bias is dangerous towards the outsiders, some mechanisms of punishment are necessary to set up and share broader ethical social norms across groups. Nevertheless other authors, by incorporating social preferences into the economic approach, show that some individuals do help others even when their help is not in their interest (Camerer & Fehr, 2004). In other words, other-regarding preferences do drive individuals (at least some of them) in their choices towards the well-being of others, as well as fairness and reciprocity (Camerer, 2003; Charness & Rabin, 2002; Fehr & Schmidt, 1999). In this work, we try to enlighten some additional insights about reputation-based in-group favoritism. We investigate through a dynamical model the effect of a share of in-group members with a pro-social attitude towards the outsiders. In particular, we analyze a situation where two groups in society must cooperate to realize a joint project, which needs the effort of both groups to be completed successfully in a given period. Some typical examples of this situation are the cooperation among sectors in the same organization, departments of a University, ethnic majorities and minorities (e.g. migrants) living in the same country, and so on.

As the returns can be different from the two groups, taking too much care just to the in-group reputation may be counterproductive, leading to low efforts to cooperate with the outside group. This "narrow rep thinking" will result in a less cohesive social aggregation, preventing the realization of a multicultural social paradigm. Nevertheless, according to the literature on other-regarding preferences, when there exists at least a minimal share of in-group members, reputational concerns can be driven in the direction of a more frequent cooperation with external groups.

The paper is divided into five Sections, including Introduction and Conclusions. In Section 2 we introduce the basic features of our dynamical model and discusses the primary variables affecting between and within group cooperation. In part 3 we explore how the existence of this minimal share can boost specific between-group efforts for the realization of a joint project. We find that two effects are at work: it can either boost between group cooperation and outsiders free-riding. This suggests a more cautious altruism requiring a threshold for the effort from the outsiders to activate this minimal share of altruistic people. In Section 4 we also compare this effect with the case of coercive cooperation by applying a penalty on the returns of the less cooperative agent in the first group. We find that this way to boost cooperation is usually inefficient in the presence of the significant sized majorities or feelings of resentments. Section 5 concludes.

2. The Model

We consider two groups of n1 and n2 individuals, who are engaged in a common project which needs the joint efforts of both groups to be completed in a given period of time T . Each group is characterized by his social identity, which is stronger in the first one so that we may assume that only the first group utility depends on reputational concerns.
We assume that two new individuals entering in the project must decide how much effort put in the in-group or between groups cooperation, as the full return from the project for each new agent can be measured as the sum of shares of the single additional returns r1 and r2, due to their marginal contribution, related to within groups cooperation. The action of making an effort has clearly a cost, which we assume to be constant over time, denoted by a1 and a2 for a representative agent in the first and in the second group, respectively. We also assume in time that u1 and u2 are the maximum exogenous level of effort for the agent in each group.

According to these assumptions we solve a dynamical game between the agents in the two groups where the first one solves:

In the above equation c and k are positive constants, while δis the depreciation rate. The last measures the individual's needs for distinctiveness and differentiation from others (For instance, according to Brewer (1991) individuals also avoid self-construals that are either too personalized or too inclusive and instead define themselves in terms of distinctive category memberships) and also the needs to be awarded for their own skills and original contribution w.r.t. the project, or to avoid alienation, etc. This equation is inspired by the club goods theory (see Shapiro (1982)) and Winfree and McCluskey (2005)) where reputation is built over time, people adjust reputation partially in response to observed variables of choice. Therefore it is time-sensitive, implying an adaptive process.

The term ku2 measures the disruption effect on the Social identity of the first group.

In other words, we assume that a social identity is disrupted when the other group is highly cooperative, so that integration and reciprocity towards it is higher and there is the threat   of rejection for those in the majority who on turn do not reciprocate with the outsider.  This   is a sort of bridging (or inclusive) social capital (Putnam, 2000) that,  unlike the bonding  social capital which may be more inward looking and have a tendency to reinforce exclusive identities and homogeneous groups, instead it may be more outward-looking and encompass people across different social divides.

The second player solves

where again the state R is ruled by (2).

We remark that for the second agent we subtract n1R since it represents an opportunity cost that is not necessary for the project itself.

We consider the Hamiltonian function, given u2 associated to (1) (2)

Therefore, the optimal level of the effort for the first group to cooperate with their own members mainly depends:

• Positively on the own group size (and negatively on the outside group size). In the literature, the exact effect of group size is controversial enough. According to some theoretical predictions, large numbers facilitate free-riding behaviors among their members. In this case, the enhancement of cooperation with outsiders and the reduction of reputation concerns could be more probable. On the contrary, other experimental results show that for large sized majorities the social identity is stronger and cooperation within group higher, while it is nearly impossible to be well integrated for minorities (see Sheremeta (2015)) for a survey on both theoretical and empirical analyses). Our theoretical result shows a positive effect of the group size which strongly depends on the relevance of reputation concerns if returns from the first group are consistent.
• Positively on the effort of the other group proportionally to their full returns. This means that the more is Group 2 cooperative with Group 1, the higher is the incentive of the latter to reciprocate and produce within this group while the cooperation with the outside group lessens. We can hence see how the outsiders are considered as a support to the improvement of the "small closed society" rather than a possible partner to cooperate.
• As the maximum possible effort of the second group increases, the optimal u1 decreases, because when it is high enough the first group can free ride when cooperating with the second one, also when returns r2 are large enough.
• Positively on the reputational concerns c through the shadow price λ.

The co-state λsolves:

• Notice that for the representative agent in the second group, for high values of n2r2, the highest costs of effort may reduce the amount of it that the player wants to provide. In general, these costs can be very high for minorities due to extrinsic factors (for in- stance diverse infrastructure for people in different countries, difference in language for minorities ethnical groups, etc.). Removing these unfair costs, through a multicultural paradigm promoting more integration, could be convenient for the organization when better minority performance can recover them and realize a more common profitable project.
• Moreover, for higher values of r2 (e the group size n2) the outsider may free ride w.r.t his own group and contribute more with the other one.
• On the other side, the disruption effect k is negative because it needs less effort to have great influence on the behaviour of the first group and weaken his social identity.
• Finally, as expected, the positive effect of c on the optimal level of the efforts, concerns both the  first and the second group and this last through the shadow price µ.  For  high values of r1 and n1, people in the minority group must contribute with higher  efforts with the  first group to be integrated but u2 increases also because that group   is working hard only with their peers and cooperation with them become the main way to realize the project, instead of working with their own group.

3. Cooperation among Groups May be Boosted by Small Shares of Individuals With Pro-Social Attitudes towards the Outsiders.

As the possible maximum effort of the second group increases, the optimal u1 decreases, because when it is high enough the first group can free ride when cooperating with the second one, even if returns r2 are large enough.

Cooperation among groups may be boosted by small shares of individuals with pro-social attitudes towards the outsiders.

Our results found in the previous Section show that also when the second group is highly productive, and it could be efficient to cooperate more with it, this cooperation may be sharply limited because of the first group reputational concerns c. If we imagine a more cohesive society, with a hypothetical social planner who sets up the levels of the efforts that both the agents should provide to realize a shared pro table project, it will solve a simple intertemporal maximization problem without any small hint of motion on R. In this way, the optimal values of u1 and u2 would depend only on the returns, costs and size of both groups(more details on these results are available upon request).

According to some authors (see for instance Sheremeta (2015)) and bibliography therein cited) when this is the case, a mechanism of punishment could enforce cooperation between groups. In this paper we want to explore another less coercive solution to build a more cohesive society where the only relevant concerns are about the project realization by each member. In particular, we aim to investigate a case where cooperation among groups can be pushed up by small shares of individuals with pro-social attitudes towards the outsiders.

Specifically, in this Section, we analyze how the dissemination of good practices between groups by a small proportion of individuals with other-regarding preferences can boost cooperation through a kind of "imitation effect". Such a result may arise since it pushes collaboration with different types of stakeholders (Becchetti & Gianfreda, 2007). This promotion towards hybrid collaboration can happen at the corporate level, i.e. the so-called "social market enterprises" like fair traders (Becchetti & Gianfreda, 2007); but also at a social level when migration policies or social incentives are put in place to smooth social conflicts and push individuals towards integration. Similarly, in our case we can see how this share of individuals with other- regarding preferences can limit the reputational concerns, hence encouraging collaboration with outsiders.

Under these assumptions, in this revised version of the model the two players solve the same maximization problems as in the previous Section but where R is now ruled by:

where s is a positive constant measuring the share or the sensitivity of the first group members to the cooperation with the outsiders(between group reciprocity).

Proceeding as in the previous Section, we find:

where the costate variable λ is the given by (8), the constants C1, C2 are defined in (5) and recalling C3 After some manipulations we get

1. The share s  of  altruistic  agents  in  the  group  boosts cooperation  towards  the  outsiders: as expected, it reduces the optimal value of u1 through a decrease in the reputational concerns by an amount c−s, and consequently it increases the optimal effort to contribute with the outside group.
2.  The share s of altruistic agents may also boost outsiders' free-riding. The effect of a reduced u1 is controversial as it depends also on the reaction of the other agent, that is on the new optimal level of the endogenous variable u2. This depends on the existence of free-riding towards the first group. In other words, the effect of the share s on the optimal value of u2 is not unambiguous: this value decreases as (c−s) raises. Therefore the ousider representative agent contributes less in the first group, as the effect of reputational concerns and social identity is weaker and he can free-ride from the others' augmented cooperation-integration (in this context by free-riding we mean that the outsider becomes more competitive in the first group and more cooperative with his own group). This effect is lower for low values of the disruption variable k.

The issue of the free riding problem inside a group has been broadly analyzed in Olson (1965) who claims that free riding in a group is more likely to happen if a group endeavors collective action to provide public goods. On the contrary, individuals will not freeride against the e offs of f the others, if the group will procure benefits to active members. Indeed, when freeriding happens, collective action is impossible, even if there are common interests and dependance among group members. Olson identified incentives like selective rewards and coercive participation as the only ways to solve out the free-riding problem.

In this Section instead, we show that there exists the possibility to have a freeriding issue between groups. In fact, we see that the share of altruistic people in the first group may boost free-riding behavior from the outsiders towards the first group itself. Our results show that the full effect depends on two combined forces acting in the same direction, i.e. increased cooperation from the first group versus the outsiders can count either on the "imitation effect" of the generous people or the increased free-riding from the second one. The first effect prevails for very low values of k, holding other things constant.

Otherwise, it seems suitable, in such context, to follow Olson (1965) to be altruistic only with the outsider who will in turn procure benefits to the active members.  In our model this implies to add an activation constraint such that the law of motion of R becomes

The first group sets the value of u ~2  considering the possibility and necessary contribution to request from the outsiders to be integrated in the first group and to realize the return r1 from the common project (for instance it could be ).

4 Coercive Vs. Altruistic Cooperation towards the Outsiders

In this Section, we consider that different solutions have been proposed to solve out the problem of free riding. In Olson (1965) even coercive participation may be a way to remediate to the free-riding problem. Therefore, we wonder if a similar solution can be applied to enhance cooperation from the first group towards the second one.

The usual two different approaches are the coercive and cooperative enforcement strategies. In general, human societies are constituted by a set of collaborative and coercive orders that coexist, sometimes in a harmonic way, other times they conflict. For example, social and institutional innovations that allow individuals to improve their living conditions often develop without the need of coercive power to impose the respect of particular rules. Individuals are aware of the benefits and spontaneously accept them. On the contrary, sometimes a cooperative society cannot exist without a coercive power able to impose some new rules. Furthermore sometimes the intervention of coercive power is necessary to re- move barriers to new joint and social stability cooperation (Montani, 2008). Functionally, punishment, also referred to as negative reciprocity, coercion, etc., is likely to be essential for maintaining cooperation. Third-party punishment, where the costs are borne by the individual but the benefits accrue to the group, has received recent theoretical interest and has been suggested to be essential to human cooperation (e.g. (Bowles & Gintis, 2003; Boyd, Gintis, Bowles, & Richerson, 2003; Fehr & Fischbacher, 2003; Fehr & Gachter, 2000) ). This effective punishment can be a fundamental pivot of non-kin cooperation when considering an occurrance  that belongs only to human beings. On one hand, positive sentiments - empathy, generosity and so on can lead individuals to prosocial actions towards strangers; on the other, negative feelings, like spite and sensitivity to unfairness, can play a pivotal role too. In fact, they can push individuals to punishment against free-riders (e.g. (Boyd et al., 2003; Fehr & Fischbacher, 2003; Hill, Barton, & Hurtado, 2009; Tomasello, Carpenter, Call, Behne, & Moll, 2005) ).

Nevertheless, there is also evidence in the literature showing that commitment can make sanctions counter-productive, spreading bitter feelings among regulated facilities with the consequent unwillingness to cooperate with regulators in the future (Burby & Paterson, 1993). In many contexts, a coercive response to these events may breed strong resentments. Burby and Paterson (1993) study the best strategy to increase compliance either with the performance standard or with specification standard.  According to their findings, a cooperative approach produces a higher impact regarding the compliance with the performance standard. In several cases it is possible to create a collaborative order without coercion but through sequential iterative processes (Axelrod & Hamilton, 1981) which lead to the manifestation of planned and desired cooperative behavior. Also, we can hypothesize situations, described by the prisoner's dilemma, some cases where individuals can change from noncooperation to cooperation by looking at the advantages from the reciprocal cooperation and reach an agreement. Batson (1991) finds prosocial feelings, e.g. empathy, a fundamental driver of prosocial actions towards newcomers.

To take into account these considerations, we propose a slightly revised version of the model, where a penalty φu1 is applied on the total return going to the subject of group one, proportional to the effort he puts into his group, and that according to a hypothetical Social Planner it would have to go instead to another group. Hence now the player one solves the same equation as in (1) but where now we also subtract the coercive term φu1, where φis a positive constant and the state R is still ruled by (2).

By solving this slightely different maximization problema and assuming that u2 is given and fixed  at the necessary level to no make free riding, we get:

The above inequality, for huge sized majority groups, i.e. for n1 big enough, always holds true. Hence, in this case, it is satisfied also for very small values of s.

Finally, notice that to take into account also the possibility of resentment, making coercion counterproductive, i.e. the penalty generates aversion towards the outsiders who are indirectly responsible for that, in our model this would imply that k decreases proportionally. That is the case where a coercive approach may even be counterproductive by engendering intransigence the low of motion of R becomes

which, by raising the optimal value of u∗2, will clearly reduce the optimal amount of u1 the first player would provide to the outsiders.

5. Conclusions

In an increasingly globalized and multicultural society, the promotion of diversity seems a necessary prerequisite when the goal is to create a joint project that requires the effort of  different groups.

Using a dynamical analysis, we show that several factors should be taken into consideration to boost cooperation between groups when reputational concerns and social identity play an important role for the majority group. In particular, when minorities have low performance and desire to integrate and high propensity to free riding, then it could be more  efficient to choose a low-inclusive model which could also be more effective if the fulfillment  of the group is very high. Nevertheless, it is more common to encounter the case where cohesion among group members is pivotal for high performance at work while drifting to a group think mechanism (Whyte, 1952) can be dangerous. In particular, when the minority has better performance, it would be better to enhance outside group cooperation. Some organizations operate more efficiently with a homogeneous workforce while others are more efficient with a heterogeneous workforce.

Our results suggest that when this is the case strategies for managing multicultural organizations, and a new social multicultural paradigm are necessary. In particular, we show that the existence of a tiny share of prosocial players in the rst group can create a sort of "imitation effect" such that each new individual in the group gives less emphasis to personal reputation inside the group and puts more effort in cooperating with the outsiders. On the other side, to avoid free-riding effrt should be conditional to the other's commitment.

Also show that this way to boost cooperation is more efficient than a coercive strategy in the presence of the significant sized majorities or feelings of resentments. Therefore, according to our analysis, it could be appropriate, under some circumstances to stimulate a multicultural paradigm so as to increase the share of prosocial people in the majority group.

There is substantial literature debating diverse groups and organizations performance advantages and disadvantages (see White (1999)). For instance, this usually happens when multicultural organizations: i)attract the highest quality of human resources, uphold and further the highest percentage of workers from diverse cultural backgrounds; ii) can under- stand and penetrate wider and enhanced (foreign) markets; iii) display greater creativity and innovation and a better problem-solving ability; iv) are better able to adapt to change and exhibit more organizational flexibility). Apparently, despite the compelling advantages possessed by the multicultural organizations, when diversity is not managed in an efficient manner, it can result in dysfunctional outcomes.

nadequate communication, weak team spirit, long decision timing and less team cohesion are more likely to arise in such cases. This mechanism has the higher probability to happen when the social identity of the first group is unyielding. Therefore cross-cultural training is necessary to enable culturally diverse groups to live up to their potential and overcome communication difficulties. Therefore, managers in an organization or social planners in society should reconcile ambitious objectives, sustain the identity of minority group members, scatter power in a representative way. Additionally, they should take actions when having abundant resources, and cultural diversities are understood. Furthermore, in a society where diverse groups coincide there is the need for strategies and programs to value and manage diversity through an acculturation process that emphasizes two-way learning, adaptation, interdependence, and mutual appreciation of different cultures. Also, an efficient communication needs to exchange new ideas, grievances, input and feedback. Before new paradigms can be created, however, the cultural environment has to experience greater acceptance, especially for evaluating in a positive way the inevitable change, as well as the challenge of an increasingly diverse society.

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