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From defection to ingroup favoritism to cooperation: simulation analysis of the social dilemma in dynamic networks

  • Hirofumi TakesueEmail author
Research Article
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Abstract

The emergence of cooperation in social dilemmas is a core question in the social sciences, the proposed solution being ingroup favoritism, a conditional strategy where individuals only cooperate with members of their own group. However, empirical literature has suggested that ingroup favoritism prevents one from realizing profitable interactions with outgroup members. Such an observation calls for a theoretical analysis that would help in understanding what factors affect the transition from ingroup favoritism to unconditional cooperation. Here, we conducted computational experiments in which agents located in social networks imitate traits of successful neighbors or sever social ties with defecting neighbors and connect them to other agents. The results of our simulation showed transitions from unconditional defection to ingroup favoritism to unconditional cooperation with a more frequent occurrence of link rewiring. This indicates the usefulness of the dynamic-networks framework in understanding the reason why cooperation is achieved in specific societies and why different types of cooperation are observed in different ones.

Keywords

Ingroup favoritism Prisoner’s dilemma Agent-based modeling 

Introduction

Social scientists have long pondered on the tension between individual selfishness and social efficiency. Mutual cooperation helps people achieve collective benefits and enhances social efficiency to an extent unachievable by solitary individuals. However, cooperation is vulnerable to free riders who benefit from cooperative behavior without bearing its costs. This temptation to free ride on others can damage general cooperation and reduce social welfare. Addressing this issue involves different classic social science statements, such as the tragedy of commons [29], the collective-action problem [49], and social dilemma [35]. Societies often do achieve cooperation despite the challenges faced, and predictions of simple models do not perfectly fit the current society. This predicament has motivated scholars working in various disciplines to work on how cooperation emerges and why it is so widely observed in society [6, 45, 66].

An answer to the question of emergence of cooperation is conditional strategy, which restricts cooperation among members of the same group: if their partners are ingroup members, they tend to cooperate; otherwise, they tend to defect. This tendency is called ingroup favoritism that is distinct from unconditional cooperation, which does not distinguish between ingroup and outgroup members. Empirical studies show the relation between ingroup favoritism and prosocial behavior, such as fair allocation of resources [70] and punishment of norm violators [8]. Previous research has also shown that ingroup favoritism helps achieve efficient coordination [15]. Theoretical models exhibit the understanding that ingroup favoritism can support the emergence of cooperation [42].

However, ingroup favoritism can also lead to socially undesirable outcomes. People may fail to construct cooperative relationships with outgroup members and tend to assess the character of ingroup members favorably [31], which can diminish fruitful interactions with outgroup members. Empirical studies have shown that heterogeneity, or the existence of multiple salient groups, causes socially undesirable outcomes, such as the underprovision of public goods in ethnically diverse societies [2, 3]. This may be because people expect cooperation only from ingroup members, which hinders the achievement of a favorable equilibrium [28]. Research has also shown that ingroup favoritism is deeply rooted in human psychology and triggers social conflict; thus, mechanisms sustaining ingroup favoritism have attracted significant scholarly interest. Moreover, the drawbacks to ingroup favoritism suggest the importance of clarifying the factors that help us understand the circumstances in which ingroup favoritism and unconditional cooperation become customary in societies.

In this study, we conduct computational simulation [40] to investigate the emergence of ingroup favoritism and unconditional cooperation, specifically focusing on the mobility of social relationships. In some societies, people can easily end their ongoing social relationships and find new social partners; in other societies, social relationships are more stable, and people tend to retain lasting ones. Empirical studies have shown that the ease of modifying current social ties and constructing new ones affects human psychology and behavior [81]. Because the success and failure in achieving cooperation can influence the productivity of (potential) social relationships, the opportunity of modifying current relationships can shape behavioral strategy in social dilemmas.

Considering the role of relational mobility, we introduce a coevolutionary framework where behavior in games influences social ties among individuals and vice versa. In this simulation, agents in social networks participate in social games and imitate the traits of their successful neighbors. Agents also sever social ties with their defecting neighbors and connect those links to other agents, introducing dynamics into the network structure. Simply put, we examine how social behavior emerges in dynamic networks. Previous empirical studies have shown that dynamic networks can enhance cooperation in social dilemmas (see [58] for a review).

One goal of this study is to examine whether the dynamic-networks framework allows the evolution of ingroup favoritism. Previous studies have indicated the existence of a cooperative regime under high relational mobility and a non-cooperative regime under low relational mobility, and it is not sure if there is room for ingroup favoritism in this framework. Furthermore, a previous study that adopted coevolutionary rules showed that network dynamics only supported the evolution of ingroup favoritism and that unconditional cooperation hardly prevailed [34]. Therefore, whether a different rule of network evolution supports unconditional cooperation is also an important question. This paper examines if the coevolutionary framework can explain different types of cooperation (ingroup favoritism and unconditional cooperation) and the transition between them.

Our simulation results show that the speed of network link rewiring determines the prevalent strategy in societies. Static networks (i.e., networks with no link rewiring) lead to a proliferation of defectors. However, with an increase in link-rewiring frequency, we observe a transition from defection to ingroup favoritism. Furthermore, even faster link-rewiring results in the transition from ingroup favoritism to unconditional cooperation. Meanwhile, our study suggests that the dynamic-networks framework can be applied not only to simple social dilemmas but also to situations where people can condition their behavior because of their group membership.

This paper is organized as follows. In the next section, we review the literature examining ingroup favoritism from a theoretical perspective and explain the motivation to introduce dynamic networks. Next, we propose a coevolutionary model in which agents’ traits and social ties in networks evolve through mutual influence. Then, we explain the numerical simulation results. Finally, we discuss the implications of the simulation results.

Ingroup favoritism from a theoretical perspective

Formal analysis of the emergence of ingroup favoritism began with a seminal paper by Riolo et al. [57]. Their model differentiated agents by tags; where two agents have the similar tag, the function is analogous to belonging to the same group. They particularly assumed that each individual in a population has a tolerance level. Here, an agent cooperates in prisoner’s dilemma games when the dissimilarity between his/her tag and that of his/her partner’s is smaller than his/her tolerance level. Agents who have higher tolerance levels are more likely to cooperate, and those who tolerate maximum dissimilarity are equivalent to unconditional cooperators. The study’s simulation showed a characteristic cyclic behavior in tolerance level, and some long-term cooperation was maintained. This model was proposed as an alternative explanation to the evolution of cooperation that did not rely on other cooperation-enhancing mechanisms, such as repeated interactions [6] or reputation formation [47]. This model was combined with other mechanisms, such as social networks [33, 76], assortative matching [26], and reputation formation [41, 43].

While Riolo et al.’s seminal model was proposed as another method of illustrating how cooperation evolved, it was quite different from unconditional cooperation, where ingroup and outgroup members are not distinct. Where ingroup members are favored, cooperation can be achieved within groups, and problems do not occur as long as social interactions are kept within narrow social groups. However, opportunity costs accompany ingroup favoritism [79]. Conditional behavior can prevent individuals from realizing potentially cooperative and profitable relationships with those in different groups. Ingroup favoritism can also influence economic activities associated with particular groups: people sometimes dismiss profitable opportunities if they would also benefit outgroup members [44]. Thus, the relation between ingroup favoritism and unconditional cooperation is an important research topic. Simply put, any framework that can deal with unconditional cooperation, unconditional defection, and ingroup favoritism would be desirable.

This paper introduces a framework for the coevolution of individual traits and network structure to examine how ingroup favoritism and unconditional cooperation developed in the context of social dilemma. The last few decades have seen a surge in interest in social networks from social science research [12, 77]. Using the social-network framework has provided new insights into understanding social behavior, including voting participation [9], social movement [11, 65, 67], opinion formation [7, 10, 18, 19, 20], and information diffusion [27, 37, 78]. Social networks can also shape individual behavior when facing social dilemmas. Instead of presuming that randomly selected individuals are interacting, the network model assumes that fixed neighbors interact in games and imitate strategy. Previous literature has found that introducing a network structure greatly enhances cooperation [46, 52, 59].

More importantly, the introduction of dynamic networks provides additional contributions to disseminating cooperation. Here, not only game strategy but also social ties in networks are modeled as evolving through mutual influence (coevolution of strategy and networks). Network dynamics allow individuals to sever links with their current neighbors and recreate those links with other ones. The coevolutionary model reflects the intuition that people have the desire to sever social relationships with uncooperative individuals and construct new ones with others. Here, the severance of social relationships can serve as a sanction against social deviation [17, 50, 69, 71]. Both theoretical [16, 21, 23, 51, 53, 54, 60, 68, 80, 82] and experimental [4, 13, 24, 56, 58, 64] studies have found that dynamic networks enhance cooperation much more than static networks. The coevolutionary model is also used to investigate the development of other types of social behavior, such as fairness [25, 73].

Dynamic social networks have a deep association with the notion of relational mobility in empirical literature [81]. Social relationships are dynamic, and people can easily construct new relationships in societies with high relational mobility, whereas relationships tend to be stable and fixed in those with low relational mobility. Relational mobility levels differ between [1, 38, 75] and within [63] societies. These varying levels can affect how people construct relationships with others. For example, studies have shown that high relational mobility is correlated with higher self-disclosure to friends [63] and lower privacy concerns [74]. More important for this study, recent large-scale cross-cultural studies have also shown the positive correlation between relational mobility and trust [75], the basic psychological trait related to cooperation [79]. These studies suggest that examining the relation between relational mobility and cooperation can be a fruitful research topic, and dynamic networks are a natural way to explore this phenomenon.

We introduce the dynamic-networks framework to examine how ingroup favoritism and unconditional cooperation emerge. In their study, Kim and Hanneman [34] also considered a coevolutionary model for ingroup favoritism, in which agents try to sever social relationships with their neighbors if they themselves do not choose cooperation with the neighbors, and a new link with another agent is created if both agents choose to cooperate.1 While it showed that link-rewiring speed did not significantly enhance unconditional cooperation, the study concluded that dynamic networks supported more tolerant (cooperative) individuals.

Our paper introduces a different network evolution rule while conceptually replicating Kim and Hanneman’s simulation [34], and it compares the effects of different rules. Contrasting with the model in their work, we assume that agents sever social ties with neighbors who defect, and these severed ties are then connected to a randomly selected agent. We examine if this different coevolutionary rule leads to the enhancement of unconditional cooperation in contrast to Kim and Hanneman’s findings [34]. In addition, the model has to allow ingroup favoritism to evolve in order to understand the transition between different types of cooperation. The two rules rely on different types of information and lead to qualitatively different evolutionary outcomes, as shown below.

Model

We consider a social network composed of N nodes, each being occupied by one agent. All agents have two traits, game strategy and tag, indicating their membership to a particular group. Agents connected by links are called neighbors, and they participate in games with their neighbors and imitate their traits. Because we are considering a coevolutionary model in which the agents’ traits and social ties are mutually influential, lists of neighbors can also be modified during simulation.

First, we explain the game that agents play with their direct neighbors. We represent the social dilemma with the widely adopted prisoner’s dilemma game, where agents can choose one of two actions: cooperation (C) and defection (D). If both agents choose cooperation they gain the payoff R, and they gain P if they choose defection. If one agent cooperates and the other defects, the cooperator acquires S, and the defector acquires T. Because of the order of the payoffs’ size (\(T> R> P > S\)), individuals can achieve larger payoffs by choosing D regardless of their partners’ decision. In particular, the simulation uses the following payoff matrix [80]:where b is the temptation to defect. This parameterization represents the harshness of the social dilemma with one parameter and simplifies the analysis.

In the game, agents have two traits that determine their behavior: tag and strategy. Tags represent the group agents belong to, and they can modify their actions depending on the tag of their opposite number. This model has four strategies [42]. In unconditional cooperation (UC), agents ignore their counterparts’ tags and always choose C. Meanwhile, in unconditional defection (UD), agents can always choose D regardless of the tags. Agents who adopt the next two strategies modify their behavior depending on their tag and that of their neighbors. In ingroup favoritism (IF), agents choose C when they and their neighbors share a tag but choose D otherwise. By contrast, in outgroup favoritism (OF), agents choose C only when they and their neighbors have different tags. Unlike the original model of Riolo and others [57], which assumed cooperation toward ingroup members, this formulation does not presume the tendency to cooperate with similar individuals. Incorporating the OF strategy addresses the criticism that assuming cooperation toward similar agents restricts the model’s relevance [22].

Agents’ traits and ties within networks are initialized at the outset of the simulations. Four strategies are randomly assigned to each agent. Additionally, G tags (groups) exist, with each agent belonging to one randomly selected group. In the initial network, each agent has an equal number of neighbors (k), and each link is connected to a randomly selected individual [61]. Beginning with this initial state, agents’ traits and network edges are updated. Trait evolution occurs in trait-updating events, whereas network evolution occurs in partner-switching events. During each period, one of these two events occurs.

Trait-updating events occur with the probability \(1-w\), and the agent can imitate the traits (strategy and tag) of one neighbor. The basic procedure of trait updating is basically the same as that in Kim and Hanneman’s model. Here, one link is randomly selected, one connected agent becomes a focal agent, and the other agent becomes a role agent [72]. The role of the two agents is randomly determined. Both focal and role agents participate in prisoner’s dilemma games with all their neighbors and accumulate the payoffs \(\varPi _f\) or \(\varPi _r\).

Next, these two agents’ payoffs are compared, and the focal agent may imitate the role agent’s traits. The probability of imitation depends on the difference in payoff, and the focal agent is more likely to imitate the role agent’s traits if the role agent has earned a larger payoff. Specifically, trait imitation occurs with the following probability [34]:
$$\begin{aligned} P(f \leftarrow r) = [1 + \exp (\beta (\varPi _f - \varPi _r))]^{-1}, \end{aligned}$$
(2)
where \(\beta\) represents the intensity of the payoff difference effect on imitation probability. For example, \(\beta \rightarrow \infty\) implies deterministic imitation, in which the focal agent always imitates the role agent’s traits if the role agent acquired a larger payoff. By contrast, \(\beta \rightarrow 0\) implies a random adoption of traits, and many types of agents will survive in the population. In this paper, we basically set the value of \(\beta\) to 0.01, assuming a relatively weak impact to the payoff and analyzing which strategy prevails while allowing many types of agents to coexist in the population.

Also, in trait-updating events, trait mutations can occur with low probability. When such mutations occur, the focal agent randomly adopts a strategy and/or a tag regardless of the outcome of the above-mentioned imitation process. Because of the random nature of these mutations, strategy and tag mutations occur independently, with the possibility that a mutation of only one trait occurs. A mutation prevents the results of a simulation from being shaped by the random extinction of a strategy or tag, although its effect is not the direct object of examination [30, 39]. In the simulation, the independent mutation of each trait takes place with the probability \(\mu\) after the imitation process.

Partner-switching events occur with the probability w. At the outset of such an event, one link is randomly selected, with one of the connected agents becoming a focal agent, who may cut his/her link with a connected neighbor if that neighbor chooses defection for him/her [23]. That link can be transferred to another agent randomly selected from the whole population. This rule simply states that agents tend to halt interactions with defecting neighbors and seek more profitable relationships. If a neighbor chooses to cooperate with the focal agent, nothing takes place during the partner-switching event. We imposed the restriction where agents with only one neighbor did not lose their link to that neighbor since individual isolation can artificially affect the simulation results [25]. The parameter w controls the speed of network evolution, and \(w = 0\) recovers the results with static networks.

For comparison, we also implemented a modified version of the link-rewiring rule of an important precursor study by Kim and Hanneman [34]. Compliance with this rule comprises two parts. First, a focal agent decides whether to sever his/her link with a neighbor; the agent then cuts the link if he/she chooses defection for that neighbor.2 Next, the severed link is transferred to one randomly selected agent if both agents choose to cooperate with each other.3 Network evolution takes place only when the focal agent severs a link and when both focal agent and potential neighbor mutually accept the new link. If these conditions are not met, then nothing occurs in this partner-switching event.

Network evolution can be better understood if the two partner-switching rules are compared in terms of their required information. We call the first rule “behavior-based partner-switching” because the agents sever their link with a defecting neighbor. Behavior-based partner-switching requires knowledge of one’s neighbor’s tag and strategy because neighbors determine their behavior on the given strategy and tag. Unlike this behavior-based rule, the partner-switching rule introduced by Kim and Hanneman only requires information on the neighbor’s tag to sever a current link. This is because the focal agent determines the continuation of the link based on his/her own behavior toward his/her neighbor. We call this compared rule “tag-based partner-switching”; it requires less information on the neighbor.

Behavior-based partner-switching, for its part, requires less information; it only requires local knowledge. With a behavior-based rule, agents only have to know the behavior of their neighbors. Meanwhile, tag-based partner-switching (potentially) requires knowledge of all agents’ tags, as they rely on the tag information of potential new neighbors to form new links. Thus, partner-switching rules requiring various information on other agents are compared, although we mainly intend to investigate the effects of novel mechanisms (behavior-based partner-switching).

Results

We conducted Monte Carlo experiments to examine the frequency of various strategies in a population of \(N = 10^3\) agents.4 Each simulation run was iterated \(N \times 10^4\) periods, and the strategy frequencies of the following \(4N\times 10^6\) periods were sampled for better statistical accuracy. For each parameter combination, we conducted four independent simulation runs and reported the average values for these.

For an overview of the simulation results, we report the resulting strategy frequency as a function of the temptation to defect (b). The left panel of Fig. 1 reports the results where no partner-switching (\(w = 0\)) took place. When b is small, agents adopting IF and UD coexist in a population. As expected, UD frequency increases monotonically with higher defection temptation. The frequencies of UC and OF remain small regardless of b value.
Fig. 1

The strategy frequency in stationary states as a function of the temptation to defect (b). a The results for static networks (\(w = 0\)), and the frequency of UD (IF) grows (shrinks) as the value of b increases. b The results for behavior-based partner-switching (\(w = 0.7\)), and UC (UD) flourishes with small (large) b. Ingroup favoritism has the largest frequency for an intermediate b value. c The results for tag-based partner-switching (\(w = 0.7\)), and network evolution somehow reduces cooperation. Fixed parameters: \(N = 1000, k = 8, G = 4, \beta = 0.01, \mu = 0.001\)

The center panel shows the behavior-based partner-switching results (\(w = 0.7\)). UC is clearly abundant, unlike the case of static networks, and is the most abundant strategy with small values of b, and its frequency decreases as defection temptation grows. By contrast, UD frequency increases with higher b value. These two strategies have frequencies indicating monotonic relationships with b, but the frequency of IF has a nonmonotonic pattern and increases its frequency by replacing UC, becoming the most abundant strategy for intermediate b values. However, additional stronger temptation to defect diminishes the IF strategy frequency and leads to the predominance of the UD strategy.

The right panel reports the results for tag-based partner-switching (\(w = 0.7\)). Link-rewiring opportunities are shown to have a small effect on strategy frequencies. When compared with the left panel, the right panel shows that tag-based partner-switching slightly reduces cooperation, especially for small b values.

This overview in Fig. 1 shows that introducing dynamic networks affects evolutionary outcomes. To explain the effects of partner-switching, Fig. 2 reports each strategy’s proportion as a function of behavior-based partner-switching frequency (w). The left panel shows that, for small partner-switching frequencies, the results are similar to those in static networks, and UD is the most abundantly encountered strategy in the population. By contrast, for large w values, agents who adopt UC proliferate in the population. Between these two contrasting results, the IF frequency achieves a maximum value with intermediate partner-switching speeds. This result shows that dynamic networks can explain the transition between different types of cooperation as well as those between cooperation and defection if we adopt this coevolutionary rule.
Fig. 2

The proportion of each strategy in stationary states as functions of behavior-based partner-switching frequency (w). a The results with \(b = 0.16\), and b UC and IF frequencies for different b values. Fast (slow) partner-switching tends to produce the UC (UD) strategy. Intermediate network evolution frequencies allow larger frequencies for agents who exhibit ingroup favoritism. The values of w required to support IF and UC depend on the value of b. Fixed parameters: \(N = 1000, k = 8, G = 4, \beta = 0.01, \mu = 0.001\)

The quantitative relationships between w and the strategy frequencies depend on values of b. The center and right panel in Fig. 2 report UC and IF frequencies with different sizes of defection temptation (b). Smaller b values imply unfavorable environments for UD, and IF achieves large frequencies even with low w values. However, the qualitative pattern remains similar: UC increases its frequency almost monotonically, whereas IF achieves maximum frequency with intermediate w values.

Many previous studies have reported on how network dynamics are more advantageous for unconditional cooperators than unconditional defectors [53]. This phenomenon has been understood to be the result of two main mechanisms: network reciprocity and degree heterogeneity (see [5, 14, 55] for the empirical relevance of these mechanisms). These mechanisms also enhance the qualitative understanding of the observed patterns in the simulation.

Network reciprocity is a major cooperation-supporting mechanism [45]. Defecting agents acquire higher payoffs on average when agents randomly interact with any other agent. Meanwhile, in a networked population, interaction partners are limited to neighbors, prompting cooperative agents to form a cooperator cluster, and enjoy the benefits of cooperation. Behavior-based partner-switching helps cooperators (including IF strategy practitioners) form cooperative clusters because the links between cooperative and defecting agents are severed.

The second mechanism here is degree heterogeneity (i.e., heterogeneity in the number of neighbors), part of the ubiquitous nature of social networks [32], which can play a central role in evolutionary processes. As indicated in a seminal study, hub nodes (i.e., larger-degree nodes) tend to stably choose cooperation [62], which prevails in the population. Network evolution clearly promotes degree heterogeneity, as defecting agents tend to lose links, whereas those among cooperative agents remain stable. This point is important in understanding the transition from the IF to UC strategies. Although both can form a cluster of mutual cooperation, their difference in cooperativeness gives UC a higher advantage. A more cooperative UC strategy realizes a larger degree relative to the IF strategy, which offsets the disadvantage of never exploiting other agents through free riding.

Comments may be needed here on the OF strategy, which never spreads widely. The IF and OF strategies differ starkly in terms of network reciprocity, although both intermediate strategies lie between UC and UD. When the traits of an agent with an IF strategy are copied, both the copying and copied agents become part of the same group and begin to cooperate. For the OF strategy, however, where two agents belong to the same group, mutual defection occurs. As a result, OF tends to be dismissed from ultimately successful strategies.

The behavior of the system can be clarified by observing the time evolution of strategy frequencies of a specific simulation run (Fig. 3). The small arrows in the figure indicate that the tag that achieves maximum frequency was replaced by a different tag. The left panel which shows the result with small w indicates that the increase in the frequency of IF strategy accompanies the emergence of the new largest group, which suggests that spatial reciprocity within a group contributes to the expansion of ingroup favoritism. In contrast, UD spreads with the same tag remain the largest group, which means that defectors who share the same tag invade the population (of IF-adopting agents). Although agents who adopt OF also invade the IF population, they fail to dominate the population and only observe the spread of agents who adopt IF and a different tag. This pattern changes drastically in the right panel, which shows the result with large w, and the competition between UC and IF was observed.
Fig. 3

Time evolution of strategy frequencies. Arrows indicate the alternation of the largest group. The pattern of competition varies drastically depending on the speed of link rewiring (w). In this figure, the value of \(\beta\) is set to 0.1 to make the frequency evolution less noisy. In addition, the value of G was set at 8 to facilitate the invasion of new groups. Other parameters: \(N = 1000, k = 8, b = 0.16, \mu = 0.001\)

Compared with the behavior-based rule, tag-based partner-switching supports neither cooperation nor ingroup favoritism. In Fig. 4, it appears that faster partner-switching leads to gradual decreases (increases) in the number of individuals with the IF (UD) strategy. The center and right panel show that this pattern can be observed with different b values. Tag-based partner-switching does not fully exploit the two cooperation-enhancing mechanisms we noted above. For example, links with defecting neighbors may not be severed, as agents determine the continuation of social ties on the basis of their own behavior. This fact prevents cooperative clusters from forming. Furthermore, defectors’ links are not severed as a result of their strategy, preventing the emergence of cooperative hubs.
Fig. 4

The proportion of each strategy in stationary states as a function of the frequency of tag-based partner-switching (w). The value of b is fixed at 0.04 in the left panel. The UD strategy frequency increases with faster partner-switching, whereas other strategies decrease in frequency in that case. Fixed parameters: \(N = 1000, k = 8, G = 4, \beta = 0.01, \mu = 0.001\)

Next, we report the results of different payoff impact intensities (\(\beta\)). Figure 5 reports the results with stronger impact. The left panel shows the results of behavior-based partner-switching (\(\beta = 0.6\)), which replicates the pattern observed in Fig. 2: transitions from the UD to IF to UC strategy regimes are observed with higher w. Where \(\beta\) is large, the strategy abundance transitions become sharper. In addition, unconditional cooperators increase its frequency with smaller w. Please note that this pattern remains the same with larger values of \(\beta\), such as 2 and 5.
Fig. 5

The proportion of the strategies in stationary states as a function of partner-switching frequency (w). The left panel shows the results for behavior-based partner-switching (\(\beta = 0.6\), \(b = 0.16\)), whereas the center and right panel shows the results for tag-based partner-switching (\(\beta = 0.6\), \(b = 0.04\) and \(\beta = 2\), \(b = 0.04\), respectively). The left panel replicates the basic pattern for Fig. 2. The center and right panel shows that tag-based partner-switching also helps cooperative strategies. Fixed parameters: \(N = 1000, k = 8, G = 4, \mu = 0.001\)

The center and right panels show that tab-based partner-switching also supports cooperation with \(b = 0.04\). A small partner-switching probability diminishes the frequency of the UD strategy, and a mixed population of the UC and IF strategies is observed to persist. Additionally, enormous w values induce further increases in the frequency of UC strategy individuals, and UC becomes the most abundant strategy when \(\beta = 2\). A comparison between Fig. 4 and this figure clearly indicates that the effects of tag-based partner-switching are moderated by \(\beta\) values. Kim and Hanneman [34] assumed large values for \(\beta\) and reported that link modification supported cooperative strategies. The result in Fig. 5 is consistent with this outcome. Additionally, the right panel shows the small effects of partner-switching frequency in a wide range of w once the network evolution is introduced, which is also consistent with the simulation results of Kim and Hanneman’s study [34].

Figure 6 confirms this pattern by presenting the results as a function of \(\beta\). Intermediate values of \(\beta\) often reduce the frequency of cooperative strategies by reducing the noise in the imitation process. However, strong payoff impact leads to the flourishing of cooperative strategies; large and small w support the proliferation of UC and IF strategies, respectively.
Fig. 6

The proportion of the strategies in stationary states as a function of payoff impact (\(\beta\)). Strong payoff impact supports cooperative strategies, and large (small) w supports unconditional cooperation (ingroup favoritism). Fixed parameters: \(N = 1000, k = 8, G = 4, \mu = 0.001\)

How are the changes in strategy frequency related to tag diversity in the population?5 Figure 7 shows that with behavior-based rule, tag diversity decreases as IF strategy increases its frequency (please refer to Figs. 2, 4, and 5 for corresponding strategy frequencies), and this tendency is prominent with large \(\beta\). The decline of the tag diversity implies that ingroup favoritism increases by exploiting spatial reciprocity within same-group members. This tendency is reversed with the prevalence of UC strategy, which does not rely on tags in maintaining cooperation. Because agents adopting IF strategy sever relationships with outgroup members, the fast link rewiring can also prevent the dissemination of a specific tag. Although tag diversity can assume lower values because of the prevalence of IF strategy and large \(\beta\) (compared to when \(\beta = 0.01\)) with the tag-based rule, its value also increases with large w.
Fig. 7

Tag diversity in stationary states as a frequency of partner switching (w). Increase in the frequency of IF (UC) is related with the loss (enhancement) of tag diversity. The value of b is 0.16 for behavior-based rule (bx) and 0.04 for tag-based rule (tag). Fixed parameters: \(N = 1000, k = 8, G = 4, \mu = 0.001\)

Throughout the remainder of the results presentation, we vary the other parameter values to determine robustness. First, Fig. 8 reports the frequency of IF and tag diversity as a function of the number of groups (G). When the environment is harsh for cooperative strategies (large b or small w for the behavior-based rule and small \(\beta\) for the tag-based rule), larger G values increase IF frequency (empty points in upper panels), and tag diversity increases in tandem (empty points in lower panels). However, this joint evolution of tag diversity and ingroup favoritism does not occur once the cooperative strategies prevail in the population, and we observe that larger G does not give IF advantage against UC (filled points in the upper panels) while the enhancement of tag diversity is observed in some cases (filled points in the lower panels).
Fig. 8

The IF strategy frequency and tag diversity in stationary states as a function of number of groups (G). The left panels show the results for behavior-based partner-switching (\(\beta = 0.01\)), and the right panels show the results for tag-based partner-switching (\(b = 0.04\)). The larger number of groups urges joint evolution of ingroup favoritism and tag diversity when the evolutionary process does not favor cooperative strategies. However, values of G hardly affect IF frequency once cooperative strategies prevail. Fixed parameters: \(N = 1000, k = 8, \mu = 0.001\)

Next, we modified the average degree values (k) and examined how neighborhood size affects evolutionary outcomes. As we reported in Fig. 9, larger k values induce defecting strategies. The left panel shows that UC (UD) strategy frequency monotonically decreases (increases) with larger k values. The frequency of the IF strategies first increases for the sacrifice of the UC strategy, but also decreases with larger k. A similar pattern is observed with tag-based partner switching: the center panel shows that the UD strategy dominates the population as neighborhood size increases. The right panel shows that this pattern can be replicated with different b values, and UD becomes more frequent as k takes larger values (we omitted the results of tag-based partner-switching because an increase in b leads to the predominance of UD even with smaller k values). These results are consistent with studies in which larger neighborhoods were found to be harmful to the evolution of cooperation in prisoner’s dilemma games with two strategies: UC and UD [48].
Fig. 9

The frequency of strategies in stationary states as a function of average degree (k). The left panel shows the results for behavior-based partner-switching (\(b = 0.16\), \(\beta = 0.01\)), whereas the center panel shows the results for tag-based partner-switching (\(b = 0.04\), \(\beta = 2\)). The right panel reports the frequency of UD strategy with the behavior-based rule and different b values (\(\beta = 0.01\)). Larger neighborhoods induce the UD strategy. Fixed parameters: \(N = 1000, G = 4, w = 0.7, \mu = 0.001\)

Discussion

We created and ran simulations of the social dilemma, emphasizing ingroup favoritism and unconditional cooperation. Besides the basic strategies of the prisoner’s dilemma (UC and UD), agents could modify their behavior in response to the groups they and their partners belonged to (IF and OF). This paper introduced a dynamic-network framework and allowed agents’ traits and social ties to coevolve. Following previous studies, agents in the network could choose whether to continue social relationships based on their neighbor’s behavior: they could sever their link with their neighbor if the said neighbor chooses defection (behavior-based partner-switching). For comparison, we also examined cases in which agents could modify links based on tags (tag-based partner-switching). We investigated how partner-switching frequencies affected agent strategies.

The simulation results showed transitions from unconditional defection to ingroup favoritism to unconditional cooperation as the frequency of behavior-based partner-switching increased. As was found in the literature, we observed that rapid behavior-based partner-switching supported cooperation, whereas static networks favored defection. In addition, nonmonotonic behavior was observed in the ingroup favoritism frequency: ingroup favoritism was at its maximum where there was a modest partner-switching speed. This pattern was confirmed with different intensities of the impact of the payoff on the imitation probability. By contrast, the effect of tag-based partner-switching was equivocal: rapid partner-switching reduced cooperation with a weak payoff impact, whereas patterns observed in the previous study were replicated with a strong payoff impact. The equivocal patterns observed with the tag-based rules may be because those rules do not completely exploit network reciprocity or degree heterogeneity, which have been shown to support cooperation in coevolving social-dilemma games.

Our results suggest that the quality of information on the behavior of other agents can effectively sustain cooperation. Although tag-based partner-switching used global knowledge of tags (i.e., information on all agents’ tags), it supported cooperation only for strong payoff impact and small defection temptation. Behavior-based link rewiring that required local information about the strategy and tag (of the neighbor) highly supported cooperation. Of course, the different patterns observed in the simulation do not negate the relevance of each partner’s switching mechanism. These two rules model different information conditions (which may be exogenously determined). The comparison of the actual behavior in experiments in different information environments can help illustrate the validity of the two models.

Our simulation results, especially those for behavior-based partner-switching, clearly indicated the prevalence of each strategy under different circumstances. Although proposed as an alternative route for the evolution of cooperation, ingroup favoritism represents a distinct strategy that provokes different behaviors and social outcomes. Our simulation distinguished different phases where different strategies thrived: in many cases, we observed transitions from defection to ingroup favoritism to cooperation. This result shows that the dynamic-networks framework can explain the transition between different types of cooperation. Other studies showed that ingroup favoritism evolved with an intermediately effective reputation mechanism (i.e., visibility of behavior) [43]. Ingroup favoritism has also been found to proliferate between full cooperation with perfect visibility and full defection with no visibility. The framework of our network also implied an intermediate nature of conditions favoring ingroup favoritism.

Because the simulation in this research has deep connections with the relational mobility framework in empirical literature, this study can be applied to empirical studies. Relational mobility measures the ease of modifying one’s social relationships, which is directly connected to link-rewiring speed in our model’s social networks: high (low) frequency of link rewiring corresponds to high (low) relational mobility. The empirical observation that the level of (generalized) trust differs across societies also suggests the applicability of our model [75, 79]. Our model may be useful in understanding why cooperation is achieved in specific societies and why different types of cooperation are observed in different ones.

Footnotes

  1. 1.

    Because cooperation with ingroup members was presumed in this study, this rule was similar to the assumption that agents cut relationships with outgroup members.

  2. 2.

    In [34], the dissimilarity between agents i and j (\(d_{ij}\)) is calculated by the two agents’ tags. Agent i compares this dissimilarity with his/her tolerance level (\(t_i\)) and chooses cooperation if the neighbor (j) is sufficiently similar (i.e., \(d_{ij} < t_{i}\)). The focal agent cuts the link to his/her neighbor if he/she chooses defection (i.e., \(d_{ij} \ge t_i\)).

  3. 3.

    In [34], a link forms between the focal agent (i) and a potential new neighbor (n) if \(d_{in} < t_{i}\) and \(d_{in} < t_{n}\). This notation mimics that of footnote 2.

  4. 4.

    The replication material of the simulation will be available on the author’s web page.

  5. 5.

    We computed the effective number of tags in the population as the index of tag diversity (see [36], for this index). Specifically, I computed \(1/\sum _{g} p_g^2\), where \(p_g\) is the proportion of agents who adopt tag g. For example, this index takes the value of n when the population is occupied by n equal-sized groups.

Notes

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© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Faculty of Political Science and EconomicsWaseda UniversityTokyoJapan

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