Participation costs dismiss the advantage of heterogeneous networks in evolution of cooperation

Evolutionary multigame with conformists and profiteers based on dynamic complex networks

Evolutionary game on complex networks provides a new research framework for analyzing and predicting group decision-making behavior in an interactive environment, in which most researchers assumed players as profiteers. However, current studies have shown that players are sometimes conformists rather than profit-seeking in society, but most research has been discussed on a simple game without considering the impact of multiple games. In this paper, we study the influence of conformists and profiteers on the evolution of cooperation in multiple games and illustrate two different strategy-updating rules based on these conformists and profiteers. Different from previous studies, we introduce a similarity between players into strategy-updating rules and explore the evolutionary game process, including the strategy updating, the transformation of players' type, and the dynamic evolution of the network structure. In the simulation, we implement our model on scale-free and regular networks and provide some explanations from the perspective of strategy transition, type transition, and network topology properties to prove the validity of our model.

Evolution of cooperation in costly institutions exhibits Red Queen and Black Queen dynamics in heterogeneous public goods

Public goods are often subject to heterogeneous costs, such as the necessary costs to maintain the public goods infrastructure. However, the extent to which heterogeneity in participation cost can affect groups' ability to provide public goods is unclear. Here, by introducing a mathematical model, I show that when individuals face a costly institution and a free institution to perform a collective action task, the existence of a participation cost promotes cooperation in the costly institution. Despite paying for a participation cost, costly cooperators, who join the costly institution and cooperate, can outperform defectors who predominantly join a free institution. This promotes cooperation in the costly institution and can facilitate the evolution of cooperation in the free institution. For small profitability of the collective action, cooperation in a costly institution but not the free institution evolves. However, individuals are doomed to a winnerless red queen dynamics in which cooperators are unable to suppress defection. For large profitabilities, cooperation in both the costly and the free institution evolves. In this regime, cooperators with different game preferences complement each other to efficiently suppress defection in a black queen dynamic.

Networks of reliable reputations and cooperation: a review

Reputation has been shown to provide an informal solution to the problem of cooperation in human societies. After reviewing models that connect reputations and cooperation, we address how reputation results from information exchange embedded in a social network that changes endogenously itself. Theoretical studies highlight that network topologies have different effects on the extent of cooperation, since they can foster or hinder the flow of reputational information. Subsequently, we review models and empirical studies that intend to grasp the coevolution of reputations, cooperation and social networks. We identify open questions in the literature concerning how networks affect the accuracy of reputations, the honesty of shared information and the spread of reputational information. Certain network topologies may facilitate biased beliefs and intergroup competition or in-group identity formation that could lead to high cooperation within but conflicts between different subgroups of a network. Our review covers theoretical, experimental and field studies across various disciplines that target these questions and could explain how the dynamics of interactions and reputations help or prevent the establishment and sustainability of cooperation in small- and large-scale societies.

This article is part of the theme issue ‘The language of cooperation: reputation and honest signalling’.

Bilateral costly expulsions resolve the public goods dilemma

Expulsion has been found to promote cooperation in social dilemmas, but only if it does not incur costs or is applied unilaterally. Here, we show that removing both conditions leads to a spontaneous resolution of the costly expulsion problem. Namely, by studying the public goods game where cooperators and defectors can expel others at a personal cost, we find that public cooperation thrives as expulsion costs increase. This is counterintuitive, as the cost of other-regarding behaviour typically places an additional burden on cooperation, which is in itself costly. Such scenarios are referred to as second-order free-rider problems, and they typically require an additional mechanism, such as network reciprocity, to be resolved. We perform a mean field analysis of the public goods game with bilateral costly expulsion, showing analytically that the expected payoff difference between cooperators and defectors increases with expulsion costs as long as players with the same strategy have, on average, a higher frequency to interact with each other. As the latter condition is often satisfied in social networks, our results thus reveal a fascinating new path to public cooperation, and they show that the costs of well-intended actions need not be low for them to be effective.

Peer pressure induced punishment resolves social dilemma on interdependent networks

Despite the fruitful evidence to support the emergence of cooperation, irrational decisions are still an essential part of promoting cooperation. Among the many factors that affect human rational decision-making, peer pressure is unique to social organisms and directly affects individual cooperative behaviors in the process of social interaction. This kind of pressure psychologically forces individuals to behave consistently with their partners, and partners with inconsistent behaviors may suffer psychological blows. As feedback, this psychological harm may in turn affect individual cooperative decisions. There is evidence that when peer pressure exists, partnerships can reduce free-riding in enterprise. Based on interdependent networks, this paper studies the impact of peer pressure on cooperation dynamics when the strategies of corresponding partners from different layers of the networks are inconsistent. We assume that when individuals are under peer pressure, their payoffs will be compromised. The simulation results show that the punishment effect will force the expulsion of partners with different strategies, which will further reduce the proportion of partners with inconsistent strategies in the system. However, in most cases, only moderate fines are most conductive to the evolution of cooperation, and the punishment mechanisms can effectively promote the interdependent network reciprocity. The results on the small world and random network prove the robustness of the result. In addition, under this mechanism, the greater the payoff dependence between partners, the better the effect of interdependent network reciprocity.

Quantifying stakeholder learning in climate change adaptation across multiple relational and participatory networks

Responding to accelerating climate change impacts requires broad and effective engagement with stakeholders, at multiple geographic and governance levels. Stakeholder participation has been hailed as a facilitated approach in climate change adaptation that supports social learning, depolarization of perceptions, and fosters collective action. But stakeholder participation remains loosely interpreted and evaluating measures are limited. This study employs social network analysis (SNA) to investigate how social relations among stakeholders, which emerge as a result of participation, are associated with stakeholder learning, as changes in perceptions of climate change. We hypothesized that reciprocal ties of understanding, respect, and influence can predict changes in perceptions of climate change. This approach was applied to a case study in Deal Island Peninsula, Maryland (USA) where local residents, scientists, and government officials met from 2016 to 2018 to collaboratively manage the impacts of sea-level rise in their communities. We found that social relations based on mutual understanding, respect,and influence are positively associated with perceptions of climate change. We provide a detailed conceptualization and implementation of a network-based approach that may serve as a potential quantitative performance measure of stakeholder participation processes in climate change adaptation. Overall, this study provides empirical evidence of the role that emerging social relations have on enhancing or constraining social learning among stakeholders in the Deal Island Peninsula project.

Evolutionary dynamics of cooperation with the celebrity effect in complex networks

How long-term cooperation is maintained in a society is an important and interesting question. The evolutionary game theory is often used as the basic framework to study this topic. The social status of game participants has an important influence on individual decision-making. Enlightened by this thought, we present a classification imitation model where the mechanisms of the celebrity effect and incomplete egoism are presented. The celebrity effect is reflected in each strategy update process to probe how individual decision-making is dynamically adjusted by comparing the social status of both parties in the game. The incomplete egoism refers to the irrational imitation of celebrities while self-interest is ignored. With this model, the group cooperation decision-making mechanism led by celebrities is revealed. Large-scale Monte Carlo simulations show that the incomplete egoism of individuals cannot stimulate cooperation but guarantee the stable existence of cooperation. Furthermore, the scale-free and community structure of the network enables cooperation to spread widely and maintains long-term survival. Our conclusion might provide practically new insight into the understanding and controlling of cooperation in the complex social systems.

Evolutionary Constraints on Connectivity Patterns in the Mammalian Suprachiasmatic Nucleus

The mammalian suprachiasmatic nucleus (SCN) comprises about 20,000 interconnected oscillatory neurons that create and maintain a robust circadian signal which matches to external light cues. Here, we use an evolutionary game theoretic framework to explore how evolutionary constraints can influence the synchronization of the system under various assumptions on the connection topology, contributing to the understanding of the structure of interneuron connectivity. Our basic model represents the SCN as a network of agents each with two properties-a phase and a flag that determines if it communicates with its neighbors or not. Communication comes at a cost to the agent, but synchronization of phases with its neighbors bears a benefit. Earlier work shows that when we have "all-to-all" connectivity, where every agent potentially communicates with every other agent, there is often a simple trade-off that leads to complete communication and synchronization of the system: the benefit must be greater than twice the cost. This trade-off for all-to-all connectivity gives us a baseline to compare to when looking at other topologies. Using simulations, we compare three plausible topologies to the all-to-all case, finding that convergence to synchronous dynamics occurs in all considered topologies under similar benefit and cost trade-offs. Consequently, sparser, less biologically costly topologies are reasonable evolutionary outcomes for organisms that develop a synchronizable oscillatory network. Our simulations also shed light on constraints imposed by the time scale on which we observe the SCN to arise in mammals. We find two conditions that allow for a synchronizable system to arise in relatively few generations. First, the benefits of connectivity must outweigh the cost of facilitating the connectivity in the network. Second, the game at the core of the model needs to be more cooperative than antagonistic games such as the Prisoner's Dilemma. These results again imply that evolutionary pressure may have driven the system towards sparser topologies, as they are less costly to create and maintain. Last, our simulations indicate that models based on the mutualism game fare the best in uptake of communication and synchronization compared to more antagonistic games such as the Prisoner's Dilemma.

Mixing protocols in the public goods game

If interaction partners in social dilemma games are not selected randomly from the population but are instead determined by a network of contacts, it has far reaching consequences for the evolutionary dynamics. Selecting partners randomly leads to a well-mixed population, where pattern formation is essentially impossible. This rules out important mechanisms that can facilitate cooperation, most notably network reciprocity. In contrast, if interactions are determined by a lattice or a network, then the population is said to be structured, where cooperators can form compact clusters that protect them from invading defectors. Between these two extremes, however, there is ample middle ground that can be brought about by the consideration of temporal networks, mobility, or other coevolutionary processes. The question that we here seek to answer is, when does mixing on a lattice actually lead to well-mixed conditions? To that effect, we use the public goods game on a square lattice, and we consider nearest-neighbor and random mixing with different frequencies, as well as a mix of both mixing protocols. Not surprisingly, we find that nearest-neighbor mixing requires a higher frequency than random mixing to arrive at the well-mixed limit. The differences between the two mixing protocols are most expressed at intermediate mixing frequencies, whilst at very low and very high mixing frequencies the two almost converge. We also find a near universal exponential growth of the average size of cooperator clusters as their fraction increases from zero to one, regardless of whether this increase is due to increasing the multiplication factor of the public goods, decreasing the frequency of mixing, or gradually shifting the mixing from random to nearest neighbors.

Parent-preferred dispersal promotes cooperation in structured populations

Dispersal is a key process for the emergence of social and biological behaviours. Yet, little attention has been paid to dispersal's effects on the evolution of cooperative behaviour in structured populations. To address this issue, we propose two new dispersal modes, parent-preferred and offspring-preferred dispersal, incorporate them into the birth-death update rule, and consider the resultant strategy evolution in the prisoner's dilemma on random-regular, small-world, and scale-free networks, respectively. We find that parent-preferred dispersal favours the evolution of cooperation in these different types of population structures, while offspring-preferred dispersal inhibits the evolution of cooperation in homogeneous populations. On scale-free networks when the strength of parent-preferred dispersal is weak, cooperation can be enhanced at intermediate strengths of offspring-preferred dispersal, and cooperators can coexist with defectors at high strengths of offspring-preferred dispersal. Moreover, our theoretical analysis based on the pair-approximation method corroborates the evolutionary outcomes on random-regular networks. We also incorporate the two new dispersal modes into three other update rules (death-birth, imitation, and pairwise comparison updating), and find that similar results about the effects of parent-preferred and offspring-preferred dispersal can again be observed in the aforementioned different types of population structures. Our work, thus, unveils robust effects of preferential dispersal modes on the evolution of cooperation in different interactive environments.

Identification of influential invaders in evolutionary populations

The identification of the most influential nodes has been a vibrant subject of research across the whole of network science. Here we map this problem to structured evolutionary populations, where strategies and the interaction network are both subject to change over time based on social inheritance. We study cooperative communities, which cheaters can invade because they avoid the cost of contributions that are associated with cooperation. The question that we seek to answer is at which nodes cheaters invade most successfully. We propose the weighted degree decomposition to identify and rank the most influential invaders. More specifically, we distinguish two kinds of ranking based on the weighted degree decomposition. We show that a ranking strategy based on negative-weighted degree allows to successfully identify the most influential invaders in the case of weak selection, while a ranking strategy based on positive-weighted degree performs better when the selection is strong. Our research thus reveals how to identify the most influential invaders based on statistical measures in dynamically evolving cooperative communities.

Invasion-controlled pattern formation in a generalized multispecies predator-prey system

Rock-scissors-paper game, as the simplest model of intransitive relation between competing agents, is a frequently quoted model to explain the stable diversity of competitors in the race of surviving. When increasing the number of competitors we may face a novel situation because beside the mentioned unidirectional predator-prey-like dominance a balanced or peer relation can emerge between some competitors. By utilizing this possibility in the present work we generalize a four-state predator-prey-type model where we establish two groups of species labeled by even and odd numbers. In particular, we introduce different invasion probabilities between and within these groups, which results in a tunable intensity of bidirectional invasion among peer species. Our study reveals an exceptional richness of pattern formations where five quantitatively different phases are observed by varying solely the strength of the mentioned inner invasion. The related transition points can be identified with the help of appropriate order parameters based on the spatial autocorrelation decay, on the fraction of empty sites, and on the variance of the species density. Furthermore, the application of diverse, alliance-specific inner invasion rates for different groups may result in the extinction of the pair of species where this inner invasion is moderate. These observations highlight that beyond the well-known and intensively studied cyclic dominance there is an additional source of complexity of pattern formation that has not been explored earlier.

Phenotype affinity mediated interactions can facilitate the evolution of cooperation

We study the coevolutionary dynamics of the diversity of phenotype and the evolution of cooperation in the Prisoner's Dilemma. Rather than pre-assigning zero-or-one interaction rate, we diversify the rate of interaction by associating it with phenotypes. Individuals each carry a set of potentially expressible traits and expresses a number of such traits at a cost proportional to the number. The set of traits expressed constitutes phenotype. Phenotypes and thus the rate of interaction are evolvable over time. Our results show that nonnegligible cost of expressing traits restrains phenotype diversity, and the evolutionary race mainly proceeds on between cooperative strains and defective strains who express a very few traits. It pays for cooperative strains to express a very few traits. Though such a low level of expression weakens reciprocity between cooperative strains, it decelerates the rate of interaction between cooperative strains and defective strains to a larger degree, leading to the predominance of cooperative strains over defective strains. We also find that evolved diversity of phenotype can occasionally destabilize due to the invasion of defective mutants, implying that cooperation and diversity of phenotype can mutually reinforce each other. Our results may help better understand the coevolution of cooperation and the diversity of phenotype.

Heterogeneous indirect reciprocity promotes the evolution of cooperation in structured populations

A fundamental aspect of all biological system is cooperation. Human society is based to a large extent on mechanisms that favor cooperation. Indirect reciprocity is a crucial factor for the emergence and maintenance of cooperation in evolutionary games. In this work, we introduce a mechanism of heterogeneous indirect reciprocity in the prisoner's dilemma game, where an altruistic attribute is considered. The so-called altruistic attribute refers to the trait that when an altruistic individual cooperates, its neighbors, regardless of their strategies, can gain additional benefits. Intuitively, altruistic cooperative agents seem to do only what is beneficial for others, but in fact, their neighbors tend to cooperate in order to maintain the cooperative strategies of altruistic individuals. In this way, the neighbors of cooperative altruistic individuals not only guarantee their own additional benefits, but also indirectly protect the income of altruistic individuals, which strengthens the links between cooperative individuals, thus promoting the evolution of cooperation. The robustness of the results is verified on homogeneous and heterogeneous networks. Moreover, the results of individual heterogeneity corroborate the existing evidence that heterogeneity, almost irrespective of its origin, promotes cooperative actions. Our conclusions might provide additional insights into understanding the roots of cooperation in social systems.

Inequality as information: Wealth homophily facilitates the evolution of cooperation

Free-riding produces inequality in the prisoners’ dilemma: cooperators suffer costs that defectors avoid, thus putting them at a material disadvantage to their anti-social peers. This inequality, accordingly, conveys information about a social partner’s choices in past game play and raises the possibility that agents can use the aggregation of past payoffs—i.e. wealth—to identify a social partner who uses their same strategy. Building on these insights, we study a computational model in which agents can employ a strategy—when playing multiple one-shot prisoners’ dilemma games per generation—in which they view other agents’ summed payoffs from previous games, choose to enter a PD game with the agent whose summed payoffs most-closely approximate their own, and then always cooperate. Here we show that this strategy of wealth homophily—labelled COEQUALS (“CO-operate with EQUALS”)—can both invade an incumbent population of defectors and resist invasion. The strategy succeeds because wealth homophily leads agents to direct cooperation disproportionately toward others of their own type—a phenomenon known as “positive assortment”. These findings illuminate empirical evidence indicating that viewable inequality degrades cooperation and they show how a standard feature of evolutionary game models—viz. the aggregation of payoffs during a generation—can double as an information mechanism that facilitates positive assortment.

Aspiration-based coevolution of link weight promotes cooperation in the spatial prisoner's dilemma game

In this article, we propose an aspiration-based coevolution of link weight, and explore how this set-up affects the evolution of cooperation in the spatial prisoner's dilemma game. In particular, an individual will increase the weight of its link to its neighbours only if the payoff received via this interaction exceeds a pre-defined aspiration. Conversely, if the received payoff is below this aspiration, the link weight with the corresponding neighbour will decrease. Our results show that an appropriate aspiration level leads to a high-cooperation plateau, whereas too high or too low aspiration will impede the evolution of cooperation. We explain these findings with a comprehensive analysis of transition points and with a systematic analysis of typical configuration patterns. The presented results provide further theoretical insights with regards to the impact of different aspiration levels on cooperation in human societies.

Competitions between prosocial exclusions and punishments in finite populations

Prosocial punishment has been proved to be a powerful mean to promote cooperation. Recent studies have found that social exclusion, which indeed can be regarded as a kind of punishment, can also support cooperation. However, if prosocial punishment and exclusion are both present, it is still unclear which strategy is more advantageous to curb free-riders. Here we first study the direct competition between different types of punishment and exclusion. We find that pool (peer) exclusion can always outperform pool (peer) punishment both in the optional and in the compulsory public goods game, no matter whether second-order sanctioning is considered or not. Furthermore, peer exclusion does better than pool exclusion both in the optional and in the compulsory game, but the situation is reversed in the presence of second-order exclusion. Finally, we extend the competition among all possible sanctioning strategies and find that peer exclusion can outperform all other strategies in the absence of second-order exclusion and punishment, while pool exclusion prevails when second-order sanctioning is possible. Our results demonstrate that exclusion is a more powerful strategy than punishment for the resolution of social dilemmas.

Invasion of Cooperation in Scale-Free Networks: Accumulated versus Average Payoffs

It is well known that cooperation cannot be an evolutionarily stable strategy for a non-iterative game in a well-mixed population. In contrast, structured populations favor cooperation, since cooperators can benefit each other by forming local clusters. Previous studies have shown that scale-free networks strongly promote cooperation. However, little is known about the invasion mechanism of cooperation in scale-free networks. To study microscopic and macroscopic behaviors of cooperators' invasion, we conducted computational experiments on the evolution of cooperation in scale-free networks where, starting from all defectors, cooperators can spontaneously emerge by mutation. Since the evolutionary dynamics are influenced by the definition of fitness, we tested two commonly adopted fitness functions: accumulated payoff and average payoff. Simulation results show that cooperation is strongly enhanced with the accumulated payoff fitness compared to the average payoff fitness. However, the difference between the two functions decreases as the average degree increases. As the average degree increases, cooperation decreases for the accumulated payoff fitness, while it increases for the average payoff fitness. Moreover, for the average payoff fitness, low-degree nodes play a more important role in spreading cooperative strategies than for the accumulated payoff fitness.

Stochastic win-stay-lose-shift strategy with dynamic aspirations in evolutionary social dilemmas

In times of plenty expectations rise, just as in times of crisis they fall. This can be mathematically described as a win-stay-lose-shift strategy with dynamic aspiration levels, where individuals aspire to be as wealthy as their average neighbor. Here we investigate this model in the realm of evolutionary social dilemmas on the square lattice and scale-free networks. By using the master equation and Monte Carlo simulations, we find that cooperators coexist with defectors in the whole phase diagram, even at high temptations to defect. We study the microscopic mechanism that is responsible for the striking persistence of cooperative behavior and find that cooperation spreads through second-order neighbors, rather than by means of network reciprocity that dominates in imitation-based models. For the square lattice the master equation can be solved analytically in the large temperature limit of the Fermi function, while for other cases the resulting differential equations must be solved numerically. Either way, we find good qualitative agreement with the Monte Carlo simulation results. Our analysis also reveals that the evolutionary outcomes are to a large degree independent of the network topology, including the number of neighbors that are considered for payoff determination on lattices, which further corroborates the local character of the microscopic dynamics. Unlike large-scale spatial patterns that typically emerge due to network reciprocity, here local checkerboard-like patterns remain virtually unaffected by differences in the macroscopic properties of the interaction network.

Individual wealth-based selection supports cooperation in spatial public goods games

In a social dilemma game group members are allowed to decide if they contribute to the joint venture or not. As a consequence, defectors, who do not invest but only enjoy the mutual benefit, prevail and the system evolves onto the tragedy of the common state. This unfortunate scenario can be avoided if participation is not obligatory but only happens with a given probability. But what if we also consider a player’s individual wealth when to decide about participation? To address this issue we propose a model in which the probabilistic participation in the public goods game is combined with a conditional investment mode that is based on individual wealth: if a player’s wealth exceeds a threshold value then it is qualified and can participate in the joint venture. Otherwise, the participation is forbidden in the investment interactions. We show that if only probabilistic participation is considered, spatially structured populations cannot support cooperation better than well-mixed populations where full defection state can also be avoided for small participation probabilities. By adding the wealth-based criterion of participation, however, structured populations are capable to augment network reciprocity relevantly and allow cooperator strategy to dominate in a broader parameter interval.

Collective Intelligence: Aggregation of Information from Neighbors in a Guessing Game

Complex systems show the capacity to aggregate information and to display coordinated activity. In the case of social systems the interaction of different individuals leads to the emergence of norms, trends in political positions, opinions, cultural traits, and even scientific progress. Examples of collective behavior can be observed in activities like the Wikipedia and Linux, where individuals aggregate their knowledge for the benefit of the community, and citizen science, where the potential of collectives to solve complex problems is exploited. Here, we conducted an online experiment to investigate the performance of a collective when solving a guessing problem in which each actor is endowed with partial information and placed as the nodes of an interaction network. We measure the performance of the collective in terms of the temporal evolution of the accuracy, finding no statistical difference in the performance for two classes of networks, regular lattices and random networks. We also determine that a Bayesian description captures the behavior pattern the individuals follow in aggregating information from neighbors to make decisions. In comparison with other simple decision models, the strategy followed by the players reveals a suboptimal performance of the collective. Our contribution provides the basis for the micro-macro connection between individual based descriptions and collective phenomena.

Evolutionary mixed games in structured populations: Cooperation and the benefits of heterogeneity

Evolutionary games on networks traditionally involve the same game at each interaction. Here we depart from this assumption by considering mixed games, where the game played at each interaction is drawn uniformly at random from a set of two different games. While in well-mixed populations the random mixture of the two games is always equivalent to the average single game, in structured populations this is not always the case. We show that the outcome is, in fact, strongly dependent on the distance of separation of the two games in the parameter space. Effectively, this distance introduces payoff heterogeneity, and the average game is returned only if the heterogeneity is small. For higher levels of heterogeneity the distance to the average game grows, which often involves the promotion of cooperation. The presented results support preceding research that highlights the favorable role of heterogeneity regardless of its origin, and they also emphasize the importance of the population structure in amplifying facilitators of cooperation.

Leaders should not be conformists in evolutionary social dilemmas

The most common assumption in evolutionary game theory is that players should adopt a strategy that warrants the highest payoff. However, recent studies indicate that the spatial selection for cooperation is enhanced if an appropriate fraction of the population chooses the most common rather than the most profitable strategy within the interaction range. Such conformity might be due to herding instincts or crowd behavior in humans and social animals. In a heterogeneous population where individuals differ in their degree, collective influence, or other traits, an unanswered question remains who should conform. Selecting conformists randomly is the simplest choice, but it is neither a realistic nor the optimal one. We show that, regardless of the source of heterogeneity and game parametrization, socially the most favorable outcomes emerge if the masses conform. On the other hand, forcing leaders to conform significantly hinders the constructive interplay between heterogeneity and coordination, leading to evolutionary outcomes that are worse still than if conformists were chosen randomly. We conclude that leaders must be able to create a following for network reciprocity to be optimally augmented by conformity. In the opposite case, when leaders are castrated and made to follow, the failure of coordination impairs the evolution of cooperation.

Sparse cliques trump scale-free networks in coordination and competition

Cooperative behavior, a natural, pervasive and yet puzzling phenomenon, can be significantly enhanced by networks. Many studies have shown how global network characteristics affect cooperation; however, it is difficult to understand how this occurs based on global factors alone, low-level network building blocks, or motifs are necessary. In this work, we systematically alter the structure of scale-free and clique networks and show, through a stochastic evolutionary game theory model, that cooperation on cliques increases linearly with community motif count. We further show that, for reactive stochastic strategies, network modularity improves cooperation in the anti-coordination Snowdrift game and the Prisoner's Dilemma game but not in the Stag Hunt coordination game. We also confirm the negative effect of the scale-free graph on cooperation when effective payoffs are used. On the flip side, clique graphs are highly cooperative across social environments. Adding cycles to the acyclic scale-free graph increases cooperation when multiple games are considered; however, cycles have the opposite effect on how forgiving agents are when playing the Prisoner's Dilemma game.

Emergence of scale-free characteristics in socio-ecological systems with bounded rationality

Socio-ecological systems are increasingly modelled by games played on complex networks. While the concept of Nash equilibrium assumes perfect rationality, in reality players display heterogeneous bounded rationality. Here we present a topological model of bounded rationality in socio-ecological systems, using the rationality parameter of the Quantal Response Equilibrium. We argue that system rationality could be measured by the average Kullback--Leibler divergence between Nash and Quantal Response Equilibria, and that the convergence towards Nash equilibria on average corresponds to increased system rationality. Using this model, we show that when a randomly connected socio-ecological system is topologically optimised to converge towards Nash equilibria, scale-free and small world features emerge. Therefore, optimising system rationality is an evolutionary reason for the emergence of scale-free and small-world features in socio-ecological systems. Further, we show that in games where multiple equilibria are possible, the correlation between the scale-freeness of the system and the fraction of links with multiple equilibria goes through a rapid transition when the average system rationality increases. Our results explain the influence of the topological structure of socio-ecological systems in shaping their collective cognitive behaviour, and provide an explanation for the prevalence of scale-free and small-world characteristics in such systems.

Costs for switching partners reduce network dynamics but not cooperative behaviour

Social networks represent the structuring of interactions between group members. Above all, many interactions are profoundly cooperative in humans and other animals. In accordance with this natural observation, theoretical work demonstrates that certain network structures favour the evolution of cooperation. Yet, recent experimental evidence suggests that static networks do not enhance cooperative behaviour in humans. By contrast, dynamic networks do foster cooperation. However, costs associated with dynamism such as time or resource investments in finding and establishing new partnerships have been neglected so far. Here, we show that human participants are much less likely to break links when costs arise for building new links. Especially, when costs were high, the network was nearly static. Surprisingly, cooperation levels in Prisoner's Dilemma games were not affected by reduced dynamism in social networks. We conclude that the mere potential to quit collaborations is sufficient in humans to reach high levels of cooperative behaviour. Effects of self-structuring processes or assortment on the network played a minor role: participants simply adjusted their cooperative behaviour in response to the threats of losing a partner or of being expelled.

Universal scaling for the dilemma strength in evolutionary games

Why would natural selection favor the prevalence of cooperation within the groups of selfish individuals? A fruitful framework to address this question is evolutionary game theory, the essence of which is captured in the so-called social dilemmas. Such dilemmas have sparked the development of a variety of mathematical approaches to assess the conditions under which cooperation evolves. Furthermore, borrowing from statistical physics and network science, the research of the evolutionary game dynamics has been enriched with phenomena such as pattern formation, equilibrium selection, and self-organization. Numerous advances in understanding the evolution of cooperative behavior over the last few decades have recently been distilled into five reciprocity mechanisms: direct reciprocity, indirect reciprocity, kin selection, group selection, and network reciprocity. However, when social viscosity is introduced into a population via any of the reciprocity mechanisms, the existing scaling parameters for the dilemma strength do not yield a unique answer as to how the evolutionary dynamics should unfold. Motivated by this problem, we review the developments that led to the present state of affairs, highlight the accompanying pitfalls, and propose new universal scaling parameters for the dilemma strength. We prove universality by showing that the conditions for an ESS and the expressions for the internal equilibriums in an infinite, well-mixed population subjected to any of the five reciprocity mechanisms depend only on the new scaling parameters. A similar result is shown to hold for the fixation probability of the different strategies in a finite, well-mixed population. Furthermore, by means of numerical simulations, the same scaling parameters are shown to be effective even if the evolution of cooperation is considered on the spatial networks (with the exception of highly heterogeneous setups). We close the discussion by suggesting promising directions for future research including (i) how to handle the dilemma strength in the context of co-evolution and (ii) where to seek opportunities for applying the game theoretical approach with meaningful impact.

Network Modularity is essential for evolution of cooperation under uncertainty

Cooperative behavior, which pervades nature, can be significantly enhanced when agents interact in a structured rather than random way; however, the key structural factors that affect cooperation are not well understood. Moreover, the role structure plays with cooperation has largely been studied through observing overall cooperation rather than the underlying components that together shape cooperative behavior. In this paper we address these two problems by first applying evolutionary games to a wide range of networks, where agents play the Prisoner's Dilemma with a three-component stochastic strategy, and then analyzing agent-based simulation results using principal component analysis. With these methods we study the evolution of trust, reciprocity and forgiveness as a function of several structural parameters. This work demonstrates that community structure, represented by network modularity, among all the tested structural parameters, has the most significant impact on the emergence of cooperative behavior, with forgiveness showing the largest sensitivity to community structure. We also show that increased community structure reduces the dispersion of trust and forgiveness, thereby reducing the network-level uncertainties for these two components; graph transitivity and degree also significantly influence the evolutionary dynamics of the population and the diversity of strategies at equilibrium.

Evolution of individual versus social learning on social networks

A number of studies have investigated the roles played by individual and social learning in cultural phenomena and the relative advantages of the two learning strategies in variable environments. Because social learning involves the acquisition of behaviours from others, its utility depends on the availability of 'cultural models' exhibiting adaptive behaviours. This indicates that social networks play an essential role in the evolution of learning. However, possible effects of social structure on the evolution of learning have not been fully explored. Here, we develop a mathematical model to explore the evolutionary dynamics of learning strategies on social networks. We first derive the condition under which social learners (SLs) are selectively favoured over individual learners in a broad range of social network. We then obtain an analytical approximation of the long-term average frequency of SLs in homogeneous networks, from which we specify the condition, in terms of three relatedness measures, for social structure to facilitate the long-term evolution of social learning. Finally, we evaluate our approximation by Monte Carlo simulations in complete graphs, regular random graphs and scale-free networks. We formally show that whether social structure favours the evolution of social learning is determined by the relative magnitudes of two effects of social structure: localization in competition, by which competition between learning strategies is evaded, and localization in cultural transmission, which slows down the spread of adaptive traits. In addition, our estimates of the relatedness measures suggest that social structure disfavours the evolution of social learning when selection is weak.

Conformity enhances network reciprocity in evolutionary social dilemmas

The pursuit of highest payoffs in evolutionary social dilemmas is risky and sometimes inferior to conformity. Choosing the most common strategy within the interaction range is safer because it ensures that the payoff of an individual will not be much lower than average. Herding instincts and crowd behaviour in humans and social animals also compel to conformity in their own right. Motivated by these facts, we here study the impact of conformity on the evolution of cooperation in social dilemmas. We show that an appropriate fraction of conformists within the population introduces an effective surface tension around cooperative clusters and ensures smooth interfaces between different strategy domains. Payoff-driven players brake the symmetry in favour of cooperation and enable an expansion of clusters past the boundaries imposed by traditional network reciprocity. This mechanism works even under the most testing conditions, and it is robust against variations of the interaction network as long as degree-normalized payoffs are applied. Conformity may thus be beneficial for the resolution of social dilemmas.

Impact of roles assignation on heterogeneous populations in evolutionary dictator game

The evolution of cooperation is a hot and challenging topic in the field of evolutionary game theory. Altruistic behavior, as a particular form of cooperation, has been widely studied by the ultimatum game but not by the dictator game, which provides a more elegant way to identify the altruistic component of behaviors. In this paper, the evolutionary dictator game is applied to model the real motivations of altruism. A degree-based regime is utilized to assess the impact of the assignation of roles on evolutionary outcome in populations of heterogeneous structure with two kinds of strategic updating mechanisms, which are based on Darwin's theory of evolution and punctuated equilibrium, respectively. The results show that the evolutionary outcome is affected by the role assignation and that this impact also depends on the strategic updating mechanisms, the function used to evaluate players' success, and the structure of populations.

Different perceptions of social dilemmas: evolutionary multigames in structured populations

Motivated by the fact that the same social dilemma can be perceived differently by different players, we here study evolutionary multigames in structured populations. While the core game is the weak prisoner's dilemma, a fraction of the population adopts either a positive or a negative value of the sucker's payoff, thus playing either the traditional prisoner's dilemma or the snowdrift game. We show that the higher the fraction of the population adopting a different payoff matrix the more the evolution of cooperation is promoted. The microscopic mechanism responsible for this outcome is unique to structured populations, and it is due to the payoff heterogeneity, which spontaneously introduces strong cooperative leaders that give rise to an asymmetric strategy imitation flow in favor of cooperation. We demonstrate that the reported evolutionary outcomes are robust against variations of the interaction network, and they also remain valid if players are allowed to vary which game they play over time. These results corroborate existing evidence in favor of heterogeneity-enhanced network reciprocity, and they reveal how different perceptions of social dilemmas may contribute to their resolution.

Optimal distribution of incentives for public cooperation in heterogeneous interaction environments

In the framework of evolutionary games with institutional reciprocity, limited incentives are at disposal for rewarding cooperators and punishing defectors. In the simplest case, it can be assumed that, depending on their strategies, all players receive equal incentives from the common pool. The question arises, however, what is the optimal distribution of institutional incentives? How should we best reward and punish individuals for cooperation to thrive? We study this problem for the public goods game on a scale-free network. We show that if the synergetic effects of group interactions are weak, the level of cooperation in the population can be maximized simply by adopting the simplest “equal distribution” scheme. If synergetic effects are strong, however, it is best to reward high-degree nodes more than low-degree nodes. These distribution schemes for institutional rewards are independent of payoff normalization. For institutional punishment, however, the same optimization problem is more complex, and its solution depends on whether absolute or degree-normalized payoffs are used. We find that degree-normalized payoffs require high-degree nodes be punished more lenient than low-degree nodes. Conversely, if absolute payoffs count, then high-degree nodes should be punished stronger than low-degree nodes.

Degree mixing in multilayer networks impedes the evolution of cooperation

Traditionally, the evolution of cooperation has been studied on single, isolated networks. Yet a player, especially in human societies, will typically be a member of many different networks, and those networks will play different roles in the evolutionary process. Multilayer networks are therefore rapidly gaining on popularity as the more apt description of a networked society. With this motivation, we here consider two-layer scale-free networks with all possible combinations of degree mixing, wherein one network layer is used for the accumulation of payoffs and the other is used for strategy updating. We find that breaking the symmetry through assortative mixing in one layer and/or disassortative mixing in the other layer, as well as preserving the symmetry by means of assortative mixing in both layers, impedes the evolution of cooperation. We use degree-dependent distributions of strategies and cluster-size analysis to explain these results, which highlight the importance of hubs and the preservation of symmetry between multilayer networks for the successful resolution of social dilemmas.

Evolutionary game dynamics in populations with heterogenous structures

Evolutionary graph theory is a well established framework for modelling the evolution of social behaviours in structured populations. An emerging consensus in this field is that graphs that exhibit heterogeneity in the number of connections between individuals are more conducive to the spread of cooperative behaviours. In this article we show that such a conclusion largely depends on the individual-level interactions that take place. In particular, averaging payoffs garnered through game interactions rather than accumulating the payoffs can altogether remove the cooperative advantage of heterogeneous graphs while such a difference does not affect the outcome on homogeneous structures. In addition, the rate at which game interactions occur can alter the evolutionary outcome. Less interactions allow heterogeneous graphs to support more cooperation than homogeneous graphs, while higher rates of interactions make homogeneous and heterogeneous graphs virtually indistinguishable in their ability to support cooperation. Most importantly, we show that common measures of evolutionary advantage used in homogeneous populations, such as a comparison of the fixation probability of a rare mutant to that of the resident type, are no longer valid in heterogeneous populations. Heterogeneity causes a bias in where mutations occur in the population which affects the mutant's fixation probability. We derive the appropriate measures for heterogeneous populations that account for this bias.

Understanding the evolution of cooperation is a persistent challenge to evolutionary theorists. A contemporary take on this subject is to model populations with interactions structured as close as possible to actual social networks. These networks are heterogeneous in the number and type of contact each member has. Our paper demonstrates that the fate of cooperation in such heterogeneous populations critically depends on the rate at which interactions occur and how interactions translate into the fitnesses of the strategies. We also develop theory that allows for an evolutionary analysis in heterogeneous populations. This includes deriving appropriate criteria for evolutionary advantage.

Spontaneous symmetry breaking in interdependent networked game

Spatial evolution game has traditionally assumed that players interact with direct neighbors on a single network, which is isolated and not influenced by other systems. However, this is not fully consistent with recent research identification that interactions between networks play a crucial rule for the outcome of evolutionary games taking place on them. In this work, we introduce the simple game model into the interdependent networks composed of two networks. By means of imitation dynamics, we display that when the interdependent factor α is smaller than a threshold value α(C), the symmetry of cooperation can be guaranteed. Interestingly, as interdependent factor exceeds α(C), spontaneous symmetry breaking of fraction of cooperators presents itself between different networks. With respect to the breakage of symmetry, it is induced by asynchronous expansion between heterogeneous strategy couples of both networks, which further enriches the content of spatial reciprocity. Moreover, our results can be well predicted by the strategy-couple pair approximation method.

Robustness of cooperation on scale-free networks under continuous topological change

In this paper, we numerically investigate the robustness of cooperation clusters in prisoner's dilemma played on scale-free networks, where the network topologies change by continuous removal and addition of nodes. Each removal and addition can be either random or intentional. We therefore have four different strategies in changing network topology: random removal and random addition (RR), random removal and preferential addition (RP), targeted removal and random addition (TR), and targeted removal and preferential addition (TP). We find that cooperation clusters are most fragile against TR, while they are most robust against RP, even for large values of the temptation coefficient for defection. The effect of the degree mixing pattern of the network is not the primary factor for the robustness of cooperation under continuous change in network topology, which is quite different from the cases observed in static networks. Cooperation clusters become more robust as the number of links of hubs occupied by cooperators increase. Our results might infer the fact that a huge variety of individuals is needed for maintaining global cooperation in social networks in the real world where each node representing an individual is constantly removed and added.

Costly advertising and the evolution of cooperation

In this paper, I investigate the co-evolution of fast and slow strategy spread and game strategies in populations of spatially distributed agents engaged in a one off evolutionary dilemma game. Agents are characterized by a pair of traits, a game strategy (cooperate or defect) and a binary ‘advertising’ strategy (advertise or don’t advertise). Advertising, which comes at a cost , allows investment into faster propagation of the agents’ traits to adjacent individuals. Importantly, game strategy and advertising strategy are subject to the same evolutionary mechanism. Via analytical reasoning and numerical simulations I demonstrate that a range of advertising costs exists, such that the prevalence of cooperation is significantly enhanced through co-evolution. Linking costly replication to the success of cooperators exposes a novel co-evolutionary mechanism that might contribute towards a better understanding of the origins of cooperation-supporting heterogeneity in agent populations.

Collective chasing behavior between cooperators and defectors in the spatial prisoner's dilemma

Cooperation is one of the essential factors for all biological organisms in major evolutionary transitions. Recent studies have investigated the effect of migration for the evolution of cooperation. However, little is known about whether and how an individuals' cooperativeness coevolves with mobility. One possibility is that mobility enhances cooperation by enabling cooperators to escape from defectors and form clusters; the other possibility is that mobility inhibits cooperation by helping the defectors to catch and exploit the groups of cooperators. In this study we investigate the coevolutionary dynamics by using the prisoner's dilemma game model on a lattice structure. The computer simulations demonstrate that natural selection maintains cooperation in the form of evolutionary chasing between the cooperators and defectors. First, cooperative groups grow and collectively move in the same direction. Then, mutant defectors emerge and invade the cooperative groups, after which the defectors exploit the cooperators. Then other cooperative groups emerge due to mutation and the cycle is repeated. Here, it is worth noting that, as a result of natural selection, the mobility evolves towards directional migration, but not to random or completely fixed migration. Furthermore, with directional migration, the rate of global population extinction is lower when compared with other cases without the evolution of mobility (i.e., when mobility is preset to random or fixed). These findings illustrate the coevolutionary dynamics of cooperation and mobility through the directional chasing between cooperators and defectors.

Evolution of emotions on networks leads to the evolution of cooperation in social dilemmas

We show that the resolution of social dilemmas in random graphs and scale-free networks is facilitated by imitating not the strategy of better-performing players but, rather, their emotions. We assume sympathy and envy to be the two emotions that determine the strategy of each player in any given interaction, and we define them as the probabilities of cooperating with players having a lower and a higher payoff, respectively. Starting with a population where all possible combinations of the two emotions are available, the evolutionary process leads to a spontaneous fixation to a single emotional profile that is eventually adopted by all players. However, this emotional profile depends not only on the payoffs but also on the heterogeneity of the interaction network. Homogeneous networks, such as lattices and regular random graphs, lead to fixations that are characterized by high sympathy and high envy, while heterogeneous networks lead to low or modest sympathy but also low envy. Our results thus suggest that public emotions and the propensity to cooperate at large depend, and are in fact determined by, the properties of the interaction network.

Short versus long term benefits and the evolution of cooperation in the prisoner's dilemma game

In this paper I investigate the evolution of cooperation in the prisoner's dilemma when individuals change their strategies subject to performance evaluation of their neighbours over variable time horizons. In the monochrome setting, in which all agents per default share the same performance evaluation rule, weighing past events strongly dramatically enhances the prevalence of cooperators. For co-evolutionary models, in which evaluation time horizons and strategies can co-evolve, I demonstrate that cooperation naturally associates with long-term evaluation of others while defection is typically paired with very short time horizons. Moreover, considering the continuous spectrum in between enhanced and discounted weights of past performance, cooperation is optimally supported when cooperators neither give enhanced weight to past nor more recent events, but simply average payoffs. Payoff averaging is also found to emerge as the dominant strategy for cooperators in co-evolutionary models, thus proposing a natural route to the evolution of cooperation in viscous populations.

Interdependent network reciprocity in evolutionary games

Besides the structure of interactions within networks, also the interactions between networks are of the outmost importance. We therefore study the outcome of the public goods game on two interdependent networks that are connected by means of a utility function, which determines how payoffs on both networks jointly influence the success of players in each individual network. We show that an unbiased coupling allows the spontaneous emergence of interdependent network reciprocity, which is capable to maintain healthy levels of public cooperation even in extremely adverse conditions. The mechanism, however, requires simultaneous formation of correlated cooperator clusters on both networks. If this does not emerge or if the coordination process is disturbed, network reciprocity fails, resulting in the total collapse of cooperation. Network interdependence can thus be exploited effectively to promote cooperation past the limits imposed by isolated networks, but only if the coordination between the interdependent networks is not disturbed.

Spreading of cooperative behaviour across interdependent groups

Recent empirical research has shown that links between groups reinforce individuals within groups to adopt cooperative behaviour. Moreover, links between networks may induce cascading failures, competitive percolation, or contribute to efficient transportation. Here we show that there in fact exists an intermediate fraction of links between groups that is optimal for the evolution of cooperation in the prisoner's dilemma game. We consider individual groups with regular, random, and scale-free topology, and study their different combinations to reveal that an intermediate interdependence optimally facilitates the spreading of cooperative behaviour between groups. Excessive between-group links simply unify the two groups and make them act as one, while too rare between-group links preclude a useful information flow between the two groups. Interestingly, we find that between-group links are more likely to connect two cooperators than in-group links, thus supporting the conclusion that they are of paramount importance.

Optimal interdependence between networks for the evolution of cooperation

Recent research has identified interactions between networks as crucial for the outcome of evolutionary games taking place on them. While the consensus is that interdependence does promote cooperation by means of organizational complexity and enhanced reciprocity that is out of reach on isolated networks, we here address the question just how much interdependence there should be. Intuitively, one might assume the more the better. However, we show that in fact only an intermediate density of sufficiently strong interactions between networks warrants an optimal resolution of social dilemmas. This is due to an intricate interplay between the heterogeneity that causes an asymmetric strategy flow because of the additional links between the networks, and the independent formation of cooperative patterns on each individual network. Presented results are robust to variations of the strategy updating rule, the topology of interdependent networks, and the governing social dilemma, thus suggesting a high degree of universality.

Building cooperative networks

We study the cooperation problem in the framework of evolutionary game theory by using the prisoner's dilemma as a metaphor of the problem. By considering the growing process of the system and individuals with imitation capacity, we show conditions that allow the formation of highly cooperative networks of any size and topology. By introducing general considerations of real systems, we reduce the required conditions for cooperation to evolve, which approaches the benefit-cost ratio r for the theoretical minimum r=1 when the mean connectivity of the individuals is increased. Throughout the paper, we distinguish different mechanisms that allow the system to maintain high levels of cooperation when the system grows by incorporation of defectors. These mechanisms require heterogeneity among individuals for cooperation to evolve. However, the required benefit-cost ratio and heterogeneities are drastically reduced as compared to those required for static networks.

Network-based criterion for the success of cooperation in an evolutionary prisoner's dilemma

We consider an evolutionary prisoner's dilemma on a random network. We introduce a simple quantitative network-based parameter and show that it effectively predicts the success of cooperation in simulations on the network. The criterion is shown to be accurate on a variety of networks with degree distributions ranging from regular to Poisson to scale free. The parameter allows for comparisons of random networks regardless of their underlying topology. Finally, we draw analogies between the criterion for the success of cooperation introduced here and existing criteria in other contexts.

Bipartite graphs as models of population structures in evolutionary multiplayer games

By combining evolutionary game theory and graph theory, “games on graphs” study the evolutionary dynamics of frequency-dependent selection in population structures modeled as geographical or social networks. Networks are usually represented by means of unipartite graphs, and social interactions by two-person games such as the famous prisoner’s dilemma. Unipartite graphs have also been used for modeling interactions going beyond pairwise interactions. In this paper, we argue that bipartite graphs are a better alternative to unipartite graphs for describing population structures in evolutionary multiplayer games. To illustrate this point, we make use of bipartite graphs to investigate, by means of computer simulations, the evolution of cooperation under the conventional and the distributed N-person prisoner’s dilemma. We show that several implicit assumptions arising from the standard approach based on unipartite graphs (such as the definition of replacement neighborhoods, the intertwining of individual and group diversity, and the large overlap of interaction neighborhoods) can have a large impact on the resulting evolutionary dynamics. Our work provides a clear example of the importance of construction procedures in games on graphs, of the suitability of bigraphs and hypergraphs for computational modeling, and of the importance of concepts from social network analysis such as centrality, centralization and bipartite clustering for the understanding of dynamical processes occurring on networked population structures.