Group-size effects on the evolution of cooperation in the spatial public goods game

Cyclical oscillations and absorbing-state probabilities in optional public goods games: Interplay of reward and group size

The optional public goods game (OPGG) is a three-strategy model in which individuals can cooperate, defect, or not participate. Despite its simplicity, this model effectively captures various social dilemmas, including those involving public services, environmental sustainability, and broader societal issues. In this study, we investigate how the reward (r) and group size of potential players (S) of public goods games influence the steady-state coexistence of these strategies or the alternation of their dominance in a rock-paper-scissors dynamic. The OPGG is simulated using Monte Carlo in a nonspatial scenario, meaning there is no topology connecting the agents, allowing any player to interact with any other player. We show that under sufficiently noisy conditions, the system consistently evolves to an absorbing state, with the prevailing strategy determined by the values of r and S. In the range 2≤r≤S, the system shows multiple stable absorbing states, with groups of size S=4 exhibiting more pronounced and transient rock-paper-scissors dynamics with longer average absorbing times. We present a thorough analysis of our results in terms of the fraction of time the system spends in rock-paper-scissor cycles, the number of cycles, and the average probability that the system relaxes to each possible absorbing state, including scenarios where the system does not reach an absorbing state at all.

Strategy evolution on higher-order networks

Cooperation is key to prosperity in human societies. Population structure is well understood as a catalyst for cooperation, where research has focused on pairwise interactions. But cooperative behaviors are not simply dyadic, and they often involve coordinated behavior in larger groups. Here we develop a framework to study the evolution of behavioral strategies in higher-order population structures, which include pairwise and multi-way interactions. We provide an analytical treatment of when cooperation will be favored by higher-order interactions, accounting for arbitrary spatial heterogeneity and nonlinear rewards for cooperation in larger groups. Our results indicate that higher-order interactions can act to promote the evolution of cooperation across a broad range of networks, in public goods games. Higher-order interactions consistently provide an advantage for cooperation when interaction hyper-networks feature multiple conjoined communities. Our analysis provides a systematic account of how higher-order interactions modulate the evolution of prosocial traits.

Replicator-mutator dynamics with evolutionary public goods game-environmental feedbacks

Feedback loops between strategies and the environment are commonly observed in socio-ecological, evolution-ecological, and psychology-economic systems. However, the impact of mutations in these feedback processes is often overlooked. This study proposes a novel model that integrates the public goods game with environmental feedback, considering the presence of mutations. In our model, the enhancement factor of the public goods game combines positive and negative incentives from the environment. By employing replicator-mutator (RM) equations, we provide an objective understanding of the system's evolutionary state, focusing on identifying conditions that foster cooperation and prevent the tragedy of the commons. Specifically, mutations play a crucial role in the RM dynamics, leading to the emergence of an oscillatory tragedy of the commons. By verifying the Hopf bifurcation condition, we establish the existence of a stable limit cycle, providing valuable insights into sustained oscillation strategies. Moreover, the feedback mechanism inherent in the public goods game model offers a fresh perspective on effectively addressing the classic dilemma of the tragedy of the commons.

Guppies in large groups cooperate more frequently in an experimental test of the group size paradox

The volunteer's dilemma, in which a single individual is required to produce a public good, predicts that individuals in larger groups will cooperate less frequently. Mechanistically, this could result from trade-offs between costs associated with volunteering and costs incurred if the public good is not produced (nobody volunteers). During predator inspection, one major contributor to the cost of volunteering is likely increased probability of predation; however, a predator also poses a risk to all individuals if nobody inspects. We tested the prediction that guppies in larger groups will inspect a predator less than those in smaller groups. We also predicted that individuals in larger groups would perceive less threat from the predator stimulus because of the protective benefits of larger groups (e.g. dilution). Contrary to prediction, we found that individuals in large groups inspected more frequently than those in smaller groups, but (as predicted) spent less time in refuges. There was evidence that individuals in intermediate-sized groups made fewest inspections and spent most time in refuges, suggesting that any link between group size, risk and cooperation is not driven by simple dilution. Extensions of theoretical models that capture these dynamics will likely be broadly applicable to risky cooperative behaviour.

Enhanced cooperation in multiplayer snowdrift games with random and dynamic groupings

An analytically tractable generalization of the N-person snowdrift (NSG) game that illustrates how cooperation can be enhanced is proposed and studied. The number of players competing within a NSG varies from one time step to another. Exact equations governing the frequency of cooperation f_{c}(r) as a function of the cost-to-benefit ratio r within an imitation strategy updating scheme are presented. For group sizes g uniformly distributed within the range g∈[1,g_{m}], an analytic formula for the critical value r_{c}(g_{m}), below which the system evolves into a totally cooperative (AllC) state, is derived. In contrast, a fixed group size NSG does not support an AllC state. The result r_{c}(g_{m}) requires the presence of sole-player groups and involves the inverse of the harmonic numbers and, more generally, the inverse first moment of the group size distribution. For r>r_{c}(g_{m}), the equation that determines the dynamical mixed states f_{c}(r) is given, with exact solutions existing for g_{m}≤5. The exact treatment allows the study of the phase boundary between the AllC state and the mixed states. The analytic results are checked against simulation results and exact agreements are demonstrated. The analytic form of the critical r_{c}(g_{m}) illustrates the necessity of having groups of a sole player in the evolutionary process. This result is supported by simulations with group sizes excluding the sole groups for which no AllC state emerges. A physically transparent picture of the importance of the sole players in inducing an AllC state is further presented based on the last surviving pattern before the AllC state is attained. The exact expression r_{c}(g_{m}) turns out to remain valid for nonuniform group-size distributions. Our analytical tractable generalization, therefore, sheds light on how a competing environment with variable group sizes could enhance cooperation and induce an AllC state.

Amplitude death in coupled replicator map lattice: Averting migration dilemma

Populations composed of a collection of subpopulations (demes) with random migration between them are quite common occurrences. The emergence and sustenance of cooperation in such a population is a highly researched topic in the evolutionary game theory. If the individuals in every deme are considered to be either cooperators or defectors, the migration dilemma can be envisaged: The cooperators would not want to migrate to a defector-rich deme as they fear of facing exploitation; but without migration, cooperation cannot be established throughout the network of demes. With a view to studying the aforementioned scenario, in this paper, we set up a theoretical model consisting of a coupled map lattice of replicator maps based on two-player-two-strategy games. The replicator map considered is capable of showing a variety of evolutionary outcomes, like convergent (fixed point) outcomes and nonconvergent (periodic and chaotic) outcomes. Furthermore, this coupled network of the replicator maps undergoes the phenomenon of amplitude death leading to nonoscillatory stable synchronized states. We specifically explore the effect of (i) the nature of coupling that models migration between the maps, (ii) the heterogenous demes (in the sense that not all the demes have the same game being played by the individuals), (iii) the degree of the network, and (iv) the cost associated with the migration. In the course of investigation, we are intrigued by the effectiveness of the random migration in sustaining a uniform cooperator fraction across a population irrespective of the details of the replicator dynamics and the interaction among the demes.

Symbiotic behaviour in the public goods game with altruistic punishment

Finding ways to overcome the temptation to exploit one another is still a challenge in behavioural sciences. In the framework of evolutionary game theory, punishing strategies are frequently used to promote cooperation in competitive environments. Here, we introduce altruistic punishers in the spatial public goods game. This strategy acts as a cooperator in the absence of defectors, otherwise it will punish all defectors in their vicinity while bearing a cost to do so. We observe three distinct behaviours in our model: i) in the absence of punishers, cooperators (who don't punish defectors) are driven to extinction by defectors for most parameter values; ii) clusters of punishers thrive by sharing the punishment costs when these are low; iii) for higher punishment costs, punishers, when alone, are subject to exploitation but in the presence of cooperators can form a symbiotic spatial structure that benefits both. This last observation is our main finding since neither cooperation nor punishment alone can survive the defector strategy in this parameter region and the specificity of the symbiotic spatial configuration shows that lattice topology plays a central role in sustaining cooperation. Results were obtained by means of Monte Carlo simulations on a square lattice and subsequently confirmed by a pairwise comparison of different strategies' payoffs in diverse group compositions, leading to a phase diagram of the possible states.

Evolutionary public goods game on the birandom geometric graph

To investigate the evolution of cooperation in spatial public goods games, this paper establishes a birandom geometric graph, in which two types of nodes, representing players and public goods respectively, are placed at random locations in the unit square. Each public good has a limit influence range and the individuals that fall into the same range engage in a public good game. In contrast to the classical network models consisting of only one type of nodes, the birandom geometric graph provides a natural way to describe the scenarios where individuals and public resources are independent of each other. Numerical simulations reveal that cooperation can be significantly promoted when the group size and the average number of groups that each player participates in are relatively small, which is at odds with the results on the square lattice, but is consistent with a body of empirical evidence reported by Ostrom and Olson et al. Analysis of the evolutionary process suggests that the facilitation of cooperation is due primarily to the formation of the cooperative clusters, which can effectively resist the invasion of the defectors.

High thresholds encouraging the evolution of cooperation in threshold public-good games

For a well-mixed population, we consider a threshold public good game where group members only obtain benefits from a public good if a sufficiently large number of them cooperates. We investigate the effect of an increase in the threshold on the level of cooperation that evolves. It is shown that for sufficiently large participation costs, the level of cooperation is higher for low and for high thresholds, than it is for intermediate thresholds – where in the latter case cooperation may not evolve at all. The counterintuitive effect where an increase in the threshold from an intermediate to a high one decreases the probability of cooperation, is related to the so-called common-enemy hypothesis of the evolution of cooperation. We further apply our analysis to assess the relative weight of different game types across the parameter space, and show that game types where either a small, or a large fraction of the population evolves as cooperators, receive more weight compared to game types where an intermediate fraction of cooperators evolves.

Does the expansion of the joint prevention and control area improve the air quality?-Evidence from China's Jing-Jin-Ji region and surrounding areas

The relationship between group size and alliance performance has long been concerned by the theoretical scholars, but their opinions vary from each other. China's air pollution prevention and control action in 2017-2018 expanded the core area of joint prevention and control from a minor alliance of "2 + 4" cities to a major alliance of "2 + 26" cities, which undoubtedly provides a good natural experiment for advancing theoretical research. In order to clarify whether the expansion of the cooperation scope of air pollution joint prevention and control will further improve the air quality, the paper uses the regression discontinuity design (RDD) to empirically analyze the original "2 + 4" cities and 22 newly added cities. The conclusions indicate that the expansion of the core area of air pollution joint prevention and control in Beijing-Tianjin-Hebei (Jing-Jin-Ji) region and surrounding areas is conducive to further improving regional air quality, and the improvement effect of the "2 + 4" cities in the original core area is greater than that of the newly added 22 cities. The improvement of air quality after the expansion of the alliance is attributed to the unified and coordinated supervision of trans-regional environmental protection agencies, which reduces the "free rider" behavior; the increase of alliance members is conducive to achieve better environmental governance and improve cooperation performance among alliance members. Therefore, it is proposed that when the joint prevention and control of air pollution area is mature, the city should expand the scope of the alliance appropriately, rationally control and strengthen the cooperation among cities within the regional alliance, and consolidate the achievements of inter-regional joint prevention and control air quality improvement.

Cooperation in Groups of Different Sizes: The Effects of Punishment and Reputation-Based Partner Choice

Reputation and punishment are two distinct mechanisms that facilitate cooperation among strangers. However, empirical research on their effectiveness is mainly limited to relatively small groups and does not address how they enhance cooperation in relatively larger groups. We address this gap in the literature by testing hypotheses from competing perspectives about the extent to which reputation-based partner choice and punishment enhance cooperation in both small and large groups. Prior work recognizes that an increase in group size is accompanied by a change in the incentive structure, which determines whether the temptation (extra benefit for each person from non-cooperation over cooperation, regardless of others’ choices) or gain (extra benefit for each person from full cooperation over full non-cooperation) remains constant or varies with group size. Thus, we first test how group size affects cooperation when temptation or gain increases with group size (Study 1), and then move on to testing predictions on the effectiveness of reputation and punishment across different group sizes (Study 2). In Study 1 (N = 820), we randomly assigned participants to play an online one-shot public goods game in groups of 4, 20, or 40, while keeping the marginal group return or marginal per capita return fixed across groups, in which case the temptation or gain increased with group size. In Study 2 (N = 1,132), we further compared a public goods situation involving a punishment or reputation mechanism with an anonymous situation across group sizes, while the marginal group return was fixed across groups. Overall, we found that when temptation increased with group size, 20-person groups cooperated significantly less than 4-person groups in one-shot interactions, and that this effect was explained by lower expectation of others’ cooperation, less perceived collective efficacy, and greater perceived conflict. However, 40-person and 4-person groups did not vary in one-shot cooperation. Importantly, reputation-based partner choice and punishment invariably promoted one-shot cooperation in groups of different sizes. These findings suggest no simple effect of group size on cooperation and underscore the utility of reputation and punishment in fostering cooperation (at least in one-shot interactions) regardless of the size of groups.

Research on an evolutionary game model and simulation of a cluster innovation network based on fairness preference

The cluster innovation network is an important part of regional economic development. In addition, the fairness preference of internal innovators in the processes of investment and benefit distribution are particularly important for curbing "free riding" and other speculative behaviors and for creating a good cooperation environment. Therefore, taking a cluster innovation network constructed by the weighted evolutionary BBV model as the research subject, this paper constructs an evolutionary game model of a cluster innovation network based on a spatial public goods game and the theory of fairness preferences, which involves the processes of investment and payoff allocation. Using simulation analysis, this paper studies the evolution of innovators’ cooperative behaviors and benefits in cluster innovation network under the conditions of a fairness preference and a return intensity. The results show that an increase in the weight coefficient, gain coefficient and degree of differentiation between the previous income and current investment can effectively promote improvements in the level of enterprise cooperation. Indeed, the greater the weight coefficient, the gain coefficient and the degree of differentiation are, the more substantial the improvement in the level of enterprise cooperation will be. Moreover, an improvement in the differentiation of the breadth and depth of enterprise cooperation has an inhibitory effect on enterprise cooperation. Furthermore, whereas increases in regulation and gain coefficients can effectively promote enterprise cooperation. However, the increase in the weight coefficient has a different effect on enterprise benefit in terms of the breadth and depth of cooperation. Finally, we hope to improve the overall cooperation level and cooperation income of the network by deeply understanding the fair preferences of innovators in the processes of investment and benefit distribution, which is helpful for promoting the evolution and development of cluster innovation networks.

Seasonal payoff variations and the evolution of cooperation in social dilemmas

Varying environmental conditions affect relations between interacting individuals in social dilemmas, thus affecting also the evolution of cooperation. Oftentimes these environmental variations are seasonal and can therefore be mathematically described as periodic changes. Accordingly, we here study how periodic shifts between different manifestations of social dilemmas affect cooperation. We observe a non-trivial interplay between the inherent spatiotemporal dynamics that characterizes the spreading of cooperation in a particular social dilemma type and the frequency of payoff changes. In particular, we show that periodic changes between two available games with global ordering best be fast, while periodic changes between global and local ordering games best be slow for cooperation to thrive. We also show that the frequency of periodic changes between two local ordering social dilemmas is irrelevant, because then the process is fast and simply the average cooperation level of the two is returned. The structure of the interaction network plays an important role too in that lattices promote local ordering, whilst random graphs hinder the formation of compact cooperative clusters. Conversely, for local ordering the regular structure of the interaction network is only marginally relevant as role-separating checkerboard patterns do not rely on long-range order.

Large-scale cooperation driven by reputation, not fear of divine punishment

Reputational considerations favour cooperation and thus we expect less cooperation in larger communities where people are less well known to each other. Some argue that institutions are, therefore, necessary to coordinate large-scale cooperation, including moralizing religions that promote cooperation through the fear of divine punishment. Here, we use community size as a proxy for reputational concerns, and test whether people in small, stable communities are more cooperative than people in large, less stable communities in both religious and non-religious contexts. We conducted a donation game on a large naturalistic sample of 501 people in 17 communities, with varying religions or none, ranging from small villages to large cities in northwestern China. We found that more money was donated by those in small, stable communities, where reputation should be more salient. Religious practice was also associated with higher donations, but fear of divine punishment was not. In a second game on the same sample, decisions were private, giving donors the opportunity to cheat. We found that donors to religious institutions were not less likely to cheat, and community size was not important in this game. Results from the donation game suggest donations to both religious and non-religious institutions are being motivated by reputational considerations, and results from both games suggest fear of divine punishment is not important. This chimes with other studies suggesting social benefits rather than fear of punishment may be the more salient motive for cooperative behaviour in real-world settings.

Group size effects and critical mass in public goods games

Understanding whether the size of the interacting group has an effect on cooperative behavior has been a major topic of debate since the seminal works on cooperation in the 1960s. Half a century later, scholars have yet to reach a consensus, with some arguing that cooperation is harder in larger groups, while others that cooperation is easier in larger groups, and yet others that cooperation attains its maximum in intermediate size groups. Here we add to this field of work by reporting a two-treatment empirical study where subjects play a Public Goods Game with a Critical Mass, such that the return for full cooperation increases linearly for early contributions and then stabilizes after a critical mass is reached (the two treatments differ only on the critical mass). We choose this game for two reasons: it has been argued that it approximates real-life social dilemmas; previous work suggests that, in this case, group size might have an inverted-U effect on cooperation, where the pick of cooperation is reached around the critical mass. Our main innovation with respect to previous experiments is that we implement a within-subject design, such that the same subject plays in groups of different size (from 5 to 40 subjects). Groups are formed at random at every round and there is no feedback. This allows us to explore if and how subjects change their choice as a function of the size of the group. We report three main results, which partially contrast what has been suggested by previous work: in our setting (i) the critical mass has no effect on cooperation; (ii) group size has a positive effect on cooperation; (iii) the most chosen option (played by about 50% of the subjects) is All Defection, followed by All Cooperation (about 10% of the subjects), whereas the rest have a slight trend to switch preferentially from defection to cooperation as the group size increases.

The role of memory in human strategy updating in optional public goods game

Most research suggests that humans can optimize their behavior by imitating other humans. However, it remains unclear whether humans actually imitate others in real-life situations. To address this question, we conducted spatial public goods experiments with voluntary participation. In direct contrast to the prevailing view, the results of our experiments show that imitation plays an insignificant role in the decision making process. Furthermore, we found that the nature of human decision making relied more on their performances in the game's earlier history rather than the performance they observed in others. The action that gained better results in one's own history had a higher chance to be adopted even if this action did not work for the others.

Investment preference promotes cooperation in spatial public goods game

It is usually assumed that each cooperator contributes equally to different public pools in spatial public goods game. However, it is more reasonable to invest differently according to individual investment preference. In this paper, an extended public goods game, in which cooperators contribute to the groups according to the investment preference, is developed. The investment preference of a cooperator is characterized by the fraction of the cooperator from his/her own memory about a group and the intensity of investment preference is represented by a tunable parameter α. The well-mixed population and the structured population are analyzed under this mechanism. It is shown that the investment preference can give rise to coordination. Moreover, the extensive numerical simulation results show that with the increasing of investment preference density or memory length, the proportion of cooperation can increase monotonously. This is because the investment preference could help cooperators resist the invasion from defectors. Compared with the basic version, the new mechanism is able to promote cooperation effectively. Our research may provide a valuable insight for further exploring the nature of cooperation in the real world.

The Beneficial Role of Mobility for the Emergence of Innovation

Innovation is a key ingredient for the evolution of several systems, including social and biological ones. Focused investigations and lateral thinking may lead to innovation, as well as serendipity and other random discovery processes. Some individuals are talented at proposing innovation (say innovators), while others at deeply exploring proposed novelties, at getting further insights on a theory, or at developing products, services, and so on (say developers). This separation in terms of innovators and developers raises an issue of paramount importance: under which conditions a system is able to maintain innovators? According to a simple model, this work investigates the evolutionary dynamics that characterize the emergence of innovation. In particular, we consider a population of innovators and developers, in which agents form small groups whose composition is crucial for their payoff. The latter depends on the heterogeneity of the formed groups, on the amount of innovators they include, and on an award-factor that represents the policy of the system for promoting innovation. Under the hypothesis that a "mobility" effect may support the emergence of innovation, we compare the equilibria reached by our population in different cases. Results confirm the beneficial role of "mobility", and the emergence of further interesting phenomena.

Think global, act local: Preserving the global commons

Preserving global public goods, such as the planet's ecosystem, depends on large-scale cooperation, which is difficult to achieve because the standard reciprocity mechanisms weaken in large groups. Here we demonstrate a method by which reciprocity can maintain cooperation in a large-scale public goods game (PGG). In a first experiment, participants in groups of on average 39 people play one round of a Prisoner's Dilemma (PD) with their two nearest neighbours on a cyclic network after each PGG round. We observe that people engage in "local-to-global" reciprocity, leveraging local interactions to enforce global cooperation: Participants reduce PD cooperation with neighbours who contribute little in the PGG. In response, low PGG contributors increase their contributions if both neighbours defect in the PD. In a control condition, participants do not know their neighbours' PGG contribution and thus cannot link play in the PD to the PGG. In the control we observe a sharp decline of cooperation in the PGG, while in the treatment condition global cooperation is maintained. In a second experiment, we demonstrate the scalability of this effect: in a 1,000-person PGG, participants in the treatment condition successfully sustain public contributions. Our findings suggest that this simple "local-to-global" intervention facilitates large-scale cooperation.

Culture and cooperation in a spatial public goods game

We study the coevolution of culture and cooperation by combining the Axelrod model of cultural dissemination with a spatial public goods game, incorporating both noise and social influence. Both participation and cooperation in public goods games are conditional on cultural similarity. We find that a larger "scope of cultural possibilities" in the model leads to the survival of cooperation, when noise is not present, and a higher probability of a multicultural state evolving, for low noise rates. High noise rates, however, lead to both rapid extinction of cooperation and collapse into cultural "anomie," in which stable cultural regions fail to form. These results suggest that cultural diversity can actually be beneficial for the evolution of cooperation, but that cultural information needs to be transmitted accurately in order to maintain both coherent cultural groups and cooperation.

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.

Antisocial pool rewarding does not deter public cooperation

Rewarding cooperation is in many ways expected behaviour from social players. However, strategies that promote antisocial behaviour are also surprisingly common, not just in human societies, but also among eusocial insects and bacteria. Examples include sanctioning of individuals who behave prosocially, or rewarding of free-riders who do not contribute to collective enterprises. We therefore study the public goods game with antisocial and prosocial pool rewarding in order to determine the potential negative consequences on the effectiveness of positive incentives to promote cooperation. Contrary to a naive expectation, we show that the ability of defectors to distribute rewards to their like does not deter public cooperation as long as cooperators are able to do the same. Even in the presence of antisocial rewarding, the spatial selection for cooperation in evolutionary social dilemmas is enhanced. Since the administration of rewards to either strategy requires a considerable degree of aggregation, cooperators can enjoy the benefits of their prosocial contributions as well as the corresponding rewards. Defectors when aggregated, on the other hand, can enjoy antisocial rewards, but due to their lack of contributions to the public good they ultimately succumb to their inherent inability to secure a sustainable future. Strategies that facilitate the aggregation of akin players, even if they seek to promote antisocial behaviour, thus always enhance the long-term benefits of cooperation.

Evolution of conditional cooperation under multilevel selection

We study the emergence of conditional cooperation in the presence of both intra-group and inter-group selection. Individuals play public goods games within their groups using conditional strategies, which are represented as piecewise linear response functions. Accordingly, groups engage in conflicts with a certain probability. In contrast to previous studies, we consider continuous contribution levels and a rich set of conditional strategies, allowing for a wide range of possible interactions between strategies. We find that the existence of conditional strategies enables the stabilization of cooperation even under strong intra-group selection. The strategy that eventually dominates in the population has two key properties: (i) It is unexploitable with strong intra-group selection; (ii) It can achieve full contribution to outperform other strategies in the inter-group selection. The success of this strategy is robust to initial conditions as well as changes to important parameters. We also investigate the influence of different factors on cooperation levels, including group conflicts, group size, and migration rate. Their effect on cooperation can be attributed to and explained by their influence on the relative strength of intra-group and inter-group selection.

Evolution of public cooperation in a monitored society with implicated punishment and within-group enforcement

Monitoring with implicated punishment is common in human societies to avert freeriding on common goods. But is it effective in promoting public cooperation? We show that the introduction of monitoring and implicated punishment is indeed effective, as it transforms the public goods game to a coordination game, thus rendering cooperation viable in infinite and finite well-mixed populations. We also show that the addition of within-group enforcement further promotes the evolution of public cooperation. However, although the group size in this context has nonlinear effects on collective action, an intermediate group size is least conductive to cooperative behaviour. This contradicts recent field observations, where an intermediate group size was declared optimal with the conjecture that group-size effects and within-group enforcement are responsible. Our theoretical research thus clarifies key aspects of monitoring with implicated punishment in human societies, and additionally, it reveals fundamental group-size effects that facilitate prosocial collective action.

Benefits of tolerance in public goods games

Leaving the joint enterprise when defection is unveiled is always a viable option to avoid being exploited. Although loner strategy helps the population not to be trapped into the tragedy of the commons state, it could offer only a modest income for nonparticipants. In this paper we demonstrate that showing some tolerance toward defectors could not only save cooperation in harsh environments but in fact results in a surprisingly high average payoff for group members in public goods games. Phase diagrams and the underlying spatial patterns reveal the high complexity of evolving states where cyclic dominant strategies or two-strategy alliances can characterize the final state of evolution. We identify microscopic mechanisms which are responsible for the superiority of global solutions containing tolerant players. This phenomenon is robust and can be observed both in well-mixed and in structured populations highlighting the importance of tolerance in our everyday life.

Group Size Effect on Cooperation in One-Shot Social Dilemmas II: Curvilinear Effect

In a world in which many pressing global issues require large scale cooperation, understanding the group size effect on cooperative behavior is a topic of central importance. Yet, the nature of this effect remains largely unknown, with lab experiments insisting that it is either positive or negative or null, and field experiments suggesting that it is instead curvilinear. Here we shed light on this apparent contradiction by considering a novel class of public goods games inspired to the realistic scenario in which the natural output limits of the public good imply that the benefit of cooperation increases fast for early contributions and then decelerates. We report on a large lab experiment providing evidence that, in this case, group size has a curvilinear effect on cooperation, according to which intermediate-size groups cooperate more than smaller groups and more than larger groups. In doing so, our findings help fill the gap between lab experiments and field experiments and suggest concrete ways to promote large scale cooperation among people.

Evolutionary dynamics of synergistic and discounted group interactions in structured populations

The emergence of cooperation between unrelated individuals enables researchers to study how the collective cooperative behavior survives in a world where egoists could get more short-term benefits. The spatial multi-player games, which invoke interactions between individuals who are not directly linked by the interactive networks, are drawing more and more attention in exploring the evolution of cooperation. Here we address the evolutionary dynamics in infinite structured populations with discounted, linear, and synergistic group interactions. The five classical scenarios are recovered from the dynamics: (i) dominating defection, (ii) dominating cooperation, (iii) co-existence, (iv) bi-stability, and (v) neutral variants. For linear interactions, the evolutionary dynamics is equivalent to that in finite as well as the well-mixed counterparts, which can be achieved by a payoff matrix transformation, and it illustrates that the more neighbors there are, the harder the cooperators survive. Yet both cooperation and defection emerge easier in finite populations than in infinite for discounted and synergistic interactions. Counterintuitively, we find that the synergistic group interactions always raise cooperators׳ barriers to occupy the population with the increase of the number of neighbors in infinite structured populations. Our results go against the common belief that synergistic interactions are necessarily beneficial for the cooperative behavior.

Spatial evolutionary public goods game on complete graph and dense complex networks

We study the spatial evolutionary public goods game (SEPGG) with voluntary or optional participation on a complete graph (CG) and on dense networks. Based on analyses of the SEPGG rate equation on finite CG, we find that SEPGG has two stable states depending on the value of multiplication factor r, illustrating how the "tragedy of the commons" and "an anomalous state without any active participants" occurs in real-life situations. When r is low (<<), the state with only loners is stable, and the state with only defectors is stable when r is high (>>). We also derive the exact scaling relation for r*. All of the results are confirmed by numerical simulation. Furthermore, we find that a cooperator-dominant state emerges when the number of participants or the mean degree, 〈k〉, decreases. We also investigate the scaling dependence of the emergence of cooperation on r and 〈k〉. These results show how "tragedy of the commons" disappears when cooperation between egoistic individuals without any additional socioeconomic punishment increases.

Group size effect on cooperation in one-shot social dilemmas

Social dilemmas are central to human society. Depletion of natural resources, climate protection, security of energy supply, and workplace collaborations are all examples of social dilemmas. Since cooperative behaviour in a social dilemma is individually costly, Nash equilibrium predicts that humans should not cooperate. Yet experimental studies show that people do cooperate even in anonymous one-shot interactions. In spite of the large number of participants in many modern social dilemmas, little is known about the effect of group size on cooperation. Does larger group size favour or prevent cooperation? We address this problem both experimentally and theoretically. Experimentally, we find that there is no general answer: it depends on the strategic situation. Specifically, we find that larger groups are more cooperative in the Public Goods game, but less cooperative in the N-person Prisoner's dilemma. Theoretically, we show that this behaviour is not consistent with either the Fehr & Schmidt model or (a one-parameter version of) the Charness & Rabin model, but it is consistent with the cooperative equilibrium model introduced by the second author.

Asymmetric interaction paired with a super-rational strategy might resolve the tragedy of the commons without requiring recognition or negotiation

Avoiding the tragedy of the commons requires that one or more individuals in a group or partnership “volunteer”, benefiting the group at a cost to themselves. Recognition and negotiation with social partners can maintain cooperation, but are often not possible. If recognition and negotiation are not always the mechanism by which cooperative partnerships avoid collective tragedies, what might explain the diverse social cooperation observed in nature? Assuming that individuals interact asymmetrically and that both “weak” and “strong” players employ a super-rational strategy, we find that tragedy of the commons can be avoided without requiring either recognition or negotiation. Whereas in the volunteer's dilemma game a rational “strong” player is less likely to volunteer to provide a common good in larger groups, we show that under a wide range of conditions a super-rational “strong” player is more likely to provide a common good. These results imply that the integration of super-rationality and asymmetric interaction might have the potential to resolve the tragedy of the commons. By illuminating the conditions under which players are likely to volunteer, we shed light on the patterns of volunteerism observed in variety of well-studied cooperative social systems, and explore how societies might avert social tragedies.

Solving the collective-risk social dilemma with risky assets in well-mixed and structured populations

In the collective-risk social dilemma, players lose their personal endowments if contributions to the common pool are too small. This fact alone, however, does not always deter selfish individuals from defecting. The temptations to free ride on the prosocial efforts of others are strong because we are hardwired to maximize our own fitness regardless of the consequences which might have for the public good. Here we show that the addition of risky assets to the personal endowments, both of which are lost if the collective target is not reached, can contribute to solving the collective-risk social dilemma. In infinite well-mixed populations, risky assets introduce new stable and unstable mixed steady states, whereby the stable mixed steady state converges to full cooperation as either the risk of collective failure or the amount of risky assets increases. Similarly, in finite well-mixed populations, the introduction of risky assets enforces configurations where cooperative behavior thrives. In structured populations cooperation is promoted as well, but the distribution of assets among the groups is crucial. Surprisingly, we find that the completely rational allocation of assets only to the most successful groups is not optimal, and this regardless of whether the risk of collective failure is high or low. Instead, in low-risk situations bounded rational allocation of assets works best, while in high-risk situations the simplest uniform distribution of assets among all the groups is optimal. These results indicate that prosocial behavior depends sensitively on the potential losses individuals are likely to endure if they fail to cooperate.

Intergroup information exchange drives cooperation in the public goods game

In this paper we explore the onset of cooperative traits in the public goods game. This well-known game involves N-agent interactions and thus reproduces a large number of social scenarios in which cooperation appears to be essential. Many studies have recently addressed how the structure of the interaction patterns influences the emergence of cooperation. Here we study how information about the payoffs collected by each individual in the different groups it participates in influences the decisions made by its group partners. Our results point out that cross-information plays a fundamental and positive role in the evolution of cooperation for different versions of the public goods game and different interaction structures.

Evolution of cooperation in spatial traveler's dilemma game

Traveler's dilemma (TD) is one of social dilemmas which has been well studied in the economics community, but it is attracted little attention in the physics community. The TD game is a two-person game. Each player can select an integer value between and () as a pure strategy. If both of them select the same value, the payoff to them will be that value. If the players select different values, say and (), then the payoff to the player who chooses the small value will be and the payoff to the other player will be . We term the player who selects a large value as the cooperator, and the one who chooses a small value as the defector. The reason is that if both of them select large values, it will result in a large total payoff. The Nash equilibrium of the TD game is to choose the smallest value . However, in previous behavioral studies, players in TD game typically select values that are much larger than , and the average selected value exhibits an inverse relationship with . To explain such anomalous behavior, in this paper, we study the evolution of cooperation in spatial traveler's dilemma game where the players are located on a square lattice and each player plays TD games with his neighbors. Players in our model can adopt their neighbors' strategies following two standard models of spatial game dynamics. Monte-Carlo simulation is applied to our model, and the results show that the cooperation level of the system, which is proportional to the average value of the strategies, decreases with increasing until is greater than the critical value where cooperation vanishes. Our findings indicate that spatial reciprocity promotes the evolution of cooperation in TD game and the spatial TD game model can interpret the anomalous behavior observed in previous behavioral experiments.

Evolutionary dynamics of group interactions on structured populations: a review

Interactions among living organisms, from bacteria colonies to human societies, are inherently more complex than interactions among particles and non-living matter. Group interactions are a particularly important and widespread class, representative of which is the public goods game. In addition, methods of statistical physics have proved valuable for studying pattern formation, equilibrium selection and self-organization in evolutionary games. Here, we review recent advances in the study of evolutionary dynamics of group interactions on top of structured populations, including lattices, complex networks and coevolutionary models. We also compare these results with those obtained on well-mixed populations. The review particularly highlights that the study of the dynamics of group interactions, like several other important equilibrium and non-equilibrium dynamical processes in biological, economical and social sciences, benefits from the synergy between statistical physics, network science and evolutionary game theory.

Inferring reputation promotes the evolution of cooperation in spatial social dilemma games

In realistic world individuals with high reputation are more likely to influence the collective behaviors. Due to the cost and error of information dissemination, however, it is unreasonable to assign each individual with a complete cognitive power, which means that not everyone can accurately realize others’ reputation situation. Here we introduce the mechanism of inferring reputation into the selection of potential strategy sources to explore the evolution of cooperation. Before the game each player is assigned with a randomly distributed parameter p denoting his ability to infer the reputation of others. The parameter p of each individual is kept constant during the game. The value of p indicates that the neighbor possessing highest reputation is chosen with the probability p and randomly choosing an opponent is left with the probability 1−p. We find that this novel mechanism can be seen as an universally applicable promoter of cooperation, which works on various interaction networks and in different types of evolutionary game. Of particular interest is the fact that, in the early stages of evolutionary process, cooperators with high reputation who are easily regarded as the potential strategy donors can quickly lead to the formation of extremely robust clusters of cooperators that are impervious to defector attacks. These clusters eventually help cooperators reach their undisputed dominance, which transcends what can be warranted by the spatial reciprocity alone. Moreover, we provide complete phase diagrams to depict the impact of uncertainty in strategy adoptions and conclude that the effective interaction topology structure may be altered under such a mechanism. When the estimation of reputation is extended, we also show that the moderate value of evaluation factor enables cooperation to thrive best. We thus present a viable method of understanding the ubiquitous cooperative behaviors in nature and hope that it will inspire further studies to resolve social dilemmas.

Velocity-enhanced cooperation of moving agents playing public goods games

In this paper we study the evolutionary dynamics of the public goods game in a population of mobile agents embedded in a two-dimensional space. In this framework, the backbone of interactions between agents changes in time, allowing us to study the impact that mobility has on the emergence of cooperation in structured populations. Our results point out that a low degree of mobility enhances cooperation in the system. In addition, we study the impact of the size of the groups in which games are played on cooperation. Again we find a rise and fall of cooperation related to the percolation point of the instant interaction networks created by the set of mobile agents.

Adaptive and bounded investment returns promote cooperation in spatial public goods games

The public goods game is one of the most famous models for studying the evolution of cooperation in sizable groups. The multiplication factor in this game can characterize the investment return from the public good, which may be variable depending on the interactive environment in realistic situations. Instead of using the same universal value, here we consider that the multiplication factor in each group is updated based on the differences between the local and global interactive environments in the spatial public goods game, but meanwhile limited to within a certain range. We find that the adaptive and bounded investment returns can significantly promote cooperation. In particular, full cooperation can be achieved for high feedback strength when appropriate limitation is set for the investment return. Also, we show that the fraction of cooperators in the whole population can become larger if the lower and upper limits of the multiplication factor are increased. Furthermore, in comparison to the traditionally spatial public goods game where the multiplication factor in each group is identical and fixed, we find that cooperation can be better promoted if the multiplication factor is constrained to adjust between one and the group size in our model. Our results highlight the importance of the locally adaptive and bounded investment returns for the emergence and dominance of cooperative behavior in structured populations.

Sustainable institutionalized punishment requires elimination of second-order free-riders

Although empirical and theoretical studies affirm that punishment can elevate collaborative efforts, its emergence and stability remain elusive. By peer-punishment the sanctioning is something an individual elects to do depending on the strategies in its neighborhood. The consequences of unsustainable efforts are therefore local. By pool-punishment, on the other hand, where resources for sanctioning are committed in advance and at large, the notion of sustainability has greater significance. In a population with free-riders, punishers must be strong in numbers to keep the "punishment pool" from emptying. Failure to do so renders the concept of institutionalized sanctioning futile. We show that pool-punishment in structured populations is sustainable, but only if second-order free-riders are sanctioned as well, and to a such degree that they cannot prevail. A discontinuous phase transition leads to an outbreak of sustainability when punishers subvert second-order free-riders in the competition against defectors.