Cooperation is essential to the prosperity of human and biological systems. Its emergence depends on not only on the structure of interactions within a single system but also the coupling between interdependent systems. While the analytical theory for strategy evolution in isolated systems is well established, the effect of coupling has been studied mainly through specific models and simulations. Here, we develop a general analytical framework for strategy evolution on interdependent networks with arbitrary within-system structures and cross-system coupling. We derive the explicit condition under which natural selection favors cooperation and show that coupling can promote cooperation even when no isolated system sustains it. Cooperation is maximized under weak but non-zero coupling, while excessively strong interdependence suppresses it. Cooperation is further enhanced when strategy updating is biased toward the larger or less cooperation-prone system. Our analysis explains when and why coupling promotes cooperation.

Understanding the emergence of cooperation in social networks has advanced through pairwise interactions, but the corresponding theory for group-based public goods games (PGGs) remains less explored. Here, we provide theoretical conditions under which cooperation thrives in PGGs on arbitrary population structures, which are accurate under weak selection. We find that a class of networks that would otherwise fail to produce cooperation, such as star graphs, are particularly conducive to cooperation in PGGs. More generally, PGGs can support cooperation on almost all networks, which is robust across all kinds of model details. This fundamental advantage of PGGs derives from self-reciprocity realized by group separations and from clustering through second-order interactions. We also apply PGGs to empirical networks, which shows that PGGs could be a promising interaction mode for the emergence of cooperation in real-world systems.

Cooperation is typically seen as the ideal outcome in a social dilemma. Because cooperators are vulnerable to exploitation, much of the literature has focused on mechanisms, such as spatial structure, that support prosocial behaviour and the production of public goods1-11. Yet the rules for distributing these goods also shape behaviour and long-term prosperity. Here we study policies for allocating public goods, comparing equitable allocation, in which returns are proportional to potential contributions, with uniform allocation, in which all individuals receive equal shares. For most social networks, we find that uniform allocation facilitates the spread of cooperation compared with equitable allocation. But this success comes with a cost. Uniform allocation concentrates resources in a small number of highly connected individuals12, whereas peripheral individuals receive fewer benefits and may even be worse off than in a non-cooperative society. We develop a theoretical analysis of the tension between cooperation and equality, and we identify this conflict across diverse empirical social networks. Our results show that inequality may be an unavoidable consequence of allocation policies designed to foster cooperation in spatially heterogeneous populations. The question of how to promote cooperation is therefore incomplete: because policies that facilitate cooperation can also generate social stratification, we must weigh the benefits of cooperation against the inequality that accompanies it.

Evolutionary game theory offers a general framework to study how behaviors evolve by social learning in a population. This body of theory can accommodate a range of social dilemmas, or games, as well as real-world complexities such as spatial structure or behaviors conditioned on reputations. Nonetheless, this approach typically assumes a deterministic payoff structure for social interactions. Here, we extend evolutionary game theory to account for random changes in the social environment, so that mutual cooperation may bring different rewards today than it brings tomorrow, for example. Even when such environmental noise is unbiased, we find it can have a qualitative impact on the behaviors that evolve in a population. Noisy payoffs can permit the stable co-existence of cooperators and defectors in the prisoner's dilemma, for example, as well as bistability in snowdrift games and stable limit cycles in rock-paper-scissors games -- dynamical phenomena that cannot occur in the absence of noise. We conclude by discussing the relevance of our framework to scenarios where the nature of social interactions is subject to external perturbations.

Cooperation is vital for the survival of living systems but is challenging due to the costs borne by altruistic individuals. Direct reciprocity, where actions are based on past encounters, is a key mechanism fostering cooperation. However, most studies assume synchronous decision-making, whereas real-world interactions are often asynchronous, with individuals acting in sequence. This asynchrony can undermine standard cooperative strategies like Tit-for-Tat and Win-Stay Lose-Shift. To better understand cooperation in real-world contexts, it is crucial to explore the theory of direct reciprocity in asynchronous interactions. To address this, we introduce a framework based on asynchronous stochastic games, incorporating asynchronous decisions and dynamic environmental feedback. We analytically derive the conditions under which strategies form cooperative Nash equilibria. Our results demonstrate that the order of interactions can significantly alter outcomes: interaction asynchrony generally inhibits cooperation, except under specific conditions where environmental feedback effectively mitigates its negative impact. When environmental feedback is incorporated, a variety of stable reciprocal strategies can be sustained. Notably, above a critical environmental threshold, any cooperative strategy can form a Nash equilibrium. Overall, our work underscores the importance of interaction order in long-term evolutionary processes and highlights the pivotal role of environmental feedback in stabilizing cooperation in asynchronous interactions.

Human decision-making is shaped by underlying motivations, which reflect both subjective well-being and fundamental biological needs. Different needs are often prioritized and traded off against one another. Here we develop a theoretical framework to study the evolution of behavioral motivations, encompassing both philanthropic (cooperating after personal needs are met) and aspirational (cooperating to fulfill personal needs) motivations. Our findings show that when the ratio of benefits to costs for cooperation exceeds a critical threshold, individuals initially driven by aspirational motivations can transition to philanthropic motivations with a low reference point for cooperation, resulting in increased cooperation. Furthermore, the critical threshold depends on the structure of the underlying social network, with network modifications capable of reversing the evolutionary trajectory of motivations. Our results reveal the complex interplay between needs, motivations, social networks, and decision-making, offering insights into how evolution shapes not only cooperative behaviors but also the motivations behind them.

The awareness of individuals regarding their social network surroundings and their capacity to use social connections to their advantage are well-established human characteristics. Economic games, incorporated with network science, are frequently used to examine social behaviour. Traditionally, such game models and experiments artificially limit players' abilities to take varied actions towards distinct social neighbours, thereby constraining their social networking agency. Here we designed an experimental paradigm that alters this agency and applied it to the prisoner's dilemma (N = 735), trust game (N = 735) and ultimatum game (N = 735) to investigate cooperation, trust and fairness. Granting participants greater network agency led to more prosocial behaviour across all three economic games, resulting in higher wealth and lower inequality compared with control groups. These findings suggest that incorporating social networking agency into experimental designs better captures the prosocial potential of human behaviour.

Human social life is shaped by repeated interactions, where past experiences guide future behavior. In evolutionary game theory, a key challenge is to identify strategies that harness such memory to succeed in repeated encounters. Decades of research have identified influential one-step memory strategies (such as Tit-for-Tat, Generous Tit-for-Tat, and Win-Stay Lose-Shift) that promote cooperation in iterated pairwise games. However, these strategies occupy only a small corner of the vast strategy space, and performance in isolated pairwise contests does not guarantee evolutionary success. The most effective strategies are those that can spread through a population and stabilize cooperation. We propose a general framework for repeated-interaction strategies that encompasses arbitrary memory lengths, diverse informational inputs (including both one's own and the opponent's past actions), and deterministic or stochastic decision rules. We analyze their evolutionary dynamics and derive general mathematical results for the emergence of cooperation in any network structure. We then introduce a unifying indicator that quantifies the contribution of repeated-interaction strategies to population-level cooperation. Applying this indicator, we show that long-memory strategies evolve to promote cooperation more effectively than short-memory strategies, challenging the traditional view that extended memory offers no advantage. This work expands the study of repeated interactions beyond one-step memory strategies to the full spectrum of memory capacities. It provides a plausible explanation for the high levels of cooperation observed in human societies, which traditional one-step memory models cannot account for.

Interactions in the real world are not limited to pairwise relationships but often involve higher-order structures. However, the evolution of cooperation in the presence of both interaction types remains poorly understood, especially regarding which mechanisms can sustain cooperation in social dilemmas. Among these, punishment is widely recognized as a regulatory mechanism that can discourage non-cooperative behavior and potentially promote collective action. In this study, we propose a multi-strategy game framework on hypergraphs to investigate how punishment promotes cooperation in structured populations facing social dilemmas, where pairwise and higher-order interactions coexist. Within this framework, pairwise and higher-order interactions are modeled as two-player and multi-player games, respectively, and individuals participate concurrently across both types. Through stability analysis, we find that punishment can trigger a secondary cooperation effect. Remarkably, cooperation can be effectively promoted regardless of how severe the punishment is—even when its cost exceeds the net loss imposed on the punished individual. Extending the analysis to finite-size populations, we observe that small groups tend to suppress cooperation, whereas large groups promote higher overall cooperation levels. We further validate our findings via numerical simulations on three empirical network structures, all of which confirm that punishment leads to enhanced cooperation. Our results demonstrate that punishment serves as an effective and scalable driver of cooperation in systems with complex interaction structures.

In large-scale networked systems composed of autonomous decision-making agents, the structure of the network determines who interacts with and learns from whom, thereby playing a pivotal role in shaping collective behaviors such as cooperation and coordination. Importantly, both interaction and learning structures are dynamic, with connections and their intensities evolving over time. Understanding how these time-varying structures influence strategy adoption and, in turn, drive collective dynamics is a challenging yet essential research problem with significant practical implications. To address this, we introduce a general framework for evolutionary games on temporal networks. This framework is highly flexible, accommodating games with any number of available strategies, temporal networks with any number of network snapshots and arbitrary structures within each snapshot. Within this framework, we establish the first mathematical condition—a structure-coefficient theorem for temporal networks—that succinctly captures the effects of all structural factors on evolutionary outcomes using only three coefficients. This theorem provides a rigorous foundation for analyzing the dynamic adoption of strategies in time-varying networks. To demonstrate the utility of our approach, we apply the framework to public goods games on temporal networks. Our results reveal that temporal network structures can significantly enhance collective cooperation compared to each static snapshot. These findings offer novel theoretical insights into how time-varying network structures drive the emergence of collective intelligence, paving the way for a deeper understanding of adaptive systems in complex and dynamic environments.

Multiagent learning is challenging when agents face mixed-motivation interactions, where conflicts of interest arise as agents independently try to optimize their respective outcomes. Recent advancements in evolutionary game theory have identified a class of "zero-determinant" strategies, which confer an agent with significant unilateral control over outcomes in repeated games. Building on these insights, we present a comprehensive generalization of zero-determinant strategies to stochastic games, encompassing dynamic environments. We propose an algorithm that allows an agent to discover strategies enforcing predetermined linear (or approximately linear) payoff relationships. Of particular interest is the relationship in which both payoffs are equal, which serves as a proxy for fairness in symmetric games. We demonstrate that an agent can discover strategies enforcing such relationships through experience alone, without coordinating with an opponent. In finding and using such a strategy, an agent ("enforcer") can incentivize optimal and equitable outcomes, circumventing potential exploitation. In particular, from the opponent's viewpoint, the enforcer transforms a mixed-motivation problem into a cooperative problem, paving the way for more collaboration and fairness in multiagent systems.

A collaborative group can often outperform a single individual in complex problem solving, even when information is limited. This phenomenon, called collective intelligence, can be achieved by engineering a central planner who assigns subtasks distributed across the group. But such algorithms cannot explain how natural populations, which often lack sophisticated central control, can nonetheless evolve collective intelligence. In fact, the process of social learning by imitating successful peers will typically reduce diversity and inhibit collective intelligence. Here, we consider a prediction task where the true outcome each round is a continuous quantity that depends linearly on a large number of random causal factors. Each individual can observe only one factor, and the collective prediction is generated by aggregating personal predictions across individuals. We propose two classes of reward structures that guarantee the emergence of collective intelligence through social learning. One scheme provides greater rewards to those individuals (called experts) whose personal predictions are more accurate. The other scheme provides greater rewards to those individuals (called reformers) whose predictions have greater potential to reduce the collective error, even though their personal predictions may be far from the truth. Although both of these payoff structures can provably maintain diversity and establish collective intelligence, we show that rewards based on collective error are more robust to diverse problem settings than rewards based on personal accuracy. Our results show that identifying reformers is more effective than identifying experts in promoting the emergence of collective intelligence.

Cooperation on social networks is crucial for understanding human survival and development. Although network structure has been found to significantly influence cooperation, human experiments have observed different cooperation phenomena under similar conditions. While evidence suggests that these differences arise from human exploration, our understanding of its impact mechanisms and characteristics remains limited. Here, we seek to formalize human exploration as an individual learning process involving trial and reflection, and integrate social learning to examine how their interdependence shapes cooperation. We find that individual learning can alter neighbor imitation tendencies, and the resulting shifts in the local cooperative environment feed back into the experiential cognition that guides individual learning. This coupled dynamic makes the ability of social networks to promote cooperation largely dependent on whether individuals focus on long-term payoffs, and exhibits a series of characteristics that can explain previously unexplained and seemingly contradictory cooperation phenomena. Surprisingly, individual learning can promote cooperation more than social learning when its probability is negatively correlated with payoffs, a mechanism rooted in the psychological tendency to avoid trial-and-error when individuals are satisfied with their current payoffs. These results explain the contradictory cooperation phenomenon by accounting for decision preferences and cognitive processes underlying exploration, bridging the gap between theoretical research and reality.

Efficient allocation and use of limited resources are fundamental to advancing collective welfare and achieving long-term societal sustainability. This challenge involves not only how policymakers distribute scarce resources among individuals, but also how individuals strategically utilize them. The complexity deepens when individuals are embedded in networks of social interactions, where outcomes are interdependent and future decisions are shaped by a dynamic tension between cooperation driven by collective long-term benefit and self-interest motivated by short-term personal gain. Here, we introduce a novel framework of generalized public goods games on hypergraphs to capture the multifaceted nature of real-world social interactions. Using Nash equilibrium analysis, we reveal how full cooperation (all individuals contribute all their resources to maximize collective benefit) emerges from the interplay between resource allocation strategies, individual usage behaviors, and the structure of interactions. We find that equal resource distribution enhances cooperation in homogeneous networks but may suppress it in heterogeneous ones, indicating that equity in allocation does not universally lead to optimal collective outcomes. To address this, we propose two complementary optimization strategies: one to guide policymakers in designing effective resource allocation schemes, and the other to support individuals in making sustainable use decisions. We validate the effectiveness of both approaches across a range of synthetic and empirical cases. Our findings provide actionable insights for designing governance frameworks and resource management policies that promote sustainable cooperation in complex socio-environmental systems.

The orderly behaviors observed in large-scale groups, such as fish schooling and the organized movement of crowds, are both ubiquitous and essential for the survival and stability of these systems. Understanding how such complex collective behaviors emerge from simple local interactions and behavioral adjustments is a significant scientific challenge. Historically, research has predominantly focused on imitation and social learning, where individuals adopt the strategies of more successful peers to refine their behavior. However, in recent years, an alternative learning approach based on self-exploration and introspective learning has garnered increasing attention. In this paradigm, individuals assess their own circumstances and select strategies that best align with their specific conditions. Two examples are coordination and anti-coordination, where individuals align with and diverge from the local majority, respectively. In this study, we analyze networked systems of coordinating and anti-coordinating individuals, exploring the combined effects of system dynamics, network structure and behavioral patterns. We address several practical questions, including the number of equilibria, their characteristics, the equilibrium time and the resilience of the system. We find that the number of equilibrium states can be extremely large, even increasing exponentially with minor alterations to the network structure. Moreover, the network structure has a significant impact on the average equilibrium time. Despite the complexity of these findings, we find that variations can be captured by a single, simple network characteristic (the average path length), which we illustrate in both synthetic and empirical networks.

Network reciprocity has been recognized as a crucial mechanism for promoting the evolution of cooperation. However, prior studies on network reciprocity have predominantly focused on isolated networks, neglecting the interconnected nature of real-world systems. In reality, many large-scale systems are closely coupled, where the interaction patterns between systems differ significantly from those within individual networks. Motivated by this observation, we construct a strategy evolution model on coupled networks to explore the effects of such interconnections. Our findings reveal that coupled networks foster cooperation more effectively than isolated networks. This advantage persists across various network structures, suggesting the universality of this phenomenon. Interestingly, we discover that the strength of the coupling plays a pivotal role: weaker coupling between networks is more conducive to the emergence and stability of cooperation compared to stronger coupling. These results high-light the importance of considering cross-system interactions in understanding and designing mechanisms to enhance cooperative behavior.

Humans update their social behavior in response to past experiences and changing environments. Behavioral decisions are further complicated by uncertainty in the outcome of social interactions. Faced with uncertainty, some individuals exhibit risk aversion while others seek risk. Attitudes toward risk may depend on socioeconomic status; and individuals may update their risk preferences over time, which will feedback on their social behavior. Here, we study how uncertainty and risk preferences shape the evolution of social behaviors. We extend the game-theoretic framework for behavioral evolution to incorporate uncertainty about payoffs and variation in how individuals respond to this uncertainty. We find that different attitudes toward risk can substantially alter behavior and long-term outcomes, as individuals seek to optimize their rewards from social interactions. In a standard setting without risk, for example, defection always overtakes a well-mixed population engaged in the classic Prisoner’s Dilemma, whereas risk aversion can reverse the direction of evolution, promoting cooperation over defection. When individuals update their risk preferences along with their strategic behaviors, a population can oscillate between periods dominated by risk-averse cooperators and periods of risk-seeking defectors. Our analysis provides a systematic account of how risk preferences modulate, and even coevolve with, behavior in an uncertain social world.

Past decades have seen numerous studies about evolutionary games on networks, and most of them have been based on the assumption of exact payoff information. However, in both natural and engineering systems, interactions between agents are often subject to various perturbations, arising from environmental change and the disturbance of communications, and the loss of information. These often lead to perturbations in the payoff one derives. Here we propose a model of evolutionary games with payoff perturbations in networked systems and aim to investigate the evolutionary dynamics in the presence of perturbed payoffs. We provide an analytical condition to predict the strategy evolution. Surprisingly, compared with the evolutionary outcomes with exact payoffs, payoff perturbations relax the condition for the establishment of collective cooperation. Our work suggests that the perturbations occurring in interactions can be of great importance in establishing collective intelligence.

The concept of fitness is central to evolution, but it quantifies only the expected number of offspring an individual will produce. The actual number of offspring is also subject to demographic stochasticity-that is, randomness associated with birth and death processes. In nature, individuals who are more fecund tend to have greater variance in their offspring number. Here, we develop a model for the evolution of two types competing in a population of nonconstant size. The fitness of each type is determined by pairwise interactions in a prisoner's dilemma game, and the variance in offspring number depends upon its mean. Although defectors are preferred by natural selection in classical population models, since they always have greater fitness than cooperators, we show that sufficiently large offspring variance can reverse the direction of evolution and favor cooperation. Large offspring variance produces qualitatively new dynamics for other types of social interactions, as well, which cannot arise in populations with a fixed size or with a Poisson offspring distribution.