Information sharing between individuals is crucial to improve performance in collective tasks. However, in a competitive world, individuals may be reluctant to share information with the others, and it is still unclear how the presence of strategic behaviors affects the collective performance of a group. In this study, we introduce an evolutionary game modeling the dynamics of individual behaviors in a collective estimation task. The individuals are organized in a network and have to guess the distribution of ball colors in a box. Each of them samples a given number of balls and can strategically decide whether to share or not this information with its neighbors. We develop a framework that allows to investigate analytically how the collective performance depends on the network structure. We find that the optimal network results from a trade-off between the sharing rate and the way the information is integrated in the network. We further reveal that there exists an intermediate average degree for each type of network maximizing the collective performance. In addition to the uniform case, we consider the case of non-homogeneous allocations of the number of individual samples, showing that the largest collective performance is obtained when the number of ball extracted by an individual is inversely proportional to its degree.

Conflicts of interest arise across biology, social sciences, and artificial intelligence, where self-interested agents often struggle to resolve them, leading to socially inefficient outcomes. In game theory and evolutionary game theory, the iterated prisoner’s dilemma (IPD) has long served as the canonical model for studying such dilemmas. Yet, despite its central role, the two-action IPD oversimplifies real-world interactions. By contrast, multi-action models are better suited to capture the richness of adaptive behavior, but systematic theoretical work in this direction remains scarce, as extending the action space greatly complicates both theoretical and simulation analyses. We advance this line of research by extending the IPD framework to an n-player, three-action setting, through which we identify cooperative equilibria that emerge more readily—those sustaining fair outcomes as Nash equilibria over a wider parameter domain—and reveal a fundamental mechanism for conflict resolution. In particular, our analysis identifies conditions under which partner equilibria can be sustained more readily in three-action games than in two-action ones. We further conduct an evolutionary analysis in which agents learn by imitating higher-performing strategies, revealing how adaptive behavior enables them to align incentives with opponents and promote fair and efficient outcomes even when opponents pursue only their own payoffs. Taken together, our results advance the theoretical foundations of repeated multi-action games and offer insights into how fairness and cooperation can emerge in multiagent systems characterized by conflicting interests.

Decision-making in natural and artificial systems is often shaped by past interactions, as individuals adjust their behavior accordingly. A persistent challenge lies in identifying successful strategies and understanding how memory influences their performance, especially when individuals’ optimal actions conflict with collective outcomes. Most theoretical studies in evolutionary dynamics have focused on memory-1 strategies, and recent studies indicate that extending memory can improve strategies’ performance. Among these, all-or-none (AoNK) strategies perform well at short memory. Our analysis reveals, however, that their effectiveness deteriorates as memory extends, showing the inherent limitations of sole coordination. Building on this, we propose the adaptive coordination strategy, combining coordination and tolerance to enhance adaptability and stability. Our theoretical analysis precisely characterizes its behavior and key properties. This strategy outperforms classic memory-1 strategies and achieves optimal outcomes, providing a framework for understanding complex history-based behaviors beyond traditional memory-n models.

Uncertainty is ubiquitous in natural and engineered systems, influencing both individual decisions and collective dynamics. While robust control theory has provided powerful tools for analyzing how uncertainty affects engineered systems, the study of robustness in game theory remains limited. This paper introduces three fundamental models to capture distinct sources of uncertainty: Environmental stochasticity affecting the payoff structure, demographic fluctuations arising from stochastic reproduction, and perceptual uncertainty shaped by noisy observation and subjective risk preference, extending the concept of robustness to strategic and evolutionary systems. Together, these models reveal how uncertainty at different levels-external, internal, and cognitive-can reshape evolutionary outcomes, alter stability, and generate complex dynamical patterns such as coexistence, multistability, and oscillations. Based on these theoretical foundations, the authors further study the evolution of cooperation in variable-sized populations with mutation. The analysis shows that when the population is divided into multiple subpopulations, migration among subgroups can effectively protect cooperators from the invasion of defectors and sustain cooperation even under mutation. This result demonstrates how robustness concepts can elucidate the emergence and persistence of cooperation in uncertain and heterogeneous environments.

Cooperation is a key driver of human social progress. Studies of the evolution of cooperation typically assume a deterministic outcome for social interactions. But in real-world social interactions, interaction outcomes are often subject to stochastic perturbations arising from open environments. Individuals may show different attitudes towards such uncertainty, some are risk-seeking, while others tend to be risk-averse. Here we investigate how risk preference towards uncertain payoffs affects the evolution of cooperation on social networks, where uncertainty originates from random punishment of defectors initiated by cooperators. We provide an analytical treatment of how the distribution of risk preference among individuals alters the threshold required for cooperation. We find that, at the population level, risk-averse behavior promotes or even rescues cooperation. At the node level, variation in risk preference has a significant impact when it occurs on nodes with high degree centrality. When nodes have the same degree centrality, the nodes with lower betweenness centrality exhibit a stronger effect on strategy evolution. Our analysis reveals how risk preference, together with spatial structure, jointly shapes and potentially reverses the evolutionary dynamics of cooperation.

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.

In this paper, the problem of distributed optimization is studied via a network of agents. Each agent only has access to a noisy gradient of its own objective function, and can communicate with its neighbors via a network. To handle this problem, a distributed clipped stochastic gradient descent algorithm is proposed, and the high probability convergence of the algorithm is studied. Existing works on distributed algorithms involving stochastic gradients only consider the light-tailed noises. Different from them, we study the case with heavy-tailed settings. Under mild assumptions on the graph connectivity, we prove that the algorithm converges in high probability under a certain clipping operator. Finally, a simulation is provided to demonstrate the effectiveness of our theoretical results

In this article, the problem of distributively seeking the equilibria of aggregative games with bilevel structures is studied. Different from the traditional aggregative games, here, the aggregation is determined by the minimizer of a virtual leader's objective function in the inner level. Moreover, the global objective function of the virtual leader is formed by the sum of local functions, each of which is determined by the local action of a player. When making decisions, each player only has access to a local part of the virtual leader's objective function and can communicate with its neighbors via a connected graph. To handle this problem, first, we propose a second-order gradient-based distributed algorithm, where the Hessian matrices associated with the objective functions of the leader are involved. Under mild assumptions on the graph and cost functions, we prove that the actions of players asymptotically converge to the Nash equilibrium point. Then, for the case where the Hessian matrices associated with the objective functions of the virtual leader are not available, we propose a first-order gradient-based distributed algorithm, where a distributed estimate strategy is developed to estimate the gradients of players' cost functions in the outer level. Under the same conditions, we prove that the convergence errors of players' actions to the Nash equilibrium point are linear with respect to the estimate parameters. Finally, simulations are provided to demonstrate the effectiveness of our theoretical results.

Understanding the evolution of cooperation in structured populations remains a central challenge in multidisciplinary areas. Although previous findings suggest that structural heterogeneity in static networks hinders cooperation, real-world interactions in most natural and social systems are dynamic and best represented as temporal networks. Here, we challenge this conventional wisdom and, by developing a systematic mathematical framework, we report that structural heterogeneity in temporal networks can instead promote collective cooperation. Importantly, we reveal that such advantages depend on an often-overlooked metric-fixation time-quantifying the time required for a single cooperator to drive the entire population to cooperation. Highly heterogeneous networks accelerate this process within each subnetwork, resulting in a quantitative enhancement of cooperation in temporal networks compared to their homogeneous counterparts. By validating our results on empirical datasets through theoretical analyses and simulations, we provide a consistent framework for analysing cooperative dynamics across static and temporal networked systems.

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.

The stability of complex ecosystems, which indicates the ability of the system to recover from perturbations, is profoundly influenced by the environment. Recent findings point out that changing environments with the renewal and decay of resources can alter ecological stability by affecting species abundances and interactions. These changes in turn affect the environment, establishing a feedback loop between environments and species. However, studies on ecological stability have primarily been based on static environments, ignoring feedbacks between species and environments. Here, we study ecological stability with environmental feedbacks, considering the co-evolutionary dynamics of species abundances, interactions and environments. We find that environmental feedbacks, while increasing the complexity of ecosystems, generally enhance ecological stability by reducing the noise (variance) of species interaction strengths. Furthermore, we derive stability criteria for multiple interaction types among species and identify the optimal resource stock to promote ecological stability. Our results hold across various realistic scenarios, such as heterogeneous communities, highlighting the significant role of environmental feedbacks on ecological stability.

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.

is ubiquitous in biological and social systems, yet it faces persistent challenges from free-riding behavior. While voluntary participation has been recognized as a key mechanism for sustaining cooperation, existing studies predominantly assume static decision-making rules, namely loner strategy or fixed participation probability, overlooking the dynamic nature of human participation strategies. To address this gap, we employ the Bush-Mosteller reinforcement learning algorithm to model aspiration-driven adaptive participation in public goods games. Our results reveal that cooperation peaks when the aspiration level equals the potential maximum payoff of cooperators, with distinct evolutionary mechanisms emerging on either side of this critical value. Below the threshold, cooperators form self-organizing defensive barriers through strategic withdrawal, effectively mitigating exploitation risks. Above the threshold, enhanced reciprocity within cooperative clusters generates positive network externalities, enabling cooperative expansion through benefit radiation effects. These findings provide novel insights into how adaptive participation strategies shape the evolution of cooperation, highlighting the importance of dynamic decision-making processes in social dilemmas.

Direct reciprocity is a fundamental mechanism for sustaining cooperation in repeated interactions, where individuals adjust their behavior based on past experiences. Most previous models have focused on the prisoner's dilemma, in which individuals face a strict choice between full cooperation and complete defection. However, this dichotomy oversimplifies the complexity of real-world reciprocal interactions. To address this, we introduce additional actions between these extremes, thereby increasing action diversity. Our analysis demonstrates that a broader range of available actions fosters cooperation more effectively than a binary choice. Through evolutionary analysis, we identify which types of intermediate actions promote cooperation. Moreover, equilibrium analysis establishes the theoretical conditions underlying this effect. While the increased computational complexity makes it infeasible to simulate scenarios with an arbitrarily large number of actions, our theoretical analysis remains applicable to settings with more actions, offering broader insights into the role of action diversity in promoting cooperation. These findings deepen our understanding of direct reciprocity and highlight the importance of action diversity in shaping cooperative behavior.

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.

Fairness plays a key role in collective decision-making such as international trade, burden sharing in global climate change, and prey allocation after hunting. The two-player ultimatum game has facilitated a large body of research on fair behavior. But multiplayer group interactions, group decision rules, and their effects on the evolution of fair behavior remain largely unexplored. We aim to study the evolutionary dynamics of collective resource allocation interactions, and recently advanced hypergraphs can efficiently describe the group interaction relationships. Each hyperlink represents a group and the members participate in multiplayer ultimatum games. Each member acts as proposer one time, and accordingly, all other members act as responders. For one game, when all responders agree upon the proposer's scheme, the scheme succeeds and all members accrue payoffs; otherwise, the scheme fails and no payoff is produced. First, we use adaptive dynamics to demonstrate that the strategy evolves toward rational and unfair solutions for uniform random hypergraphs. However, by incorporating empathy into decision-making, the population is stabilized with fair allocation schemes. Increasing the order of uniform random hypergraphs generally promotes group fairness as long as the group size is not too large. This enhancing effect is further reinforced on heterogeneous hypergraphs, where individuals with higher degrees play a leading role in strategy propagation and thus hold sway over the population dynamics. Our work enriches the literature on the evolutionary dynamics of multiplayer ultimatum games in structured populations.

AIMS:With the rapid growth of cardiac implantable electronic device (CIED) implantations, lead management has become increasingly challenging. Transvenous lead extraction (TLE) is now a primary approach for addressing lead-related complications. However, data on TLE in patients with abandoned leads remain limited. This study evaluates the safety and effectiveness of TLE in patients with abandoned leads. METHODS AND RESULTS:Consecutive patients undergoing TLE between January 2013 and January 2021 were enrolled and stratified into two groups: those with abandoned leads presence (Group 1) and those with active leads only (Group 2). The primary safety endpoint was major complications, and the effectiveness endpoint was clinical success of lead extraction. Among 1109 patients (median age 68 years; 70.7% male), 293 (26.4%) had abandoned leads (Group 1), while 816 (73.6%) did not (Group 2). A total of 2222 leads were extracted. Group 1 exhibited a higher number of leads per patient [2.0 (2.0-3.0) vs. 2.0 (2.0-2.0), P < 0.001], longer dwell time of the oldest lead [120.0 (72.0-174.0) vs. 48.0 (16.0-96.0) months, P < 0.001], a higher prevalence of CIED infection (94.9 vs. 87.4%, P < 0.001) and venous stenosis (51.5 vs. 38.8%, P = 0.001), and a higher proportion of MB scores ≥5 (45.1 vs. 15.0%, P < 0.001) and EROS 3 stratification (23.2 vs. 5.0%, P < 0.001). Clinical success was achieved in 1080 patients (97.4%), with major complications observed in 19 patients (1.7%). No significant differences were found between groups in major complication rates (2.4 vs. 1.5%, P = 0.299) or clinical success rates (96.2 vs. 97.8%, P = 0.154). Group 1 had lower manual traction success rate (8.9 vs. 37.5%, P < 0.001), required more advanced tools or multi-venous approaches (8.5 vs. 3.2%, P < 0.001), and showed higher rate of surgical extraction (3.1 vs. 1.1%, P = 0.030) and emergency thoracotomy (1.4 vs. 0.1%, P = 0.019). The dwell time of the oldest lead and number of leads removed mediated procedural risks. Multivariate analysis identified the dwell time of the oldest lead [odds ratio (OR) 1.15, 95% confidence interval (CI) 1.08-1.21, P < 0.001] and the number of leads removed (OR 2.04, 95% CI 1.29-3.21, P = 0.002) as independent predictors of failure. Similarly, the dwell time of the oldest lead (OR 1.12, 95% CI 1.07-1.18, P < 0.001) and the number of leads removed (OR 1.54, 95% CI 1.04-2.28, P = 0.033) independently predicted complications. CONCLUSION:Transvenous lead extraction in patients with abandoned leads is characterized by clinical feasibility and effectiveness. However, the presence of abandoned leads is associated with increased procedural complexity, necessitating more frequent utilization of advanced extraction tools. The adverse consequences provide a clinical rationale for prophylactic removal of non-functional leads.

Using past behaviors to guide future actions is essential for fostering cooperation in repeated social dilemmas. Traditional memory-based strategies that focus on recent interactions have yielded valuable insights into the evolution of cooperative behavior. However, as memory length increases, the complexity of analysis grows exponentially, since these strategies need to map every possible action sequence of a given length to subsequent responses. Due to their inherent reliance on exhaustive mapping and a lack of explicit information processing, it remains unclear how individuals can handle extensive interaction histories to make decisions under cognitive constraints. To fill this gap, we introduce coordinated reciprocity strategies (CORE), which incrementally evaluate the entire game history by tallying instances of consistent actions between individuals without storing round-to-round details. Once this consistency index surpasses a threshold, CORE prescribes cooperation. Through equilibrium analysis, we derive an analytical condition under which CORE constitutes an equilibrium. Moreover, our numerical results show that CORE effectively promotes cooperation between variants of itself, and it outperforms a range of existing strategies including memory-1, memory-2, and those from a documented strategy library in evolutionary dynamics. Our work thus underscores the pivotal role of cumulative action consistency in enhancing cooperation, developing robust strategies, and offering cognitively low-burden information processing mechanisms in repeated social dilemmas.

Graph rigidity theory is an important tool for examining the solvability of sensor network localization (SNL) problems, and ensuring global convergence of localization algorithms. Along this direction, diverse measurements such as signed angle (SA) and ratio of distance (RoD) have been considered. However, little is known about how the bipartition of nodes based on perceptual abilities affects the rigidity property of the network. In this paper, we study the rigidity and localization of networks with heterogeneous nodes, namely, two types of sensors measuring SA and RoD, respectively. Interestingly, the rigidity property is shown to be strongly dependent on the bipartition of nodes, and exhibits a duality. Moreover, an SA-RoD constrained network can be uniquely determined up to uniform rotations, translations, and scalings (global SA-RoD rigidity) even if it is neither SA rigid nor RoD rigid. A scalable approach to construction of globally SA-RoD rigid frameworks is proposed. Localizability analysis and localization algorithm synthesis are both conducted based on weaker network topology conditions, compared with SA- or RoD-based SNL approaches. Numerical simulations are worked out to validate the theoretical results.