Chaoli Wang

Over the past three decades, evolutionary game theory in structured populations has witnessed explosive progress. Counterintuitive cooperative phenomena discovered in simulations have stimulated the development of rigorous mathematical theories to describe the underlying dynamics. In this paper, we review these theoretical advancements, including pair approximation and coalescent theory, which offer analytical solutions to evolutionary games on networks. We summarize the key results obtained by these theories, while also discussing their limitations, such as high computational complexity and constrained flexibility when applied to complex models. We therefore suggest that mathematical approaches provide a complementary tool for understanding cooperation in structured populations alongside simulations.

Reinforcement learning provides a framework for studying how individuals adjust their behavior through repeated interaction and feedback in social dilemmas. In Q-learning, exploration controls how often agents choose actions other than those favored by their current learned Q-values. Yet, the existing models usually treat the exploration rate as a constant parameter. In systems with social evaluation, however, trial-and-error behavior carries different costs and opportunities for agents with different reputations, making exploration dependent on social standing rather than uniform across agents. Herein, we develop a spatial prisoner’s dilemma model in which Q-learning agents adapt their exploration rates according to local reputation differences, while reputation is updated through an asymmetric, state-dependent rule. The results show that adaptive exploration and asymmetric reputation updating each promote cooperation, but their combination produces a stronger reinforcing effect than either mechanism alone. Low-reputation agents explore more and can recover reputation through cooperation, while high-reputation agents explore less and avoid reputation losses caused by defection. This mechanism also reorganizes cooperation in space, producing a stable checkerboard-like coexistence at intermediate reputation concern. In addition, cooperation is most vulnerable at intermediate baseline exploration rates, whereas stronger asymmetric reputation updating mitigates this exploration-induced disruption. These results suggest that reputation can act not only as a record of past behavior but also as a dynamic signal that regulates exploratory behavior during learning and thereby stabilizes cooperation.

At present, in the optimal control problem of nonzero-sum games, the designed event-triggered controller needs to continuously detect and calculate the triggered error, which will increase the communication burden of the network. In addition, the persistent excitation (PE) conditions in the design are difficult to verify, and the issue of input saturation is not considered. The optimal control designed in this paper takes these problems into consideration. This paper proposes a self-triggered control method to address the issue of continuous detection and calculation of triggered errors. First, the control input saturation is solved by using a nonquadratic cost function. Then the learning rules of the critic neural network are constructed by using the experience replay technique and a stabilization operator, which not only can get the approximate optimal control, but also can obtain the PE condition that is easier to verify, and remove the assumption of the initial admissible control. Theoretical proof is provided to establish that the proposed self-triggered adaptive control law stabilizes the closed-loop system. Finally, the efficacy of this method is validated through numerical simulation.

Safety is a fundamental issue in autonomous mobile robot design. This paper presents a safety-critical control framework for nonholonomic tractor-trailer vehicles subject to both safety and physical constraints. Control barrier functions (CBFs) are employed to enforce hard safety constraints, generating a safe set that ensures obstacle avoidance. These CBFs are integrated with control Lyapunov functions (CLFs) to enable trajectory tracking while maintaining safety. To reduce conflicts between CBFs and CLFs and improve obstacle avoidance performance, we optimize their decay rates. Leveraging the concept of differential flatness, we design a controller that enables the system to follow the desired safe trajectory. Finally, the effectiveness and performance of the proposed method are demonstrated through numerical simulations and real-world experiments.

The so-called trust game models asymmetric interactions between trustors, who may offer a certain amount, and trustees who respond by keeping either the entire amount or just a portion of it. Although payoffs depend on these interactions, the members of both groups learn strategy behavior from similar peers. The separation between gaming and learning can be modeled by a bilayer framework, in which individuals learn from neighbors within their own layer but play the game with individuals from the other layer. Realistically, trustee behaviours are heterogeneous. Even if they cooperate and return the donation, its portion could be significantly different. Such diversity can sustain trust even at relatively low levels of trustworthiness. This mechanism is driven by a percolation phenomenon where large clusters of trustworthiness promote trust formation inside the clusters but impede outward diffusion. In contrast, an intermixed distribution facilitates the spread of trust while hindering its local emergence. Consequently, a moderate degree of spatial clustering of trustworthiness maximizes overall trust by balancing connectivity and nucleation. These findings provide delicate insights into the self-organization of spatial and dynamical mechanisms in large-scale populations characterized by asymmetric interactions.

Trust is one of the cornerstones of human society. One of the evolutionary pressure mechanisms that may have led to its emergence is the presence of incentives for trustworthy behavior. However, this type of reward has received relatively little attention in the context of spatial trust games, which are often used to build models in evolutionary game theory. To fill this gap, we introduce an inter-role reward mechanism in the spatial trust game, so that an investing trustor can choose to pay an extra cost to reward a trustworthy trustee. With extensive numerical simulations, we find that this type of reward does not always promote trust. Rather, while moderate rewards break the dominance of mistrust, thereby favoring investment, excessive rewards eventually stimulate a nonreturn strategy, ultimately suppressing the evolution of trust. Additionally, lower reward costs do not necessarily promote trust. Instead, more costly, but not excessive, rewards enhance the advantage of the original investment, consolidating the clusters of rewarders and improving trust. Our model thus provides evidence about the counterintuitive nature of the relationship between trust and rewards in a complex society.

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.

From ant–acacia mutualism to performative conflict resolution among Inuit, dedicated punishments between distinct subsets of a population are widespread and can reshape the evolutionary trajectory of cooperation. Existing studies have focused on punishments within a homogeneous population, paying little attention to cooperative dynamics in a situation where belonging to a subset is equally important to the actual strategy represented by an actor. To fill this gap, we here study a bipartite population where cooperator agents in a public goods game penalize exclusively those defectors who belong to the alternative subset. We find that cooperation can emerge and remain stable under symmetric intergroup punishment. In particular, at low punishment intensity and at a small value of the enhancement factor of the dilemma game, intergroup punishment promotes cooperation more effectively than a uniformly applied punishment. Moreover, intergroup punishment in bipartite populations tends to be more favorable for overall social welfare. When this incentive is balanced, cooperators can collectively restrain defectors of the alternative set via aggregate interactions in a randomly formed working group, offering a more effective incentive. Conversely, breaking the symmetry of intergroup punishment inhibits cooperation, as the imbalance creates an Achilles’ heel in the enforcement structure. Our work, thus, reveals symmetry in intergroup punishment as a unifying principle behind cooperation across human and biological systems.