Insight into the so-called spatial reciprocity

Donations triggered by inequality tolerance affect the evolution of cooperation in spatial public goods game

Generosity through donation plays a crucial role in reducing inequality and influencing human behavior. However, previous research on donation has overlooked individuals' acceptance of the extent of inequality, which acts as a trigger for donation. To address this gap, this paper systematically explores the impact of donation based on inequality tolerance on the evolution of cooperation in spatial public goods game. Specifically, donation occurs only when an individual's payoff advantage exceeds her inequality tolerance. The results show that donation patterns are crucial for the emergence and stability of cooperation. In the enduring period, the defector-to-cooperator donation pattern helps to form cooperative clusters. In the expanding period, cooperator-to-cooperator, defector-to-defector, and defector-to-cooperator donation patterns create a stable three-layer structure through self-organization, providing a payoff advantage to boundary cooperators. As donation ratio increases, the three-layer structure provides a greater payoff advantage to boundary cooperators, leading to an increase in cooperation. As inequality tolerance increases, changes in donation patterns weaken the three-layer structure, causing cooperation to decrease or disappear through discontinuous phase transitions. Subsequently, all critical points of discontinuous phase transitions are identified by specific spatial configurations. In addition, the influence of donation patterns on the evolution of cooperation is robust, even in heterogeneous small-world networks. This paper offers valuable insights into the dynamics of cooperation evolution and the role of donation in shaping behavior.

Temporal interaction and its role in the evolution of cooperation

This research investigates the impact of dynamic, time-varying interactions on cooperative behavior in social dilemmas. Traditional research has focused on deterministic rules governing pairwise interactions, yet the impact of interaction frequency and synchronization in groups on cooperation remains underexplored. Addressing this gap, our work introduces two temporal interaction mechanisms to model the stochastic or periodic participation of individuals in public goods games, acknowledging real-life variances due to exogenous temporal factors and geographical time differences. We consider that the interaction state significantly influences both game payoff calculations and the strategy updating process, offering new insights into the emergence and sustainability of cooperation. Our results indicate that maximum game participation frequency is suboptimal under a stochastic interaction mechanism. Instead, an intermediate activation probability maximizes cooperation, suggesting a vital balance between interaction frequency and inactivity security. Furthermore, local synchronization of interactions within specific areas is shown to be beneficial, as time differences hinder the spread of cross-structures but promote the formation of dense cooperative clusters with smoother boundaries. We also note that stronger clustering in networks, larger group sizes, and lower noise increase cooperation. This research contributes to understanding the role of node-based temporality and probabilistic interactions in social dilemmas, offering insights into fostering cooperation.

Impact of discrepant accumulations strategy on collective cooperation in multilayer networks

Understanding large-scale cooperation among related individuals has been one of the largest challenges. Since humans are in multiple social networks, the theoretical framework of multilayer networks is perfectly suited for studying this fascinating aspect of our biology. To that effect, we here study the cooperation in evolutionary game on interdependent networks. Importantly, a part of players are set to adopt Discrepant Accumulations Strategy. Players employing this mechanism not only use their payoffs in the multilayer network as the basis for the updating process as in previous experiments, but also take into account the similarities and differences in strategies across different layers. Monte Carlo simulations demonstrate that considering the similarities and differences in strategies across layers when calculating fitness can significantly enhance the cooperation level in the system. By examining the behavior of different pairing modes within cooperators and defectors, the equilibrium state can be attributed to the evolution of correlated pairing modes between interdependent networks. Our results provide a theoretical analysis of the group cooperation induced by the Discrepant Accumulations Strategy. And we also discuss potential implications of these findings for future human experiments concerning the cooperation on multilayer networks.

Asymmetry of individual activity promotes cooperation in the spatial prisoner's dilemma game

We consider an aspiration-based asymmetric individual activity co-evolutionary prisoner's dilemma game model on the square lattice. In detail, each player has an individual weight that evolves with its strategy. We introduce an asymmetric setting that only μ fractions of players in the network chosen to be active can update their individual weights according to whether their payoffs satisfy their aspirations. Therefore, our individual weights can be regarded as a type of intrinsic motivation satisfaction. Through Monte Carlo simulations on the square lattice, our co-evolution mechanism has been proven to significantly promote cooperation. Furthermore, we find that the moderate μ leads to the highest cooperation level. By dividing the players into four types according to their strategies and weights, we prove that the inversion of the dominant relationship between cooperators and defectors with low weights leads to a conversion cycle of the four types of players, which enhances spatial reciprocity to promote cooperation. Moderate μ results in an appropriate number of low-weight players in this cycle. Interestingly, our results show that higher weight heterogeneity does not lead to higher levels of cooperation, which is contrary to the intuition formed in the previous work.

Dual effects of conformity on the evolution of cooperation in social dilemmas

Human beings are easily impacted by social influences, due to their social nature. As an essential manifestation of social influences, conformity is associated with the frequency witnessed in others' behavior, involving normative conformity and informational conformity according to the reaction of individuals. The former comes from the fear of a normative environment, while the latter means most behaviors are followed due to information asymmetry. Normative conformity significantly enhances network reciprocity, producing optimal cooperation at a moderate proportion, which induces within-cluster behavioral homogeneity and between-cluster behavioral diversity. On the contrary, informational conformity has an inhibitory effect on the evolution of cooperation for a low proportion of the conformity population, which contributes to the formation of defectors' clusters. The symmetry and duality of the two types of conformity on cooperation evolution provide an interesting and unexplored approach for future research, revealing the mechanism of conformity in evolutionary games.

Small bots, big impact: solving the conundrum of cooperation in optional Prisoner's Dilemma game through simple strategies

Cooperation plays a crucial role in both nature and human society, and the conundrum of cooperation attracts the attention from interdisciplinary research. In this study, we investigated the evolution of cooperation in optional Prisoner's Dilemma games by introducing simple bots. We focused on one-shot and anonymous games, where the bots could be programmed to always cooperate, always defect, never participate or choose each action with equal probability. Our results show that cooperative bots facilitate the emergence of cooperation among ordinary players in both well-mixed populations and a regular lattice under weak imitation scenarios. Introducing loner bots has no impact on the emergence of cooperation in well-mixed populations, but it facilitates the dominance of cooperation in regular lattices under strong imitation scenarios. However, too many loner bots on a regular lattice inhibit the spread of cooperation and can eventually result in a breakdown of cooperation. Our findings emphasize the significance of bot design in promoting cooperation and offer useful insights for encouraging cooperation in real-world scenarios.

Towards preferential selection in the prisoner's dilemma game

In previous works, the choice of learning neighbor for an individual has generally obeyed pure random selection or preferential selection rules. In this paper, we introduce a tunable parameter ε to characterize the strength of preferential selection and focus on the transition towards preferential selection in the spatial evolutionary game by controlling ε to guide the system from pure random selection to preferential selection. Our simulation results reveal that the introduction of preferential selection can hugely alleviate social dilemmas and enhance network reciprocity. A larger ε leads to a higher critical threshold of the temptation b for the extinction of cooperators. Moreover, we provide some intuitive explanations for the above results from the perspective of strategy transition and cooperative clusters. Finally, we examine the robustness of the results for noise K and different topologies, find that qualitative features of the results are unchanged.

A game theoretical model for the stimulation of public cooperation in environmental collaborative governance

Digital technologies provide a convenient way for the public to participate in environmental governance. Therefore, by means of a two-stage evolutionary model, a new mechanism for promoting public cooperation is proposed to accomplish environmental collaborative governance. Interactive effects of government-enterprise environmental governance are firstly explored, which is the external atmosphere for public behaviour. Second, the evolutionary dynamics of public behaviour is analysed to reveal the internal mechanism of the emergence of public cooperation in environmental collaborative governance projects. Simulations reveal that the interaction of resource elements between government and enterprise is an important basis for environmental governance performance, and that governments can improve this as well as public cooperation by increasing the marginal governance propensity. Similarly, an increase in the government's fixed expenditure item of environmental governance can also significantly improve government-enterprise performance and public cooperation. And finally, the effect of government's marginal incentive propensity on public environmental governance is moderated by enterprises' marginal environmental governance propensity, so that simply increasing the government's marginal incentive propensity cannot improve the evolutionary stable state of public behaviour under the scenario where enterprises' marginal environmental governance propensity is low.

The Polarization of the Coupling Strength of Interdependent Networks Stimulates Cooperation

We introduce a mixed network coupling mechanism and study its effects on how cooperation evolves in interdependent networks. This mechanism allows some players (conservative-driven) to establish a fixed-strength coupling, while other players (radical-driven) adjust their coupling strength through the evolution of strategy. By means of numerical simulation, a hump-like relationship between the level of cooperation and conservative participant density is revealed. Interestingly, interspecies interactions stimulate polarization of the coupling strength of radical-driven players, promoting cooperation between two types of players. We thus demonstrate that a simple mixed network coupling mechanism substantially expands the scope of cooperation among structured populations.

Emergence of cooperation through chain-reaction death

We generalize the Bak-Sneppen model of coevolution to a game model for evolutionary dynamics which provides a natural way for the emergence of cooperation. Interaction between members is mimicked by a prisoner's dilemma game with a memoryless stochastic strategy. The fitness of each member is determined by the payoffs π of the games with its neighbors. We investigate the evolutionary dynamics using a mean-field calculation and Monte Carlo method with two types of death processes, fitness-dependent death and chain-reaction death. In the former, the death probability is proportional to e^{-βπ} where β is the "selection intensity." The neighbors of the death site also die with a probability R through the chain-reaction process invoked by the abrupt change of the interaction environment. When a cooperator interacts with defectors, the cooperator is likely to die due to its low payoff, but the neighboring defectors also tend to disappear through the chain-reaction death, giving rise to an assortment of cooperators. Owing to this assortment, cooperation can emerge for a wider range of R values than the mean-field theory predicts. We present the detailed evolutionary dynamics of our model and the conditions for the emergence of cooperation.

Non-Hermitian topology in rock-paper-scissors games

Non-Hermitian topology is a recent hot topic in condensed matters. In this paper, we propose a novel platform drawing interdisciplinary attention: rock–paper–scissors (RPS) cycles described by the evolutionary game theory. Specifically, we demonstrate the emergence of an exceptional point and a skin effect by analyzing topological properties of their payoff matrix. Furthermore, we discover striking dynamical properties in an RPS chain: the directive propagation of the population density in the bulk and the enhancement of the population density only around the right edge. Our results open new avenues of the non-Hermitian topology and the evolutionary game theory.

State-of-art review of information diffusion models and their impact on social network vulnerabilities

With the development of information society and network technology, people increasingly depend on information found on the Internet. At the same time, the models of information diffusion on the Internet are changing as well. However, these models experience the problem due to the fast development of network technologies. There is no thorough research in regards to the latest models and their applications and advantages. As a result, it is essential to have a comprehensive study of information diffusion models. The primary goal of this research is to provide a comparative study on the existing models such as the Ising model, Sznajd model, SIR model, SICR model, Game theory and social networking services models. We discuss several of their applications with the existing limitations and further categorizations. Vulnerabilities and privacy challenges of information diffusion models are extensively explored. Furthermore, categorization including strengths and weaknesses are discussed. Finally, limitations and recommendations are suggested with diverse solutions for the improvement of the information diffusion models and envisioned future research directions.

Chiral edge modes in evolutionary game theory: A kagome network of rock-paper-scissors cycles

We theoretically demonstrate the realization of a chiral edge mode in a system beyond natural science. Specifically, we elucidate that a kagome network of rock-paper-scissors (K-RPS) hosts a chiral edge mode of the population density which is protected by the nontrivial topology in the bulk. The emergence of the chiral edge mode is demonstrated by numerically solving the Lotka-Volterra (LV) equation. This numerical result can be intuitively understood in terms of the cyclic motion of a single rock-paper-scissors cycle, which is analogous to the cyclotron motion of fermions. Furthermore, we point out that a linearized LV equation is mathematically equivalent to the Schrödinger equation describing quantum systems. This equivalence allows us to clarify the topological origin of the chiral edge mode in the K-RPS; a nonzero Chern number of the payoff matrix induces the chiral edge mode of the population density, which exemplifies the bulk-edge correspondence in two-dimensional systems described by evolutionary game theory.

Dynamic aspiration based on Win-Stay-Lose-Learn rule in spatial prisoner's dilemma game

Prisoner’s dilemma game is the most commonly used model of spatial evolutionary game which is considered as a paradigm to portray competition among selfish individuals. In recent years, Win-Stay-Lose-Learn, a strategy updating rule base on aspiration, has been proved to be an effective model to promote cooperation in spatial prisoner’s dilemma game, which leads aspiration to receive lots of attention. In this paper, according to Expected Value Theory and Achievement Motivation Theory, we propose a dynamic aspiration model based on Win-Stay-Lose-Learn rule in which individual’s aspiration is inspired by its payoff. It is found that dynamic aspiration has a significant impact on the evolution process, and different initial aspirations lead to different results, which are called Stable Coexistence under Low Aspiration, Dependent Coexistence under Moderate aspiration and Defection Explosion under High Aspiration respectively. Furthermore, a deep analysis is performed on the local structures which cause defectors’ re-expansion, the concept of END- and EXP-periods are used to justify the mechanism of network reciprocity in view of time-evolution, typical feature nodes for defectors’ re-expansion called Infectors, Infected nodes and High-risk cooperators respectively are found. Compared to fixed aspiration model, dynamic aspiration introduces a more satisfactory explanation on population evolution laws and can promote deeper comprehension for the principle of prisoner’s dilemma.

Adaptive willingness resolves social dilemma in network populations

Cooperation is an effective manner to enable different elements of complex networks to work well. In this work, we propose a coevolution mechanism of learning willingness in the network population: an agent will be more likely to imitate a given neighbor's strategy if her payoff is not less than the average performance of all her neighbors. Interestingly, increase of learning willingness will greatly promote cooperation even under the environment of extremely beneficial temptation to defectors. Through a microscopic analysis, it is unveiled that cooperators are protected due to the appearance of large-size clusters. Pair approximation theory also validates all these findings. Such an adaptive mechanism thus provides a feasible solution to relieve social dilemmas and will inspire further studies.

The influence of heterogeneous learning ability on the evolution of cooperation

In this paper, we design a simple coevolution model to investigate the role of heterogeneous learning ability on the evolution of cooperation. The model weakens the winner's learning ability in order to keep its current advantage. Conversely, it strengthens the loser's learning ability for increasing the chance to update its strategy. In particular, we consider this coevolutionary model separately applying to both cooperators and defectors (rule I), only cooperators (rule II), as well as only defectors (rule III) in spatial prisoner's dilemma game. Through numerical simulations, we find that cooperation can be promoted in rule II, whereas, cooperation is hampered in rule I and rule III. We reveal its potential reason from the viewpoint of enduring and expanding periods in game dynamics. Our results thus provide a deeper understanding regarding the heterogeneous learning ability on game theory.

Aspiration induced interdependence leads to optimal cooperation level

How to couple different networks is a key issue in interdependent networks, where information sharing and payoff coupling are two frequently used methods. Unlike previous studies, in this paper, we propose a new coupling mode and test its performance in interdependent networks. Specifically, a player tends to seek additional support on another network only if his environment (defined as the proportion of holding different strategies in the neighborhood) is worse enough and exceeds an aspiration level. Conversely, it turns to the traditional version on single network if his environment is pleasing enough (the value of environment is small). Whether to establish additional support will directly influence the range of selecting fittest learning objects. As we can see from numerical results, moderate aspiration introduces diversity into the system and cooperation evolves with the support of network coupling. Besides, we also demonstrate that players with external links on the boundary of cooperative clusters protect internal cooperators and attract more players to cooperate under preferential selection rule.

Synergistic third-party rewarding and punishment in the public goods game

We study the evolution of cooperation in the spatial public goods game in the presence of third-party rewarding and punishment. The third party executes public intervention, punishing groups where cooperation is weak and rewarding groups where cooperation is strong. We consider four different scenarios to determine what works best for cooperation, in particular, neither rewarding nor punishment, only rewarding, only punishment or both rewarding and punishment. We observe strong synergistic effects when rewarding and punishment are simultaneously applied, which are absent if neither of the two incentives or just each individual incentive is applied by the third party. We find that public cooperation can be sustained at comparatively low third-party costs under adverse conditions, which is impossible if just positive or negative incentives are applied. We also examine the impact of defection tolerance and application frequency, showing that the higher the tolerance and the frequency of rewarding and punishment, the more cooperation thrives. Phase diagrams and characteristic spatial distributions of strategies are presented to corroborate these results, which will hopefully prove useful for more efficient public policies in support of cooperation in social dilemmas.

Dynamic resettlement as a mechanism of phase transitions in urban configurations

We study formation and growth of human settlements as thermodynamic phenomena and focus on critical regimes associated with these processes. In doing so we develop a common thermodynamic perspective on modeling urban structures driven by "fast" and "slow" underlying dynamics of human resettlement and infrastructure development. The unifying perspective is illustrated by a comparative analysis of two qualitatively and quantitatively different models of dynamic resettlement. We demonstrate that the considered urban systems undergo phase transitions, irrespective of the particular details of the dynamic models. This suggests that the observed critical behavior is intrinsic to resettlement process.

Swarm intelligence inspired cooperation promotion and symmetry breaking in interdependent networked game

The evolution of a cooperative strategy on multilayer networks is arousing increasing concern. Most of the previous studies assumed that agents can only choose cooperation or defection when interacting with their partners, whereas the actual provisions in real world scenarios might not be discrete, but rather continuous. Furthermore, in evolutionary game, agents often make use of their memory which keeps the most successful strategy in the past, as well as the best current strategy gained by their directed neighbors, to find the best available strategies. Inspired by these observations, we study the impact of the particle swarm optimization (PSO) algorithm on the evolution of cooperation on interdependent networks in the continuous version of spatial prisoner's dilemma games. Following extensive simulations of this setup, we can observe that the introduction of the PSO mechanism on the interdependent networks can promote cooperation strongly, regardless of the network coupling strength. In addition, we find that the increment of coupling strength is more suitable for the propagation of cooperation. More interestingly, we find that when the coupling strength is relatively large, a spontaneous symmetry breaking phenomenon of cooperation occurs between the interdependent networks. To interpret the symmetry breaking phenomenon, we investigate the asynchronous expansion of heterogeneous strategy couples between different networks. Since this work takes cooperation from a more elaborate perspective, we believe that it may provide a deep understanding of the evolution of cooperation in social networks.

Evolution of Conformity Dynamics in Complex Social Networks

Conformity is a common phenomenon among people in social networks. In this paper, we focus on customers’ conformity behaviors in a symmetry market where customers are located in a social network. We establish a conformity model and analyze it in ring network, random network, small-world network, and scale-free network. Our simulations shown that topology structure, network size, and initial market share have significant effects on the evolution of customers’ conformity behaviors. The market will likely converge to a monopoly state in small-world networks but will form a duopoly market in scale networks. As the size of the network increases, there is a greater possibility of forming a dominant group of preferences in small-world network, and the market will converge to the monopoly of the product which has the initial selector in the market. Also, network density will become gradually significant in small-world networks.

Coevolution of Vertex Weights Resolves Social Dilemma in Spatial Networks

In realistic social system, the role or influence of each individual varies and adaptively changes in time in the population. Inspired by this fact, we thus consider a new coevolution setup of game strategy and vertex weight on a square lattice. In detail, we model the structured population on a square lattice, on which the role or influence of each individual is depicted by vertex weight, and the prisoner’s dilemma game has been applied to describe the social dilemma of pairwise interactions of players. Through numerical simulation, we conclude that our coevolution setup can promote the evolution of cooperation effectively. Especially, there exists a moderate value of δ for each ε that can warrant an optimal resolution of social dilemma. For a further understanding of these results, we find that intermediate value of δ enables the strongest heterogeneous distribution of vertex weight. We hope our coevolution setup of vertex weight will provide new insight for the future research.

Evolution of cooperative imitators in social networks

Many evolutionary game models for network reciprocity are based on an imitation dynamics, yet how semirational imitators prevail has seldom been explained. Here we use a model to investigate the coevolutionary dynamics of cooperation and partnership adjustment in a polygenic population of semirational imitators and rational payoff maximizers. A rational individual chooses a strategy best responding to its neighbors when updating strategy and switches to a new partner who can bring it the maximal payoff from all candidates when adjusting the partnership. In contrast, a semirational individual imitates its neighbor's strategy directly and adjusts its partnership based upon a simple reputation rule. Individual-based simulations show that cooperation cannot evolve in a population of all best responders even if they can switch their partners to somebody who can reward them best in game playing. However, when imitators exist, a stable community that consists of cooperative imitators emerges. Further, we show that a birth-death selection mechanism can eliminate all best responders, cultivating a social regime of all cooperative imitators. Compared with parallel simulations that assume fixed networks, cooperative imitators are evolutionarily favored, provided they are able to adjust their partners.

Impact of Social Reward on the Evolution of the Cooperation Behavior in Complex Networks

Social reward, as a significant mechanism explaining the evolution of cooperation, has attracted great attention both theoretically and experimentally. In this paper, we study the evolution of cooperation by proposing a reward model in network population, where a third strategy, reward, as an independent yet particular type of cooperation is introduced in 2-person evolutionary games. Specifically, a new kind of role corresponding to reward strategy, reward agents, is defined, which is aimed at increasing the income of cooperators by applying to them a social reward. Results from numerical simulations show that consideration of social reward greatly promotes the evolution of cooperation, which is confirmed for different network topologies and two evolutionary games. Moreover, we explore the microscopic mechanisms for the promotion of cooperation in the three-strategy model. As expected, the reward agents play a vital role in the formation of cooperative clusters, thus resisting the aggression of defectors. Our research might provide valuable insights into further exploring the nature of cooperation in the real world.

Evidence Combination From an Evolutionary Game Theory Perspective

Dempster-Shafer evidence theory is a primary methodology for multisource information fusion because it is good at dealing with uncertain information. This theory provides a Dempster's rule of combination to synthesize multiple evidences from various information sources. However, in some cases, counter-intuitive results may be obtained based on that combination rule. Numerous new or improved methods have been proposed to suppress these counter-intuitive results based on perspectives, such as minimizing the information loss or deviation. Inspired by evolutionary game theory, this paper considers a biological and evolutionary perspective to study the combination of evidences. An evolutionary combination rule (ECR) is proposed to help find the most biologically supported proposition in a multievidence system. Within the proposed ECR, we develop a Jaccard matrix game to formalize the interaction between propositions in evidences, and utilize the replicator dynamics to mimick the evolution of propositions. Experimental results show that the proposed ECR can effectively suppress the counter-intuitive behaviors appeared in typical paradoxes of evidence theory, compared with many existing methods. Properties of the ECR, such as solution's stability and convergence, have been mathematically proved as well.

Spatial prisoner's dilemma games with zealous cooperators

The existence of a zealot who stays a cooperator irrespective of the result of an interaction has been reported to add "social viscosity" to a population and thereby helps increase the cooperation level in prisoner's dilemma games, which premises the so-called well-mixed situation of a population. We found that this is not always true when a spatial structure, i.e., connecting agent, is introduced. Deploying zealots is counterproductive, especially when the underlying topology is homogenous, similar to that of a lattice. Our simulation reveals how the existence of never-converting cooperators destroys rather than boosts cooperation. We explain detailed mechanisms behind this interesting finding by referring to our previously presented concepts with respect to evolutionary dynamic processes for spatial games under the names enduring and expanding periods.

Solving Potential Games With Dynamical Constraint

We solve N -player potential games with dynamical constraint in this paper. Potential games with stable dynamics are first considered followed by one type of potential games without inherently stable dynamics. Different from most of the existing Nash seeking methods, we provide an extremum seeking-based method that does not require explicit information on the game dynamics or the payoff functions. Only measurements of the payoff functions are needed in the game strategy synthesis. Lie bracket approximation is used for the analysis of the proposed Nash seeking scheme. A singularly semi-globally practically uniformly asymptotically stable result is presented for potential games with stable dynamics and an ultimately bounded result is provided for potential games without inherently stable dynamics. For first-order perturbed integrator-type dynamics, we employ an extended-state observer to deal with the disturbance such that better convergence is achievable. Stability of the closed-loop system is proven and the ultimate bound is quantified. Numerical examples are presented to verify the effectiveness of the proposed methods.

A novel framework of classical and quantum prisoner's dilemma games on coupled networks

Evolutionary games on multilayer networks are attracting growing interest. While among previous studies, the role of quantum games in such a infrastructure is still virgin and may become a fascinating issue across a myriad of research realms. To mimick two kinds of different interactive environments and mechanisms, in this paper a new framework of classical and quantum prisoner's dilemma games on two-layer coupled networks is considered. Within the proposed model, the impact of coupling factor of networks and entanglement degree in quantum games on the evolutionary process has been studied. Simulation results show that the entanglement has no impact on the evolution of the classical prisoner's dilemma, while the rise of the coupling factor obviously impedes cooperation in this game, and the evolution of quantum prisoner's dilemma is greatly impacted by the combined effect of entanglement and coupling.

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.

Experience Replay for Optimal Control of Nonzero-Sum Game Systems With Unknown Dynamics

In this paper, an approximate online equilibrium solution is developed for an N -player nonzero-sum (NZS) game systems with completely unknown dynamics. First, a model identifier based on a three-layer neural network (NN) is established to reconstruct the unknown NZS games systems. Moreover, the identifier weight vector is updated based on experience replay technique which can relax the traditional persistence of excitation condition to a simplified condition on recorded data. Then, the single-network adaptive dynamic programming (ADP) with experience replay algorithm is proposed for each player to solve the coupled nonlinear Hamilton- (HJ) equations, where only the critic NN weight vectors are required to tune for each player. The feedback Nash equilibrium is provided by the solution of the coupled HJ equations. Based on the experience replay technique, a novel critic NN weights tuning law is proposed to guarantee the stability of the closed-loop system and the convergence of the value functions. Furthermore, a Lyapunov-based stability analysis shows that the uniform ultimate boundedness of the closed-loop system is achieved. Finally, two simulation examples are given to verify the effectiveness of the proposed control scheme.

Coupled disease-behavior dynamics on complex networks: A review

It is increasingly recognized that a key component of successful infection control efforts is understanding the complex, two-way interaction between disease dynamics and human behavioral and social dynamics. Human behavior such as contact precautions and social distancing clearly influence disease prevalence, but disease prevalence can in turn alter human behavior, forming a coupled, nonlinear system. Moreover, in many cases, the spatial structure of the population cannot be ignored, such that social and behavioral processes and/or transmission of infection must be represented with complex networks. Research on studying coupled disease-behavior dynamics in complex networks in particular is growing rapidly, and frequently makes use of analysis methods and concepts from statistical physics. Here, we review some of the growing literature in this area. We contrast network-based approaches to homogeneous-mixing approaches, point out how their predictions differ, and describe the rich and often surprising behavior of disease-behavior dynamics on complex networks, and compare them to processes in statistical physics. We discuss how these models can capture the dynamics that characterize many real-world scenarios, thereby suggesting ways that policy makers can better design effective prevention strategies. We also describe the growing sources of digital data that are facilitating research in this area. Finally, we suggest pitfalls which might be faced by researchers in the field, and we suggest several ways in which the field could move forward in the coming years.

Percolation in spatial evolutionary prisoner's dilemma game on two-dimensional lattices

We study the spatial evolutionary prisoner's dilemma game with updates of imitation max on triangular, hexagonal, and square lattices. We use the weak prisoner's dilemma game with a single parameter b. Due to the competition between the temptation value b and the coordination number z of the base lattice, a greater variety of percolation properties is expected to occur on the lattice with the larger z. From the numerical analysis, we find six different regimes on the triangular lattice (z=6). Regardless of the initial densities of cooperators and defectors, cooperators always percolate in the steady state in two regimes for small b. In these two regimes, defectors do not percolate. In two regimes for the intermediate value of b, both cooperators and defectors undergo percolation transitions. The defector always percolates in two regimes for large b. On the hexagonal lattice (z=3), there exist two distinctive regimes. For small b, both the cooperators and the defectors undergo percolation transitions while only defectors always percolate for large b. On the square lattice (z=4), there exist three regimes. Combining with the finite-size scaling analyses, we show that all the observed percolation transitions belong to the universality class of the random percolation. We also show how the detailed growth mechanism of cooperator and defector clusters decides each regime.

Modelling and computation in the valuation of carbon derivatives with stochastic convenience yields

The anthropogenic greenhouse gas (GHG) emission has risen dramatically during the last few decades, which mainstream researchers believe to be the main cause of climate change, especially the global warming. The mechanism of market-based carbon emission trading is regarded as a policy instrument to deal with global climate change. Although several empirical researches about the carbon allowance and its derivatives price have been made, theoretical results seem to be sparse. In this paper, we theoretically develop a mathematical model to price the CO₂ emission allowance derivatives with stochastic convenience yields by the principle of absence of arbitrage opportunities. In the case of American options, we formulate the pricing problem to a linear parabolic variational inequality (VI) in two spatial dimensions and develop a power penalty method to solve it. Then, a fitted finite volume method is designed to solve the nonlinear partial differential equation (PDE) resulting from the power penalty method and governing the futures, European and American option valuation. Moreover, some numerical results are performed to illustrate the efficiency and usefulness of this method. We find that the stochastic convenience yield does effect the valuation of carbon emission derivatives. In addition, some sensitivity analyses are also made to examine the effects of some parameters on the valuation results.

Universal scaling for the dilemma strength in evolutionary games

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

The expansion of neighborhood and pattern formation on spatial prisoner's dilemma

The prisoner's dilemma (PD), in which players can either cooperate or defect, is considered a paradigm for studying the evolution of cooperation in spatially structured populations. There the compact cooperator cluster is identified as a characteristic pattern and the probability of forming such pattern in turn depends on the features of the networks. In this paper, we investigate the influence of expansion of neighborhood on pattern formation by taking a weak PD game with one free parameter T, the temptation to defect. Two different expansion methods of neighborhood are considered. One is based on a square lattice and expanses along four directions generating networks with degree increasing with K=4m. The other is based on a lattice with Moore neighborhood and expanses along eight directions, generating networks with degree of K=8m. Individuals are placed on the nodes of the networks, interact with their neighbors and learn from the better one. We find that cooperator can survive for a broad degree 4≤K≤70 by taking a loose type of cooperator clusters. The former simple corresponding relationship between macroscopic patterns and the microscopic PD interactions is broken. Under a condition that is unfavorable for cooperators such as large T and K, systems prefer to evolve to a loose type of cooperator clusters to support cooperation. However, compared to the well-known compact pattern, it is a suboptimal strategy because it cannot help cooperators dominating the population and always corresponding to a low cooperation level.

Network reciprocity created in prisoner's dilemma games by coupling two mechanisms

We found that a nontrivial enhancement of network reciprocity for 2 × 2 prisoner's dilemma games can be achieved by coupling two mechanisms. The first mechanism presumes a larger strategy update neighborhood than the conventional first neighborhood on the underlying network. The second is the strategy-shifting rule. At the initial time step, the averaged cooperation extent is assumed to be 0.5. In the case of strategy shifting, an agent adopts a continuous strategy definition during the initial period of a simulation episode (when the global cooperation fraction decreases from its initial value of 0.5; that is, the enduring period). The agent then switches to a discrete strategy definition in the time period afterwards (when the global cooperation fraction begins to increase again; that is, the expanding period). We explored why this particular enhancement comes about, and to summarize, the continuous strategy during the initial period relaxes the conditions for the survival of relatively cooperative clusters, and the large strategy adaptation neighborhood allows those cooperative clusters to expand easily.

Dependency links can hinder the evolution of cooperation in the prisoner's dilemma game on lattices and networks

Networks with dependency links are more vulnerable when facing the attacks. Recent research also has demonstrated that the interdependent groups support the spreading of cooperation. We study the prisoner's dilemma games on spatial networks with dependency links, in which a fraction of individual pairs is selected to depend on each other. The dependency individuals can gain an extra payoff whose value is between the payoff of mutual cooperation and the value of temptation to defect. Thus, this mechanism reflects that the dependency relation is stronger than the relation of ordinary mutual cooperation, but it is not large enough to cause the defection of the dependency pair. We show that the dependence of individuals hinders, promotes and never affects the cooperation on regular ring networks, square lattice, random and scale-free networks, respectively. The results for the square lattice and regular ring networks are demonstrated by the pair approximation.

Impact of small groups with heterogeneous preference on behavioral evolution in population evacuation

Up to now, there have been a great number of mechanisms to explain the individual behavior and population traits, which seem of particular significance in evolutionary biology and social behavior analysis. Among them, small groups and heterogeneity are two useful frameworks to the above issue. However, vast majority of existing works separately consider both scenarios, which is inconsistent with realistic cases in our life. Here we propose the evolutionary games of heterogeneous small groups (namely, different small groups possess different preferences to dilemma) to study the collective behavior in population evacuation. Importantly, players usually face completely different dilemmas inside and outside the small groups. By means of numerous computation simulations, it is unveiled that the ratio of players in one certain small group directly decides the final behavior of the whole population. Moreover, it can also be concluded that heterogeneous degree of preference for different small groups plays a key role in the behavior traits of the system, which may validate some realistic social observations. The proposed framework is thus universally applicable and may shed new light into the solution of social dilemmas.

Sociality as a natural mechanism of public goods provision

In the recent literature, several hypotheses have been offered to explain patterns of human behavior in social environments. In particular, these patterns include ‘prosocial’ ones, such as fairness, cooperation, and collective good provision. Psychologists suggest that these prosocial behaviors are driven not by miscalculations, but by salience of social identity, in-group favoritism, emotion, or evolutionary adaptations. This paper imports psychology scholarship into an economic model and results in a sustainable solution to collective action problems without any external enforcement mechanisms. This natural mechanism of public goods provision is created, analyzed, and observed in a controlled laboratory environment using experimental techniques.

Impact of roles assignation on heterogeneous populations in evolutionary dictator game

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

Modelling the effects of selection temperature and mutation on the prisoner's dilemma game on a complete oriented star

This paper models the prisoner’s dilemma game based on pairwise comparison in finite populations on a complete oriented star (COS). First, we derive a linear system on a COS for calculating the corresponding fixation probabilities that imply dependence of the selection temperature and mutation. Then we observe and analyze the effects of two parameters on fixation probability under different population sizes. In particular, it is found through the experimental results that (1) high mutation is more sensitive to the fixation probability than the low one when population size is increasing, while the opposite is the case when the number of cooperators is increasing, and (2) selection temperature demotes the fixation probability.