Referring to the social performance promotes cooperation in spatial prisoner's dilemma games

Cooperation with dynamic asymmetric evaluation in complex networks from a risk perspective

The choice of strategy exposes individuals to the risk of betrayal. This induces individuals' irrational tendencies in strategy selection, which further influences the emergence of cooperative behavior. However, the underlying mechanisms connecting risk perception and the emergence of cooperation are still not fully understood. To address this, the classic evolutionary game model on complex networks is extended. We depict the interaction between strategy imitation and payoff evaluation from two perspectives: dynamic adjustment and irrational assessment. Specifically, the probability distortion involved in the dynamic selection of imitative reference points, as well as the asymmetric psychological utility associated with reference point dependence, is emphasized. Monte Carlo simulations demonstrate that individual irrational cognition induced by the risk of strategy selection can promote the emergence of cooperative behavior. Among them, the risk sensitivity within psychological utility has the most significant moderating effect. Moreover, the promoting effect of strong heterogeneity and high clustering in the network topology on cooperation under risk scenarios has been clarified. Additionally, the influence of initial states on the emergence of cooperation follows a step-like pattern. This research offers valuable insights for further exploring the cooperation mechanisms among irrational agents, even in scenarios involving the regulation of group cooperation behavior in risky situations.

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.

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.

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.

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.

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.

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.

Impact of deterministic and stochastic updates on network reciprocity in the prisoner's dilemma game

In 2 × 2 prisoner's dilemma games, network reciprocity is one mechanism for adding social viscosity, which leads to cooperative equilibrium. This study introduced an intriguing framework for the strategy update rule that allows any combination of a purely deterministic method, imitation max (IM), and a purely probabilistic one, pairwise Fermi (Fermi-PW). A series of simulations covering the whole range from IM to Fermi-PW reveals that, as a general tendency, the larger fractions of stochastic updating reduce network reciprocity, so long as the underlying lattice contains no noise in the degree of distribution. However, a small amount of stochastic flavor added to an otherwise perfectly deterministic update rule was actually found to enhance network reciprocity. This occurs because a subtle stochastic effect in the update rule improves the evolutionary trail in games having more stag-hunt-type dilemmas, although the same stochastic effect degenerates evolutionary trails in games having more chicken-type dilemmas. We explain these effects by dividing evolutionary trails into the enduring and expanding periods defined by Shigaki et al. [Phys. Rev. E 86, 031141 (2012)].

Role of investment heterogeneity in the cooperation on spatial public goods game

Public cooperation plays a significant role in the survival and maintenance of biological species, to elucidate its origin thus becomes an interesting question from various disciplines. Through long-term development, the public goods game has proven to be a useful tool, where cooperator making contribution can beat again the free-rides. Differentiating from the traditional homogeneous investment, individual trend of making contribution is more likely affected by the investment level of his neighborhood. Based on this fact, we here investigate the impact of heterogeneous investment on public cooperation, where the investment sum is mapped to the proportion of cooperators determined by parameter α. Interestingly, we find, irrespective of interaction networks, that the increment of α (increment of heterogeneous investment) is beneficial for promoting cooperation and even guarantees the complete cooperation dominance under weak replication factor. While this promotion effect can be attributed to the formation of more robust cooperator clusters and shortening END period. Moreover, we find that this simple mechanism can change the potential interaction network, which results in the change of phase diagrams. We hope that our work may shed light on the understanding of the cooperative behavior in other social dilemmas.

Spontaneous symmetry breaking in interdependent networked game

Spatial evolution game has traditionally assumed that players interact with direct neighbors on a single network, which is isolated and not influenced by other systems. However, this is not fully consistent with recent research identification that interactions between networks play a crucial rule for the outcome of evolutionary games taking place on them. In this work, we introduce the simple game model into the interdependent networks composed of two networks. By means of imitation dynamics, we display that when the interdependent factor α is smaller than a threshold value α(C), the symmetry of cooperation can be guaranteed. Interestingly, as interdependent factor exceeds α(C), spontaneous symmetry breaking of fraction of cooperators presents itself between different networks. With respect to the breakage of symmetry, it is induced by asynchronous expansion between heterogeneous strategy couples of both networks, which further enriches the content of spatial reciprocity. Moreover, our results can be well predicted by the strategy-couple pair approximation method.

Moderate intra-group bias maximizes cooperation on interdependent populations

Evolutionary game theory on spatial structures has received increasing attention during the past decades. However, the majority of these achievements focuses on single and static population structures, which is not fully consistent with the fact that real structures are composed of many interactive groups. These groups are interdependent on each other and present dynamical features, in which individuals mimic the strategy of neighbors and switch their partnerships continually. It is however unclear how the dynamical and interdependent interactions among groups affect the evolution of collective behaviors. In this work, we employ the prisoner's dilemma game to investigate how the dynamics of structure influences cooperation on interdependent populations, where populations are represented by group structures. It is found that the more robust the links between cooperators (or the more fragile the links between cooperators and defectors), the more prevalent of cooperation. Furthermore, theoretical analysis shows that the intra-group bias can favor cooperation, which is only possible when individuals are likely to attach neighbors within the same group. Yet, interestingly, cooperation can be even inhibited for large intra-group bias, allowing the moderate intra-group bias maximizes the cooperation level.

Freezing period strongly impacts the emergence of a global consensus in the voter model

It is well known that human beings do not always change opinions or attitudes, since the duration of interaction with others usually has a significant impact on one's decision-making. Based on this observation, we introduce a freezing period into the voter model, in which the frozen individuals have a weakened opinion switching ability. We unfold the presence of an optimal freezing period, which leads to the fastest consensus, using computation simulations as well as theoretical analysis. We demonstrate that the essence of an accelerated consensus is attributed to the biased random walk of the interface between adjacent opinion clusters. The emergence of an optimal freezing period is robust against the size of the system and the number of distinct opinions. This study is instructive for understanding human collective behavior in other relevant fields.

Simultaneously selecting appropriate partners for gaming and strategy adaptation to enhance network reciprocity in the prisoner's dilemma

Network reciprocity is one mechanism for adding social viscosity, which leads to cooperative equilibrium in 2 × 2 prisoner's dilemma games. Previous studies have shown that cooperation can be enhanced by using a skewed, rather than a random, selection of partners for either strategy adaptation or the gaming process. Here we show that combining both processes for selecting a gaming partner and an adaptation partner further enhances cooperation, provided that an appropriate selection rule and parameters are adopted. We also show that this combined model significantly enhances cooperation by reducing the degree of activity in the underlying network; we measure the degree of activity with a quantity called effective degree. More precisely, during the initial evolutionary stage in which the global cooperation fraction declines because initially allocated cooperators becoming defectors, the model shows that weak cooperative clusters perish and only a few strong cooperative clusters survive. This finding is the most important key to attaining significant network reciprocity.

Influence of stochastic perturbation of both action updating and strategy updating in mixed-strategy 2×2 games on evolution of cooperation

In a mixed-strategy game framework, each agent's strategy is defined by a real number; on the other hand, in a discrete strategy game framework, only binary strategies, either cooperation or defection, are allowed. In a spatial mixed-strategy game, with respect to the process for updating action (offer), either a synchronous or an asynchronous strategy update should be presumed. This study elucidates how stochastic perturbation that results from a synchronous or an asynchronous process for updating action significantly affects the enhancement of cooperation in an evolutionary process. Especially, when a synchronous process for updating action is assumed, the extent of cooperation increases with an increase in degree.

Effect of initial fraction of cooperators on cooperative behavior in evolutionary prisoner's dilemma game

We investigate the influence of initial fraction of cooperators on the evolution of cooperation in spatial prisoner's dilemma games. Compared with the results of heterogeneous networks, we find that there is a relatively low initial fraction of cooperators to guarantee higher equilibrium cooperative level. While this interesting phenomenon is contrary to the commonly shared knowledge that higher initial fraction of cooperators can provide better environment for the evolution of cooperation. To support our outcome, we explore the time courses of cooperation and find that the whole course can be divided into two sequent stages: enduring (END) and expanding (EXP) periods. At the end of END period, thought there is a limited number of cooperator clusters left for the case of low initial setup, these clusters can smoothly expand to hold the whole system in the EXP period. However, for high initial fraction of cooperators, superfluous cooperator clusters hinder their effective expansion, which induces many remaining defectors surrounding the cooperator clusters. Moreover, through intensive analysis, we also demonstrate that when the tendency of three cooperation cluster characteristics (cluster size, cluster number and cluster shape) are consistent within END and EXP periods, the state that maximizes cooperation can be favored.

Insight into the so-called spatial reciprocity

Up to now, there have been a great number of studies that demonstrate the effect of spatial topology on the promotion of cooperation dynamics (namely, the so-called "spatial reciprocity"). However, most researchers probably attribute it to the positive assortment of strategies supported by spatial arrangement. In this paper, we analyze the time course of cooperation evolution under different evolution rules. Interestingly, a typical evolution process can be divided into two evident periods: the enduring (END) period and the expanding (EXP) period where the former features that cooperators try to endure defectors' invasion and the latter shows that perfect C clusters fast expand their area. We find that the final cooperation level relies on two key factors: the formation of the perfect C cluster at the end of the END period and the expanding fashion of the perfect C cluster during the EXP period. For deterministic rule, the smooth expansion of C cluster boundaries enables cooperators to reach a dominant state, whereas, the rough boundaries for stochastic rule cannot provide a sufficient beneficial environment for the evolution of cooperation. Moreover, we show that expansion of the perfect C cluster is closely related to the cluster coefficient of interaction topology. To some extent, we present a viable method for understanding the spatial reciprocity mechanism in nature and hope that it will inspire further studies to resolve social dilemmas.

Direct reciprocity in spatial populations enhances R-reciprocity as well as ST-reciprocity

As is well-known, spatial reciprocity plays an important role in facilitating the emergence of cooperative traits, and the effect of direct reciprocity is also obvious for explaining the cooperation dynamics. However, how the combination of these two scenarios influences cooperation is still unclear. In the present work, we study the evolution of cooperation in 2×2 games via considering both spatial structured populations and direct reciprocity driven by the strategy with 1-memory length. Our results show that cooperation can be significantly facilitated on the whole parameter plane. For prisoner's dilemma game, cooperation dominates the system even at strong dilemma, where maximal social payoff is still realized. In this sense, R-reciprocity forms and it is robust to the extremely strong dilemma. Interestingly, when turning to chicken game, we find that ST-reciprocity is also guaranteed, through which social average payoff and cooperation is greatly enhanced. This reciprocity mechanism is supported by mean-field analysis and different interaction topologies. Thus, our study indicates that direct reciprocity in structured populations can be regarded as a more powerful factor for the sustainability of cooperation.

Spontaneous scale-free structure in adaptive networks with synchronously dynamical linking

Inspired by the anti-Hebbian learning rule in neural systems, we study how the feedback from dynamical synchronization shapes network structure by adding new links. Through extensive numerical simulations, we find that an adaptive network spontaneously forms scale-free structure, as confirmed in many real systems. Moreover, the adaptive process produces two nontrivial power-law behaviors of deviation strength from mean activity of the network and negative degree correlation, which exists widely in technological and biological networks. Importantly, these scalings are robust to variation of the adaptive network parameters, which may have meaningful implications in the scale-free formation and manipulation of dynamical networks. Our study thus suggests an alternative adaptive mechanism for the formation of scale-free structure with negative degree correlation, which means that nodes of high degree tend to connect, on average, with others of low degree and vice versa. The relevance of the results to structure formation and dynamical property in neural networks is briefly discussed as well.

Evolution of cooperation in a heterogeneous graph: fixation probabilities under weak selection

It has been shown that natural selection favors cooperation in a homogenous graph if the benefit-to-cost ratio exceeds the degree of the graph. However, most graphs related to interactions in real populations are heterogeneous, in which some individuals have many more neighbors than others. In this paper, we introduce a new state variable to measure the time evolution of cooperation in a heterogeneous graph. Based on the diffusion approximation, we find that the fixation probability of a single cooperator depends crucially on the number of its neighbors. Under weak selection, a cooperator with more neighbors has a larger probability of fixation in the population. We then investigate the average fixation probability of a randomly chosen cooperator. If a cooperator pays a cost for each of its neighbors (the so called fixed cost per game case), natural selection favors cooperation if the benefit-to-cost ratio is larger than the average degree. In contrast, if a cooperator pays a fixed cost and all its neighbors share the benefit (the fixed cost per individual case), cooperation is favored if the benefit-to-cost ratio is larger than the harmonic mean of the degree distribution. Moreover, increasing the graph heterogeneity will reduce the effect of natural selection.