Spatial Evolutionary Games Research Articles

A memory-based spatial evolutionary game with the dynamic interaction between learners and profiteers

A memory-based spatial evolutionary game with the dynamic interaction between learners and profiteers

Zealous cooperation does not always promote cooperation in public goods games.

There is a conventional belief that prosocial behaviors cannot arise through selfish human nature, because defection always exploits cooperation to achieve a higher payoff at an individual level. Unyieldingly, some people hope to move society to cooperation through their zealous cooperation, regardless of payoffs. From the perspective of spatial evolutionary games, however, such zealous behavior is unnecessary because cooperation can emerge from selfish human nature by aggregating in evolution. Yet, to what extent can zealous cooperation induce others to cooperate? We assume a fraction of zealous agents in spatial public goods games who always cooperate. The results show that a moderate proportion of these zealous cooperators can diminish the cooperation level in the system, and cooperation is only promoted when zealots are many. Regarding spatial behaviors, the areas of zealous cooperation in a medium density can prevent evolutionary cooperation from passing through and aggregating. The phenomenon of zealous cooperation impeding cooperation becomes more pronounced when agents become less random and more selfish. This is because dotted zealous cooperation provides significant payoffs to neighboring defection, making them more solid in fitness. In this way, we also find that when zealous cooperators have low productivity, the neighbors receive fewer benefits by exploitation, thus allowing cooperation to spread. We also study replicator dynamics in unstructured populations where zealous cooperation always promotes cooperation, agreeing that zealous cooperation hindering cooperation is a spatial effect.

Preferential selection to promote cooperation on degree–degree correlation networks in spatial snowdrift games

Degree–degree correlation is an important topological characteristic of real-world networks and has been shown to have a significant effect on cooperation in a population. Meanwhile, the preferential selection of individuals could be exploited as an effective mechanism to improve cooperation in evolutionary games. Network structure and strategy evolution are two essential factors affecting the level of cooperation in spatial evolutionary games, and by considering both of them, proposed here is a spatial snowdrift game with preferential selection on scale-free networks with degree–degree correlation. How two main parameters—the Pearson correlation coefficient r and the intensity of preferential selection δ—affect the cooperation is investigated, and simulation results show that when preferential selection is involved in evolutionary games, then larger δ can give a higher level of cooperation. Importantly, an optimal value of r is found that gives maximum cooperation in the population, and how r and δ affect the cooperation is found to depend on the cost parameter of the game. Furthermore, corresponding microcosmic explanations are provided based on the quantity χ, the time series of links, and the strategy distribution.

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.

Evolutionary games on networks: Phase transition, quasi-equilibrium, and mathematical principles

The stable cooperation ratio of spatial evolutionary games has been widely studied using simulations or approximate analysis methods. However, sometimes such “stable” cooperation ratios obtained via approximate methods might not be actually stable, but correspond to quasi-equilibriums instead. We find that various classic game models, like the evolutionary snowdrift game, evolutionary prisoner’s dilemma, and spatial public goods game on square lattices and scale-free networks, exhibit the phase transition in convergence time to the equilibrium state. Moreover, mathematical principles are provided to explain the phase transition of convergence time and quasi-equilibrium of cooperation ratio. The findings explain why and when cooperation and defection have a long-term coexistence.

The replicator equation in stochastic spatial evolutionary games

We study the multi-type stochastic evolutionary game with death–birth updating in expanding spatial populations of size N→∞. The model is a voter model perturbation. For typical eligible populations, we require perturbation strengths satisfying 1/N≪w≪1. Under these conditions, the main results prove that the vector density processes of type obey an extended replicator equation in the limit, and the normalized fluctuations converge to a Gaussian process subject to the Wright–Fisher covariance function. In particular, we obtain a positive resolution of a conjecture from [34] that the replicator equation extends to many non-regular graphs.

Impact of strategy conformity on vaccination behaviors

In most of the studies focusing on the conformity of voluntary vaccination decisions, the conformity was always directly modeled as a conformity-driven strategy-updating rule. However, the utility of an individual can also be influenced by the group identity or discrimination behaviors associated with strategy conformity in realities. Thus, a novel utility model of the vaccination game is first formulated in which the influence of strategy conformity is considered. Then, we use the spatial evolutionary game theory to study the dynamics of individual vaccination strategies under the influence of strategy conformity on the scale-free network. The results show that moderate strategy conformity and a high herd immunity threshold have a significant positive effect on vaccination behaviors when the initial vaccination fraction is low. Moreover, for a high initial vaccination fraction, the strong strategy conformity and high herd immunity threshold are more conducive to encourage vaccination behavior. To analyze the model sensitivity, experiments are conducted in the small world network and square lattice network. In addition, we performed the sensitivity analysis on vaccination effectiveness. Finally, the generality of strategy conformity effect is investigated when the myopic strategy updating rule is adopted in the whole population. The result shows that vaccination behaviors can also be promoted under the condition of moderate strategy conformity and low initial vaccination fraction.

Modeling the social dilemma of involution on a square lattice

Involution refers to the over-generation meaningless competition in society when the total amount of social resources is fixed. With the help of spatial evolutionary game theory, this paper models the social dilemma of involution on a square lattice. It is assumed that an agent can adopt a strategy of either devoting more effort or devoting less effort. When devoting more effort, the agent can acquire more social resources by paying more than those who devote less effort. Meanwhile, the total amount of social resources obtained by the group remains unchanged. The agents synchronously update strategies. We measure the degree of involution by the fraction of the strategy of more effort in the system. The model is investigated from three perspectives: the abundance of social resources, the social temperature in resource distribution, and the cost of more effort. The results show that more abundant social resources lead to the involution, an increase in social temperature in resource distribution suppresses the involution, and an increase in the cost of more effort does not always aggravate or suppress the involution. The results in this paper provide society with qualitative conclusions to resist involution.

The spatial inheritance enhances cooperation in weak prisoner’s dilemmas with agents’ exponential lifespan

The spatial evolutionary games have been studied abundantly in the past two decades. Recently, there is research showing that the emergence of cooperators on temporal networks varies from static networks. However, few research takes players’ interaction lifespans into account, which can be regarded as one of the individuals’ characteristics in real networks. In this paper, we introduce a network model based on the M/M/∞ queueing system and perform the weak prisoner’s dilemmas, where each new individual arrives at the network by Poisson process and has an exponential lifespan. In addition, we propose a neighborhood similarity inheritance mechanism to maintain the players’ interaction requirements and reduce the impact of leaving players to the network. We suggest that cooperators emerge in the aforementioned network model on weak prisoner’s dilemmas, and the inheritance mechanism brings more cooperators to the network, which helps to overcome social dilemmas. Besides, by analyzing the evolution process of the degree distribution and the mean clustering coefficient, it is shown that the inheritance mechanism guarantees the degree heterogeneity and higher mean clustering coefficients, which are shown to promote cooperative behaviors in previous studies.

Wealth-based rule favors cooperation in costly public goods games when individual selection is inevitable

Individual selection, as an effective mechanism, is often used in spatial evolutionary games to promote cooperation. Previous research assumes that, individual selection usually occurs with people who fail to meet a certain criterion. However, individual selection is usually inevitable, regardless of whether players in population cooperate or defect. This paper studies the effects of wealth-based rule in costly public goods games when individual selection is inevitable. Specifically, we assume that only the top V individuals with relatively high cumulative payoffs in each group can be selected for costly PGG. The results show that when V is large, the increase of participation cost has slight inhibitory effects on the evolution of cooperation, but it alleviates the polarization of individuals. However, when V is small, the increase of participation cost within a certain range promotes cooperation prosperity, but it also causes an increase in the proportion of polarized individuals and a widening of the wealth gap between rich and poor individuals.

Effects of inequality on a spatial evolutionary public goods game

Effects of inequality on a spatial evolutionary public goods game

Introducing a graph topology for robust cooperation.

Identifying the conditions that support cooperation in spatial evolutionary game theory has been the focus of a large body of work. In this paper, the classical Prisoner's Dilemma is adopted as an interaction model; agents are placed on graphs and their interactions are constrained by a graph topology. A simple strategy update mechanism is used where agents copy the best performing strategy of their neighbourhood (including themselves). In this paper, we begin with a fully cooperative population and explore the robustness of the population to the introduction of defectors. We introduce a graph structure that has the property that the initial fully cooperative population is robust to any one perturbation (a change of any cooperator to a defector). We present a proof of this property and specify the necessary constraints on the graph. Furthermore, given the standard game payoffs, we calculate the smallest graph which possesses this property. We present an approach for increasing the size of the graph and we show empirically that this extended graph is robust to an increasing percentage of perturbations. We define a new class of graphs for the purpose of future work.

Self-questioning dynamical evolutionary game with altruistic behavior and sharing mechanism in scale-free network

The spatial evolutionary game is one of the efficient models to explain the emergence and maintenance of cooperation among selfish individuals. In the existing work, the relationship between self-questioning dynamical evolutionary game model and Ising model has been discussed. However, the study on the dividing lines which were used to distinguish entirely cooperative phase, entirely defective phase and cooperative and defective coexistence in the ground state is not enough. That is, the dividing lines were only considered to be suitable for regular networks before. To address this issue, a self-questioning evolutionary game model with altruistic or sharing preference is studied in scale-free Barabasi–Albert networks. Using the Ising model theory and Monte Carlo simulation, it is found that the players considering their opponents’ payoffs with probability p are equivalent to the players sharing their payoffs with probability $$p/(1+p)$$ . A further research on the relationship between the self-questioning dynamical evolutionary game model and Ising model shows that the dividing lines are in fact unrelated to network structure, which means that the dividing lines are suitable for arbitrary networks. In addition, the nodes with large degree have higher stability and robustness than those with small degree.

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.

Career choice as an extended spatial evolutionary public goods game

We propose an extended spatial evolutionary public goods game (SEPGG) model to study the dynamics of individual career choice and the corresponding social output. Based on the social value orientation theory, we study the game dynamics of players choosing from two types of work, namely the public work if it serves public interests, and the private work if it serves personal interests. Under the context of SEPGG, choosing public work is to cooperate and choosing private work is to defect. We then investigate the effects of employee productivity, human capital and external subsidies on individual career choices of the two work types, as well as the overall social welfare. From simulation results, we found that when employees have low productivity, they are more willing to enter the private sector. Although this will make both the effort level and human capital of individuals doing private work higher than those engaging in public work, the total outcome of the private sector is still lower than that of the public sector with a low level of public subsidies. When the employee productivity is higher for public work, a certain amount of subsidy can greatly improve system output. On the contrary, when employees in the public sector have low productivity, increasing the subsidy can result in a decline in social output.

Evolutionary game inspired by Cipolla's basic laws of human stupidity.

In this work we present an evolutionary game inspired by the work of Carlo Cipolla entitled The Basic Laws of Human Stupidity. The game expands the classical scheme of two archetypical strategies, collaborators and defectors, by including two additional strategies. One of these strategies is associated with a stupid player that, according to Cipolla, is the most dangerous one as it undermines the global wealth of the population. By considering a spatial evolutionary game and imitation dynamics that go beyond the paradigm of a rational player we explore the impact of Cipolla's ideas and analyze the extent of the damage that stupid players inflict on the population.

Interplay of Elementary Interactions Causing Social Traps in Evolutionary Games

In evolutionary games, pair interactions are defined by payoff matrices that can be decomposed into four types of orthogonal elementary games that represent fundamentally different interaction situations. The four classes of elementary interactions are formed by games with self- and cross-dependent payoffs, coordination games, and cyclic games. At the level of two-person games, social traps (dilemmas) can not occur for symmetric payoff matrices, which are combinations of coordination games and symmetrically paired self- and cross-dependent components, because individual and common interests coincide in them. In spatial evolutionary games that follow the logit evolutionary dynamics, however, the total payoff is still not maximized at certain noise levels in certain combinations of symmetric components. This phenomenon is similar to the appearance of partially ordered phases in solid state physics, which are stabilized by their higher entropy. In contrast, it is the antisymmetric part of their self- and cross-dependent components that is responsible for the emergence of traditional social dilemmas in games like the two-strategy donation game or the prisoner's dilemma. The general features of these social dilemmas are inherited by $n$-strategy games in the absence of cyclic components, which would prevent the existence of a potential and thus thermodynamic behavior. Using the mathematical framework of matrix decomposition, we survey the ways in which the interplay of elementary games can lead to a loss of total payoff for a society of selfish players. We describe the general features of different illustrative combinations of elementary games, including a game in which the presence of a cyclic component gives rise to the the tragedy of the commons via a paradoxical effect.

Evolution of cooperation with individual diversity on interdependent weighted networks

The social networks as a complex set of networks fully reflect internal relations of individual interactions between them. Individual as an integral part of networks can show different functions under different levels. In the vast majority of current research realm of spatial evolutionary game, individuals who are often treated as identical peers interact with the local neighbours on a single, isolated same network, even the independent networks extended the content of spatial reciprocity. However, the individual diversity, including gender, wealth and social status and so on, usually is presented within the population. Individual heterogeneity impacts on the evolution of cooperation amongst selfish individuals. With this motivation, here we consider that two forms including Prisoner’s Dilemma (PD) and Snowdrift Game (SG) take place on interdependent weighted networks via the mixed-coupling in which individuals participate in different networks of interactions, cooperative behaviour can be maintained. Remarkably, the numerical analysis shows that, as the network interdependence considering individual diversity increases, cooperation thrives on one network joining in PD, the other engaging in SG may be plagued by defectors. Meanwhile, there exists an optimal region of mixed-coupling between networks to persist in cooperation of one network. Furthermore, individual diversity may be a link between non-trivial systems across the network connection, thus probing in how to schedule heterogeneous competitive tasks and services in complex manufacturing systems.

Co-evolution of limited resources in the memory-based spatial evolutionary game

Abstract In this paper, the co-evolution of cooperation and limited resources in spatial evolutionary game is studied, where memory ability is involved in the evolutionary process. Competition for resources exists widely in nature and human society. Therefore, the allocation of limited resources is always accompanied with various kinds of games. The acquisition of resources depends on the results of the game and the number of resources owned by the players would in turn affect the choice of game strategies. Memory, or so-called experience, usually plays an important role in guiding human behaviors in the real world, which is vital to individual rational decision-making. In order to better simulate the memory ability in real life, memory mechanism with attenuation factor is introduced into the co-evolution of prisoner's dilemma game. Moreover, in order to protect the vulnerable groups, we have introduced the mechanism of minimum guarantee of resources, which simulates the social security system in reality such that the extreme polarization can be alleviated. By using strategy-independent rules, memory ability can effectively raise the level of cooperation, where both the memory length and attenuation factor would affect the co-evolutionary results. Furthermore, limited resources have fast aggregation effect and players with cooperative behaviors have easier access to resources. Meanwhile, the minimum resources guarantee mechanism can considerably affect the cooperation level. Besides, as to the complex dynamical behaviors of co-evolution, quantitative and qualitative discussions are implemented. In order to illustrate the robustness of the obtained results, the cooperative behaviors of systems in public goods game and snow game are also discussed.

On the geometric structures in evolutionary games on square and triangular lattices

We study a model of a spatial evolutionary game, based on the Prisoner’s dilemma for two regular arrangements of players, on a square lattice and on a triangular lattice. We analyze steady state distributions of players which evolve from irregular, random initial configurations. We find significant differences between the square and triangular lattice, and we characterize the geometric structures which emerge on the triangular lattice.