Snowdrift Game Research Articles

Taxation Can Significantly Enhance The Effectiveness of Reward and Punishment Mechanisms in The N-Person Snowdrift Game

We extend the reward and punishment mechanisms in the N-person evolutionary snowdrift game (NSG) model by constructing a tax-based pure punishment NSG model and a tax-based pure reward NSG model. Using replicator dynamics, we established two sets of dynamic equations that describe the evolution of the frequencies of the three strategies. Over longer time scales, both systems exhibit stable states involving two strategies, with the exclusion of the third. Introducing tax factors into the reward and punishment mechanisms significantly increases the proportion of cooperators in the steady state and eliminates defectors, with the time to reach the steady state depending on the tax rate. Moreover, we found that tax-based pure reward strategies are more conducive to fostering cooperative behavior within the system compared to tax-based pure punishment strategies. We employed numerical algorithms to simulate the replicator dynamics, and the results of the dynamic equations were in perfect agreement with the numerical simulations.

Evolutionary branching in multi-level selection models.

We study a model of group-structured populations featuring individual-level birth and death events, and group-level fission and extinction events. Individuals play games within their groups, while groups play games against other groups. Payoffs from individual-level games affect birth rates of individuals, and payoffs from group-level games affect group extinction rates. We focus on the evolutionary dynamics of continuous traits with particular emphasis on the phenomenon of evolutionary diversification. Specifically, we consider two-level processes in which individuals and groups play continuous snowdrift or prisoner's dilemma games. Individual game strategies evolve due to selection pressure from both the individual and group level interactions. The resulting evolutionary dynamics turns out to be very complex, including branching and type-diversification at one level or the other. We observe that a weaker selection pressure at the individual level results in more adaptable groups and sometimes group-level branching. Stronger individual-level selection leads to more effective adaptation within each group while preventing the groups from adapting according to the group-level games.

Evolution of cooperation with the diversity of cooperation tendencies

The complete cooperation and the complete defection are two typical strategies considered in evolutionary games in many previous works. However, in real life, strategies of individuals are full of variety rather than only two complete ones. In this work, the diversity of strategies is introduced into the weak prisoners’ dilemma game, which is measured by the diversity of the cooperation tendency. A greater diversity means more cooperation tendencies are provided. The complete cooperation strategy is the full cooperation tendency and the complete defection strategy is without any cooperation tendency. Agents with other cooperation tendencies are partial cooperative and defective. The numerical simulation shows that increasing the diversity of the cooperation tendency promotes the cooperation level, not only the number of cooperators but also the average tendency over the whole population, until the diversity reaches its saturated value. Furthermore, our work points out maintaining cooperation is based on the cooperation efficiency. When cooperator clusters can survive the cooperation efficiency slightly oscillates around its equilibrium. When cooperator clusters cannot resist the invasion of defectors the cooperation efficiency oscillates and quickly decreases to zero. The observations above is not only applied on the weak prisoner’s dilemma but also the snowdrift game. When the effect of the noise for the Femi update mechanism is considered, a greater diversity of strategies not only improves the cooperation level of the whole population but also supports the survival of more rational agents.

A social monitoring mechanism for third-party judges promotes cooperation in evolutionary games

Corruption of third-party judges seriously undermines the level of cooperation. Without intervention, more corruptors and defectors would emerge, disrupting social harmony. Therefore, introducing an anti-corruption mechanism is crucial for the evolution of cooperation. In this paper, we propose a social monitoring mechanism to monitor third-party judges so that their payoffs are affected by the proportions of cooperators. Monte Carlo simulations on periodic boundary lattices. The results show that the social monitoring mechanism is effective in promoting cooperation and inhibiting corruption, and enhances the effectiveness of zealots in promoting cooperation. This facilitation effect is not only manifested in the Prisoner's Dilemma Game but also in the Snowdrift Game, which confirms the robustness of the results. Our research provides new insights for solving social dilemmas and curbing corruption.

Memory and target payoff enhance cooperation in evolutionary social dilemmas.

We proposed a neighbor selection mechanism based on memory and target payoff, where the target payoff is the maximum value of the group's average expected payoff. According to this mechanism, individuals prioritize selecting neighbors whose average payoffs in the last M rounds are close to the target payoff for strategy learning, aiming to maximize the group's expected payoff. Simulation results on the grid-based Prisoner's Dilemma and Snowdrift games demonstrate that this mechanism can significantly improve the group's payoff and cooperation level. Furthermore, the longer the memory length, the higher the group's payoff and cooperation level. Overall, the combination of memory and target payoff can lead to the emergence and persistence of cooperation in social dilemmas as individuals are motivated to cooperate based on both their past experiences and future goals. This interplay highlights the significance of taking into account numerous variables in comprehending and promoting cooperation within evolutionary frameworks.

Adaptive payoff-driven interaction in networked snowdrift games

In social dilemmas, most interactions are transient and susceptible to restructuring, leading to continuous changes in social networks over time. Typically, agents assess the rewards of their current interactions and adjust their connections to optimize outcomes. In this paper, we introduce an adaptive network model in the snowdrift game to examine dynamic levels of cooperation and network topology, involving the potential for both the termination of existing connections and the establishment of new ones. In particular, we define the agent’s asymmetric disassociation tendency toward their neighbors, which fundamentally determines the probability of edge dismantlement. The mechanism allows agents to selectively sever and rewire their connections to alternative individuals to refine partnerships. Our findings reveal that adaptive networks are particularly effective in promoting a robust evolution toward states of either pure cooperation or complete defection, especially under conditions of extreme cost-benefit ratios, as compared to static network models. Moreover, the dynamic restructuring of connections and the distribution of network degrees among agents are closely linked to the levels of cooperation in stationary states. Specifically, cooperators tend to seek broader neighborhoods when confronted with the invasion of multiple defectors.

Co-evolution of heterogeneous cognition in spatial snowdrift game with asymmetric cost.

The emergence of the evolutionary game on complex networks provides a fresh framework for studying cooperation behavior between complex populations. Numerous recent progress has been achieved in studying asymmetric games. However, there is still a substantial need to address how to flexibly express the individual asymmetric nature. In this paper, we employ mutual cognition among individuals to elucidate the asymmetry inherent in their interactions. Cognition arises from individuals' subjective assessments and significantly influences their decision-making processes. In social networks, mutual cognition among individuals is a persistent phenomenon and frequently displays heterogeneity as the influence of their interactions. This unequal cognitive dynamic will, in turn, influence the interactions, culminating in asymmetric outcomes. To better illustrate the inter-individual cognition in asymmetric snowdrift games, the concept of favor value is introduced here. On this basis, the evolution of cognition and its relationship with asymmetry degree are defined. In our simulation, we investigate how game cost and the intensity of individual cognitive changes impact the cooperation frequency. Furthermore, the temporal evolution of individual cognition and its variation under different parameters was also examined. The simulation results reveal that the emergence of heterogeneous cognition effectively addresses social dilemmas, with asymmetric interactions among individuals enhancing the propensity for cooperative choices. It is noteworthy that distinctions exist in the rules governing cooperation and cognitive evolution between regular networks and Watts-Strogatz small-world networks. In light of this, we deduce the relationship between cognition evolution and cooperative behavior in co-evolution and explore potential factors influencing cooperation within the system.

Complex pathways to cooperation emergent from asymmetry in heterogeneous populations.

Cooperation within asymmetric populations has garnered significant attention in evolutionary games. This paper explores cooperation evolution in populations with weak and strong players, using a game model where players choose between cooperation and defection. Asymmetry stems from different benefits for strong and weak cooperators, with their benefit ratio indicating the degree of asymmetry. Varied rankings of parameters including the asymmetry degree, cooperation costs, and benefits brought by weak players give rise to scenarios including the prisoner's dilemma (PDG) for both player types, the snowdrift game (SDG), and mixed PDG-SDG interactions. Our results indicate that in an infinite well-mixed population, defection remains the dominant strategy when strong players engage in the prisoner's dilemma game. However, if strong players play snowdrift games, global cooperation increases with the proportion of strong players. In this scenario, strong cooperators can prevail over strong defectors when the proportion of strong players is low, but the prevalence of cooperation among strong players decreases as their proportion increases. In contrast, within a square lattice, the optimum global cooperation emerges at intermediate proportions of strong players with moderate degrees of asymmetry. Additionally, weak players protect cooperative clusters from exploitation by strong defectors. This study highlights the complex dynamics of cooperation in asymmetric interactions, contributing to the theory of cooperation in asymmetric games.

Effects of group size and population size on the evolutionary stability of cooperation

Evolutionary game theory has classically been developed under the implicit assumption of an infinite population. Exact analytical results for finite populations are rare, and those that exist apply to situations in which strategy sets are discrete. We rigorously analyse a standard model for the evolution of cooperation (the multi-player continuous-strategy snowdrift game) and show that in many situations in which there is a cooperative evolutionarily stable strategy (ESS) if the population is infinite , there is no cooperative ESS if the population is finite (no matter how large). In these cases, contributing nothing is a globally convergently stable finite-population ESS, implying that apparent evolution of cooperation in such games is an artefact of the infinite population approximation. The key issue is that if the size of groups that play the game exceeds a critical proportion of the population then the infinite-population approximation predicts the wrong evolutionary outcome (in addition, the critical proportion itself depends on the population size). Our results are robust to the underlying selection process.

Promotion of cooperation in evolutionary snowdrift game with heterogeneous memories

Cooperative behavior is one of main concerns in evolutionary game researches. Since memory generally plays important roles when updating strategies, many memory-based mechanisms have been proposed to promote evolution cooperation and assumed that all players have identical memory capacity. To be more realistic, a heterogeneous memory mechanism is proposed for snowdrift game in the present paper, whose core lies in: (i) The whole population is divided into two groups and each group is endowed with different memory. (ii) When updating strategies, both adopted frequency within memory interval and the payoff performance of reference strategy are taken into account. Simulation results show that the proposed mechanism results in very high cooperation levels of heterogeneous population in a wide range of the ratio r, and that the strategy updating rule can also significantly facilitate cooperation of homogeneous population. Moreover, the new mechanism reveals a “heterogeneity superiority” phenomenon, that is, the cooperation level of heterogeneous population is usually higher than the weighted sum of those of two corresponding homogeneous populations. Therefore, the proposed mechanism may have potential applications to social management.

Higher-order temporal interactions promote the cooperation in the multiplayer snowdrift game

Higher-order temporal interactions promote the cooperation in the multiplayer snowdrift game

Coupling injunctive social norms with evolutionary games

Human behaviors are strongly influenced by social norms. This work couples injunctive social norms with evolutionary games. Injunctive norms refer to behaviors which individuals should conform to and thereby entail social pressure to adhere to a particular action. Here we investigate impact of such social norms on both pairwise and multi-player games. It is shown that injunctive norms can enforce the system to either extreme, meaning that they give rise to the possibility of bi-stability (two stable equilibrium states) in evolutionary games. We derive the condition for such bi-stability in the mean-field limit. While investigating pairwise games with social norms, we observe that —besides other possibilities—Prisoner's dilemma (PD), snow-drift (SD), and Trivial (TR) games may experience the bi-stability like the stag-hunt (SH) game. Interestingly, SD game with social norms exhibits four possible asymptotic outcomes: all-cooperation (AllC), all-defection (AllD), either AllC or AllD (bi-stable), and coexistence. However, its multi-player variant does not manifest a unique coexistence equilibrium. Moreover, it demonstrates another bi-stable equilibrium, comprising either coexistence or AllC, besides other outcomes observed in the two-player case. Conversely, PD and its multi-player variant, i.e., public goods game with injunctive norms show identical evolutionary outcomes (AllD or bi-stable).

Modeling cooperative evolution in prey species using the snowdrift game with evolutionary impact on prey–predator dynamics

Cooperative evolution refers to the collaborative process by which two or more individuals or species interact and adapt together some specific behavior, leading to mutual benefits for their survival and reproduction. In this article, we employ the snowdrift game to model cooperative evolution in prey species and examine its evolutionary impact on the dynamics of prey–predator interactions. The model system consists of a replicator equation that captures the evolution of cooperative behavior in prey species, along with two growth equations of prey and predator incorporating the factors that influence cooperative evolution on prey reproduction. Our analysis reveals that in long time run, the prey population becomes polymorphic, allowing cooperative and defective individuals to coexist and persist over time. However, a very low cooperative proportion poses a risk of extinction for both prey and predator. Again if majority of prey species exhibit cooperative behavior, the predator species still faces possible extinction due to the strong group defense mechanism employed by prey. At some intermediate level of cooperative proportions, the proposed model system exhibits more rich and complex dynamics such as oscillatory patterns, bi-stable phenomena, homoclinic orbit, and saddle–node bifurcations of limit cycles. Further, we explore the transient dynamics of the proposed system through which one can anticipate the potential shifts in population size and can identify critical thresholds in promoting sustainable coexistence.

Utility coupling promotes cooperation in multiplayer snowdrift games on interdependent simplicial networks

Utility coupling promotes cooperation in multiplayer snowdrift games on interdependent simplicial networks

Facilitating cooperation in human-agent hybrid populations through autonomous agents

Cooperative AI has shown its effectiveness in solving the conundrum of cooperation. Understanding how cooperation emerges in human-agent hybrid populations is a topic of significant interest, particularly in the realm of evolutionary game theory. In this article, we scrutinize how cooperative and defective Autonomous Agents (AAs) influence human cooperation in social dilemma games with a one-shot setting. Focusing on well-mixed populations, we find that cooperative AAs have a limited impact in the prisoner's dilemma games but facilitate cooperation in the stag hunt games. Surprisingly, defective AAs can promote complete dominance of cooperation in the snowdrift games. As the proportion of AAs increases, both cooperative and defective AAs have the potential to cause human cooperation to disappear. We then extend our investigation to consider the pairwise comparison rule and complex networks, elucidating that imitation strength and population structure are critical for the emergence of human cooperation in human-agent hybrid populations.

Harmful strong agents and asymmetric interaction can promote the frequency of cooperation in the snowdrift game

The emergence and mechanism of cooperation in social dilemmas have always been fundamental issues in evolutionary game theory. In this paper, we study the snowdrift game, in which individuals in a stronger position can gain additional benefits in cooperation with weaker individuals due to differences in status. Meanwhile, innocuous-type strong individuals will not harm their partners’ interests, while harmful-type ones will. In a mixed population, the strong individuals in these two asymmetric snowdrift games are more inclined to cooperate, but the weaker ones are more willing to choose defection. The average cooperation frequency is not lower than in symmetric populations. In structured populations, on the other hand, asymmetry promotes the formation of a configuration with a cooperator cluster with a core of strong cooperators, thereby promoting cooperation among individuals. However, only within a small range of parameters the frequency of cooperation will be slightly reduced. Harmful-type strong individuals can be more conducive to promoting cooperation than innocuous-type ones in some cases. The existence of harmful and strong individuals in the snowdrift game further helps to solve the social loafing effect.

Evolution of cooperation in games with dual attribute strategy

Cooperation and defection are two common strategies in game theory. Considering that there may be a kind of middle state between the two as a third option in the real dilemma, we introduce the dual attribute strategy. Compared with cooperation, this strategy possesses the defection attribute, while compared with defection, this strategy possesses the cooperation attribute. Here, it is assumed that players holding the dual attribute strategy play snowdrift games with cooperators based on their defection attribute, meanwhile, they play another snowdrift game with defectors based on their cooperation attribute. Furthermore, we explore the evolution of the three strategies in the well-mixed population and the structured population, respectively. The results show that the dual attribute strategy is more likely to survive. Additionally, the smaller the two kinds of cost-to-benefit ratios are, the more likely cooperation is to exist. Moreover, in the well-mixed population, the existence of cooperation is independent of the temptation to defect. In the structured population, cooperation is more favorable when the cost-to-benefit ratio between the cooperation strategy and the dual attribute strategy is moderate, regardless of the temptation to defect. The dual attribute strategy is actually the embodiment of the simultaneous weakening of cooperation and defection.

Cooperative Dilemmas with Binary Actions and Multiple Players

The prisoner’s dilemma, the snowdrift game, and the stag hunt are two-player symmetric games that are often considered as prototypical examples of cooperative dilemmas across disciplines. However, surprisingly little consensus exists about the precise mathematical meaning of the words “cooperation” and “cooperative dilemma” for these and other binary-action symmetric games, in particular when considering interactions among more than two players. Here, we propose definitions of these terms and explore their evolutionary consequences on the equilibrium structure of cooperative dilemmas in relation to social optimality. We show that our definition of cooperative dilemma encompasses a large class of collective action games often discussed in the literature, including congestion games, games with participation synergies, and public goods games. One of our main results is that regardless of the number of players, all cooperative dilemmas—including multi-player generalizations of the prisoner’s dilemma, the snowdrift game, and the stag hunt—feature inefficient equilibria where cooperation is underprovided, but cannot have equilibria in which cooperation is overprovided. We also find simple conditions for full cooperation to be socially optimal in a cooperative dilemma. Our framework and results unify, simplify, and extend previous work on the structure and properties of cooperative dilemmas with binary actions and two or more players.

Reactive means in the iterated Prisoner’s dilemma

The Iterated Prisoner’s Dilemma (IPD) is a well studied framework for understanding direct reciprocity and cooperation in pairwise encounters. However, measuring the morality of various IPD strategies is still largely lacking. Here, we partially address this issue by proposing a suit of plausible morality metrics to quantify four aspects of justice. We focus our closed-form calculation on the class of reactive strategies because of their mathematical tractability and expressive power. We define reactive means as a tool for studying how actors in the IPD and Iterated Snowdrift Game (ISG) behave under typical circumstances. We compute reactive means for four functions intended to capture human intuitions about “goodness” and “fair play”. Two of these functions are strongly anticorrelated with success in the IPD and ISG, and the other two are weakly anticorrelated with success. Our results will aid in evaluating and comparing powerful IPD strategies based on machine learning algorithms, using simple and intuitive morality metrics.

Structured heterogeneity of network in evolutionary social dilemmas under assortative linking and learning

We use assortativity in linking and learning by studying social dilemmas on a dynamic network, where assortativity in linking runs faster than assortativity in learning, so that both can work together. The emergence of cooperation is based on the structured heterogeneity of a network, which includes heterogeneous interactions and a heterogeneous population structure made by assortativity. In an infinite population, cooperation is more likely to happen in the Snowdrift game when there is heterogeneous interaction, in the Stag-hunt game when there is heterogeneous population structure, and in the Prisoner’s Dilemma game when there are two different kinds of settings. In a finite population size, the heterogeneous interaction plays an important role in emergence of cooperation in all dilemmas.