بازی برف توده تکراری در میان ماموران سیار با انتظار پاداش نزدیک بین بر اساس قانون تصمیم گیری: تحقیقات عددی و تحلیلی

Iterated spatial game among mobile population is an interesting problem in the fields of biological, social and economic sciences. Inspired by some recent works, this paper concentrates on iterated snowdrift game among movable and myopic agents. Two difference decision-making schemes, namely the utility-maximum rule and the Fermi rule, are applied and examined. In the former case, cooperation is found to be enhanced by moving velocity with an upperbound. The analytical results of the model are deduced at two extreme cases when agents cannot move or move quickly. In the latter case, the influence of velocity and temptation-to-defect are much more complicate. These results allow a deeper insight of related model as well as the emergence of cooperation.

The emergence of cooperation among selfish individuals has attracted an amount of attention. In this light, game theory describes such cooperation and competition between individuals. In relation to this theory, two simple but classic games have been developed: the prisoner’s dilemma game(PDG) and the snowdrift game (SDG). In each game, an individual has two choices: to cooperate (C strategy) or to defect (D strategy). A cooperator earns reward RR if he faces another cooperator, and receives reward SS if his opponent chooses to defect (in this situation the defector earns reward TT, TT is often the highest reward, and is known as the ‘temptation-to-defect’). If both defect, then each earns reward PP. A reward matrix describes the rewards of the four different combinations as follows, View the MathML source{RSTP}. Turn MathJax on The reward matrices of PDG and SDG differ. In PDG, the reward matrix satisfies T>R>P≥ST>R>P≥S, whereas that in SDG is T>R>S>PT>R>S>P. Hence, always-defect is classically considered the best choice for both sides in PDG, and cooperators do not survive. By contrast, SDG encourages cooperation. Strategy C favors an agent when his opponent plays D, but D is the best response if his opponent plays C. Nonetheless, cooperators do not dominate the system in either case. Spatial evolutionary game theory, which was pioneered by Nowak and May [1], greatly enhances research on the emergence of cooperation. In their model, an agent plays PDG with his “neighbors” in a 2D lattice structure. Cooperators gather in groups and survive the invasion of defectors. This work indicates that repetition and spatial structure are critical to the improvement of cooperative behavior. The evolutionary PDG has since been extended to other spatial structures. In particular, the evolutionary PDG in complex networks has received much attention. Complex networks are considered good frameworks for the emergence of cooperation [2]. In addition to spatial structures, related studies also investigate the behavioral scheme of game agents. Factors such as age, individual learning abilities, and memory have been incorporated into new models to mimic real-world behaviors [3], [4], [5] and [6]. Meanwhile, SDG is also widely played in the real world. It explains many phenomena in the microbial community [7], animal population [8], and human society [9], [10] and [11]. Existing studies have unveiled some interesting characteristics. For instance, the 2D lattice structure suppresses cooperative behaviors in SDG [12], which is contrary to the PDG. Although the conclusion that SDG facilitates cooperative behavior is supported by existing studies, the cooperation ratio is sensitive to model configuration. Ref. [13] investigated the evolution of cooperative SDG behaviors in a small-world network. The results indicated that network structure was crucial to game dynamics. Ref. [14] studied SDG in several different spatial structures and reported that cooperation is either inhibited or promoted depending on the topological structure of the network, update scheme, and the rewards. Similar results can be found in other studies [15], [16], [17] and [18]. Among the mobile population, the iterated spatial game is also interesting [19], [20], [21], [22], [23] and [24]. Recently, Jia and Ma [24] proposed a spatial evolutionary SDG model for the mobile population. Their model considered several realistic constraints, including the one-to-one pairing scheme and information privacy. They found that the moving ability enhances cooperation within a large range of parameters, whereas depresses it in a small region, which are different from known results. This paper concentrates on a modified version of the model in Ref. [24], when agents are all myopic, who can only remember information from the most recent turn of game. This ‘myopic’ assumption is very popular in evolutionary game research [17], [25] and [26], and effectively describes real-world observations [27] and [28]. Based on this assumption, we propose two different models according to different decision-making rules. First, we examine the model under the ‘utility-maximum’ (UM) decision-making rule and note that cooperation is enhanced by moving velocity. However, the cooperation level has an upper-bound limit. Then the analytical results of the model are deduced when agents cannot move or move quickly. Subsequently, we investigate the model under the Fermi/Logit decision-making rule. In this model, the temptation-to-defect may promote cooperation. This finding is seldom reported in existing literature. Furthermore, we discuss the microscopic dynamics behind this observation. These results enrich understanding regarding moving-agent models and the emergence of cooperation. This paper proceeds as follows: Section 2 defines the model; Section 3 presents and analyzes the simulation results of the UM model; Section 4 investigates the Fermi model; and Section 5 concludes the paper and makes a brief discussion of the results.

In this study, we propose an evolutionary spatial snowdrift game among mobile populations. The model contains numerous agents that move randomly on a 2D surface. In each turn, every agent is paired with its nearest neighbor. Successfully paired agents then engage in snowdrift game with each other. All agents are assumed to be myopic, who decide whether to cooperate or to defect based on the expected reward of each strategy. We numerically simulate this situation under two different decision-making rules, namely, the UM and the Fermi/Logit rules. Under UM rule, cooperation is promoted by mobility throughout the entire parameter range. By investigating the state transition of a single agent, we obtained the analytical results of the immobile and high-velocity models. The low-velocity model can be regarded as a moderate case. The model under Fermi/Logit rule is richer than the UM model. vv may either prompt or suppress cooperation depending on rr. In the low-velocity region, fCfC increased with rr. This finding is seldom reported in previous research. These observations are also supported by the state transition dynamics of agents. Our model varies from existing models. Aside from the previously discussed decision-making rules, it differs in terms of interaction scheme. In most existing models, an agent constructs a relationship network with all of its neighbors. Thus, such models can be considered an iteration game model on a time-varied network structure. However, our model is a “random-pairing” game, and vv determines how often an agent changes opponents. Refs [33] andd [34] are recent results of random-pairing iterated game. Although different decision-making rules (replicator dynamics vs. expected-reward based) and reward matrices (prisoner’s dilemma vs. snowdrift) are used, our results still support their findings. They both pointed out that interaction stochastically with neighbors provides a favorable environment for cooperators to thrive, which is consistent with the conclusion drawn by the UM model. This finding also agrees partially with the Fermi model. In the Fermi model, the role of vv differs depending on rr. Jia and Ma [24] reached the same conclusion. Nonetheless, the most interesting finding in our study may be that rr does not always affect cooperation negatively. Hence, fCfC is non-monotonic in relation to the parameters. When vv is low, we can observe an optimal region in which fCfC is maximized in Fig. 10. This result is consistent with that of Fu and Nowak [35], who reported that the mobile-agent model contains an optimal parameter region. The results allow a deeper insight of the emergence of cooperation. One of potential improvements of the model in the future lies in the motion rules of agents. In the proposed model, the direction is stochastically determined, which is seldom applied by real-world individuals. Collective motion, such as the Vicsek style movements [22] and [23], or the profit-pursuing behavior [36], [37] and [38] could be studied in the future work.