To investigate the evolution of cooperation in spatial public goods games, this paper establishes a bi-random geometric graph, in which two types of nodes, representing players and public goods respectively, are placed at random locations in the unit square. Each public good has a limit influence range and the individuals that fall into the same range engage in a public good game. In contrast to the classical network models consisting of only one type of nodes, the bi-random geometric graph provides a natural way to describe the scenarios where individuals and public resources are independent of each other. Numerical simulations reveal that cooperation can be significantly promoted when the group size and the average number of groups that each player participates in are relatively small, which is at odds with the results on the square lattice, but is consistent with a body of empirical evidence reported by Ostrom (the winner of the 2009 Nobel Prize in Economics) and Olson et al. Analysis of the evolutionary process suggests that the facilitation of cooperation is due primarily to the formation of the cooperative clusters which can effectively resist the invasion of the defectors.

中文翻译：

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双随机几何图上的进化公共物品博弈

为了研究空间公益游戏中合作的演变，本文建立了一个双随机几何图，将分别代表参与者和公益的两种类型的节点放置在单位正方形中的随机位置。每个公共物品都有一个极限影响范围，并且属于同一范围的个人参与公共物品博弈。与仅由一种类型的节点组成的经典网络模型相比，双随机几何图提供了一种自然的方式来描述个人和公共资源彼此独立的情况。数值模拟显示，当每个参与者参加的小组规模和平均小组数目相对较小时，可以显着促进合作，这与方格上的结果不一致，但与Ostrom（2009年诺贝尔经济学奖获得者）和Olson等人报道的大量经验证据相一致。对进化过程的分析表明，合作的便利主要是由于可以有效抵抗叛逃者入侵的合作集群的形成。