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Hierarchical prisoner's dilemma in hierarchical game for resource competition

机译：资源竞争的分层博弈中的分层囚徒困境

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Dilemmas in cooperation are?one of the major concerns in game theory. In a public?goods game, each individual cooperates by paying a cost or defecting?without paying it, and receives a reward from the group out of the collected cost. Thus, defecting?is beneficial for each individual, while cooperation is beneficial for the group. Now, groups (say, countries) consisting of individuals also play games. To study such a multi-level game, we introduce a hierarchical game in which multiple groups compete for limited resources by utilizing the collected cost in each group,?where the power to appropriate resources increases with the population of the group. Analyzing this hierarchical game, we found a hierarchical prisoner's dilemma, in which groups choose the defecting policy (say, armament) as a Nash strategy to optimize each group's benefit, while cooperation optimizes the total benefit. On the other hand, for each individual, refusing to pay the cost (say, tax) is a Nash strategy, which turns out to be a cooperation policy for the group, thus leading to a hierarchical dilemma. Here the group reward increases with the group size. However,?we find that there exists an optimal group size that maximizes the individual payoff. Furthermore, when the population asymmetry between two groups is large, the smaller group will choose a cooperation policy (say, disarmament) to avoid excessive response from the larger group, and the prisoner's dilemma between the groups is resolved. Accordingly, the relevance of this hierarchical game on?policy selection in society and the optimal size of human or animal groups are?discussed.
机译：合作中的困境是博弈论中的主要问题之一。在公共物品博弈中，每个人通过支付费用或叛逃而不支付费用来合作，并从收集的费用中获得小组的奖励。因此，背叛对每个人都是有益的，而合作对这个群体是有益的。现在，由个人组成的团体（例如国家）也玩游戏。为了研究这种多层次的博弈，我们引入了一种分层博弈，其中多个群体通过利用每个群体中收集的成本竞争有限的资源，其中，适当资源的力量随群体的人口而增加。通过分析这种等级博弈，我们发现了一个等级囚犯的困境，在这种困境中，各团体选择叛逃政策（例如军备）作为纳什策略来优化每个团体的利益，而合作则优化了总利益。另一方面，对于每个人来说，拒绝支付费用（例如，税收）是纳什策略，这最终成为该集团的合作政策，从而导致了等级困境。在这里，小组奖励随着小组人数的增加而增加。但是，我们发现存在一个最佳的小组规模，该小组规模可以最大程度地提高个人收益。此外，当两个群体之间的人口不对称性很大时，较小的群体将选择一种合作政策（例如裁军），以避免较大的群体做出过多的反应，从而解决了群体之间的囚徒困境。因此，讨论了这种等级博弈对社会政策选择和人类或动物群体的最佳规模的相关性。

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