Here, we consider a game-theoretic model of multilevel selection in which individuals compete based on their payoff and groups also compete based on the average payoff of group members. Our focus is on multilevel social dilemmas: games in which individuals are best off cheating, while groups of individuals do best when composed of many cooperators. We analyze the dynamics of the two-level replicator dynamics, a nonlocal hyperbolic PDE describing deterministic birth–death dynamics for both individuals and groups. While past work on such multilevel dynamics has restricted attention to scenarios with exactly solvable within-group dynamics, we use comparison principles and an invariant property of the tail of the population distribution to extend our analysis to all possible two-player, two-strategy social dilemmas. In the Stag–Hunt and similar games with coordination thresholds, we show that any amount of between-group competition allows for fixation of cooperation in the population. For the prisoners’ dilemma and Hawk–Dove game, we characterize the threshold level of between-group selection dividing a regime in which the population converges to a delta function at the equilibrium of the within-group dynamics from a regime in which between-group competition facilitates the existence of steady-state densities supporting greater levels of cooperation. In particular, we see that the threshold selection strength and average payoff at steady state depend on a tug-of-war between the individual-level incentive to be a defector in a many-cooperator group and the group-level incentive to have many cooperators over many defectors. We also find that lower-level selection casts a long shadow: If groups are best off with a mix of cooperators and defectors, then there will always be fewer cooperators than optimal at steady state, even in the limit of infinitely strong competition between groups.

中文翻译：

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一般二策略社会困境的多级复制动力学分析

在这里，我们考虑多层次选择的博弈论模型，其中个人根据他们的收益进行竞争，而群体也根据群体成员的平均收益进行竞争。我们的重点是多层次的社会困境：在游戏中，个人最好不要作弊，而个人团体在由许多合作者组成时表现最好。我们分析了两级复制器动力学的动力学，这是一种描述个人和群体确定性生死动力学的非局部双曲偏微分方程。虽然过去关于这种多级动态的工作仅限于关注具有完全可解组内动态的场景，但我们使用比较原则和人口分布尾部的不变属性将我们的分析扩展到所有可能的双人、双策略社交困境。在 Stag-Hunt 和具有协调阈值的类似游戏中，我们表明，任何数量的组间竞争都允许固定人口中的合作。对于囚徒困境和 Hawk-Dove 博弈，我们描述了群体间选择的阈值水平，将群体收敛到群体内动态平衡的 delta 函数的制度与群体间的制度划分为竞争促进了支持更高水平合作的稳态密度的存在。特别是，我们看到，阈值选择强度和稳态下的平均收益取决于在多合作者群体中成为叛逃者的个人层面激励与拥有许多合作者的群体层面激励之间的拉锯战。在许多叛逃者身上。