Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, 31261, Dhahran, Saudi Arabia.
Department of Mathematics and Statistics, University of Montreal, Montreal, H3C 3J7, Canada.
J Math Biol. 2022 Sep 12;85(3):27. doi: 10.1007/s00285-022-01789-1.
We consider interactions between players in groups of size [Formula: see text] with payoffs that not only depend on the strategies used in the group but also fluctuate at random over time. An individual can adopt either cooperation or defection as strategy and the population is updated from one time step to the next by a birth-death event according to a Moran model. Assuming recurrent symmetric mutation and payoffs to cooperators and defectors according to the composition of the group whose expected values, variances, and covariances are of the same small order, we derive a first-order approximation for the average abundance of cooperation in the selection-mutation equilibrium. In general, we show that increasing the variance of any payoff for defection or decreasing the variance of any payoff for cooperation increases the average abundance of cooperation. As for the effect of the covariance between any payoff for cooperation and any payoff for defection, we show that it depends on the number of cooperators in the group associated with these payoffs. We study in particular the public goods game, the stag hunt game, and the snowdrift game, all social dilemmas based on random benefit b and random cost c for cooperation, which lead to correlated payoffs to cooperators and defectors within groups. We show that a decrease in the scaled variance of b or c, or an increase in their scaled covariance, makes it easier for weak selection to favor the abundance of cooperation in the stag hunt game and the snowdrift game. The same conclusion holds for the public goods game except that the variance of b has no effect on the average abundance of C. Moreover, while the mutation rate has little effect on which strategy is more abundant at equilibrium, the group size may change it at least in the stag hunt game with a larger group size making it more difficult for cooperation to be more abundant than defection under weak selection.
我们考虑大小为 [Formula: see text] 的群体中玩家之间的相互作用，其收益不仅取决于群体中使用的策略，而且还随时间随机波动。个体可以采用合作或背叛作为策略，并且种群根据 Moran 模型通过出生-死亡事件从一个时间步更新到下一个时间步。假设经常出现对称突变，并且根据群体组成给予合作者和背叛者报酬，其期望、方差和协方差都具有相同的小阶数，我们推导出选择-突变平衡中合作平均丰度的一阶近似值。一般来说，我们表明，增加任何背叛收益的方差或减少任何合作收益的方差都会增加合作的平均丰度。至于合作的任何收益与背叛的任何收益之间的协方差的影响，我们表明它取决于与这些收益相关的群体中的合作者数量。我们特别研究了公共物品博弈、猎鹿博弈和雪堆博弈，所有这些博弈都是基于合作的随机收益 b 和随机成本 c，这导致群体内部合作者和背叛者的相关收益。我们表明，b 或 c 的缩放方差减小，或者它们的缩放协方差增加，使得弱选择更容易有利于猎鹿博弈和雪堆博弈中合作的丰度。公共物品博弈也得出了相同的结论，除了 b 的方差对 C 的平均丰度没有影响。此外，虽然突变率对哪种策略在平衡时更丰富的影响不大，但群体大小可能会改变它，至少在猎鹿博弈中，群体越大，弱选择下合作比背叛更丰富的可能性就越小。
