Evolution of cooperation with respect to fixation probabilities in multi-player games with random payoffs

We study the effect of variability in payoffs on the evolution of cooperation ($C$) against defection ($D$) in multi-player games in a finite well-mixed population. We show that an increase in the covariance between any two payoffs to $D$, or a decrease in the covariance between any two payoffs to $C$, increases the probability of ultimate fixation of $C$ when represented once, and decreases the corresponding fixation probability for $D$. This is also the case with an increase in the covariance between any payoff to $C$ and any payoff to $D$ if and only if the sum of the numbers of $C$-players in the group associated with these payoffs is large enough compared to the group size. In classical social dilemmas with random cost and benefit for cooperation, the evolution of $C$ is more likely to occur if the variances of the cost and benefit, as well as the group size, are small, while the covariance between cost and benefit is large.