  Cooperation within asymmetric populations has garnered significant attention
in evolutionary games. This paper explores cooperation evolution in populations
with weak and strong players, using a game model where players choose between
cooperation and defection. Asymmetry stems from different benefits for strong
and weak cooperators, with their benefit ratio indicating the degree of
asymmetry. Varied rankings of parameters including the asymmetry degree,
cooperation costs, and benefits brought by weak players give rise to scenarios
including the prisoner's dilemma (PDG) for both player types, the snowdrift
game (SDG), and mixed PDG-SDG interactions. Our results indicate that in an
infinite well-mixed population, defection remains the dominant strategy when
strong players engage in the prisoner's dilemma game. However, if strong
players play snowdrift games, global cooperation increases with the proportion
of strong players. In this scenario, strong cooperators can prevail over strong
defectors when the proportion of strong players is low, but the prevalence of
cooperation among strong players decreases as their proportion increases. In
contrast, within a square lattice, the optimum global cooperation emerges at
intermediate proportions of strong players with moderate degrees of asymmetry.
Additionally, weak players protect cooperative clusters from exploitation by
strong defectors. This study highlights the complex dynamics of cooperation in
asymmetric interactions, contributing to the theory of cooperation in
asymmetric games.
