Strategy dominance in mixed games under weak selection

Although natural selection systematically favors defectors over cooperators in unstructured populations, evolutionary dynamics in structured populations reveal substantially greater complexity. Moreover, multi-game environments have emerged as a critical frontier in evolutionary game theory. Whereas prior research predominantly focused on static single-game scenarios, growing scholarly attention now addresses mixed-game frameworks that better capture the dynamical heterogeneity of real-world interactions. This work investigates strategy dominance in mixed games comprising two game types, examining both fixed and stochastically varying game mixtures. Through rigorous mathematical analysis and computational simulations, we establish that under weak selection conditions, the criterion for strategy dominance is equivalent to that of an average single game. This equivalence provides a unifying framework for simplifying complex multi-game interactions and advances mechanistic understanding of cooperation evolution in structured populations.