Cooperation is widely recognized to be challenging for the well-balanced development of human societies. The emergence of cooperation in populations has been largely studied in the context of the Prisoner's Dilemma game, where temptation to defect and fear to be betrayed by others often activate defective strategies. In this paper we analyze the decision making mechanisms fostering cooperation in the two-strategy Stag-Hunt and Chicken games, which include the mixed strategy Nash equilibrium, describing partially cooperative behavior. We find the conditions for which cooperation is asymptotically stable in both full and partial cases, and we show that the partially cooperative steady state is also globally stable in the simplex. Furthermore, we show that the last can be more rewarding than the first, thus making the mixed strategy effective, although people cooperate at a lower level with respect to the maximum allowed, as it is reasonably expected in real situations. Our findings highlight the importance of Stag-Hunt and Chicken games in understanding the emergence of cooperation in social networks.