We revisit in this paper the relation between evolution of species and the mathematical tool of evolutionary games which has been used to model and predict it. We indicate known shortcoming of this model that restricts the capacity of evolutionary games to model groups of individuals that share a common gene or a common fitness function. In this paper we provide a new concept to remedy this shortcoming in the standard of evolutionary games in order to cover this kind of behavior. Further, we explore the relationship between this new concept and Nash equilibrium or ESS. We indicate through the study of many examples as Hawk-Dove game, Stag-Hunt Game and Prisoner's Dilemma, that when taking into account a utility that is common to a group of individuals, the equilibrium structure may change dramatically.