Elucidating the mechanisms that lead to the emergence, evolution, and survival of cooperation in natural systems is still one of the main scientific challenges of current times. During the last three decades, theoretical and computational models as well as experimental data have made it possible to unveil and explain, from an evolutionary perspective, key processes underlying the dynamics of cooperation. How-ever, many common cooperative scenarios remain elusive and at odds with Darwin's natural selection theory. Here, we study evolutionary games on populations that are structured beyond pairwise interactions. Specifically, we introduce a completely new and general evolutionary approach that allows studying situations in which indirect interactions via a neighbor other than the direct pairwise connection (or via a group of neighbors), impacts the strategy of the focal player. To this end, we consider simplicial graphs that encode two-and three-body interactions. Our simplicial game framework enables us to study the competition between all possible pairs of social dilemmas, and grants us the option to scrutinize the role of three-body interactions in all the observed phenomenology. Thus, we simultaneously investigate how social dilemmas with different Nash equilibria compete in simplicial structures and how such a competition is modulated by the unbalance of 2-and 1-simplices, which in its turn reflects the relative prevalence of pairwise or group interactions among the players. We report a number of results that: (i) support that higher-order games allow for non-dominant strategists to emerge and coexist with dominant ones, a scenario that can't be explained by any pairwise schemes, no matter the network of contacts; (ii) characterize a novel transition from dominant defection to dominant cooperation as a function of the simplicial structure of the population; and (iii) demonstrate that 2-simplex interactions are a source of strategy diversity, i.e. increasing the relative prevalence of group interactions always promotes diverse strategic identities of individuals. Our study constitutes, thus, a step forward in the quest for understanding the roots of cooperation and the mechanisms that sustain it in real world and social environments. (C) 2021 Published by Elsevier Ltd.