Some infinite-player generalizations of Scarf's theorem: Finite-coalition α-cores and weak α-cores

In this paper, we first obtain some infinite-dimension versions of Scarf's theorem. Second, we provide two generalizations of Scarf (1971) to normal-form games with infinitely many players. Under the assumptions analogous to Scarf (1971), we prove the nonemptiness of the finite-coalition α-core. Furthermore, by strengthening the assumptions, we obtain the nonemptiness of the weak α-core, and show that the weak α-core coincides with the closed-coalition α-core.