A generalization of Scarf's theorem: An α-core existence theorem without transitivity or completeness

Consider an n-person non-cooperative game. A coalition is said to α-block a given social state if it has a feasible strategy with which the coalition can ensure a social state preferred by all the agents in it regardless of the strategies the other agents outside the coalition may choose. The α-core is the set of social states that cannot be α-blocked. This paper shows the non-emptiness of the α-core without the assumption of transitivity or completeness on the agents' preference relations.