Abstract
For the cooperative equilibria in α-core form, the balancedness and coalitionally C−security of games with usual utilities are extended to the games with discontinuous set payoffs. The existence of α−core for these games is established, which can deduce some typical results for normal form games. For continuous games, a quasi-concave-like set payoff is introduced. Based on the introduced set payoffs, the existence of α−core is also obtained. Examples are given to verify these results. Comparison with existing results in references shows that the introduced conditions are new to guarantee the existence of α−core for games with set payoffs.
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