Nonemptiness of the f‐Core Without Comprehensiveness

This paper analyzes the coalition structure core when coalitions have a finite number of players in atomless NTU games. Kaneko and Wooders showed that when there are finite types of players the above notion of the core (the f ‐core) is nonempty. In this paper, we provide a direct proof of the above result using Kakutani's fixed‐point theorem when the sizes of coalitions are not only finite but also bounded above. This condition simplifies the presentation of the model and the existence proof. Unlike previous work, we dispense with the comprehensiveness assumption in NTU games, thereby broadening the applicability of our result to include matching problems and hedonic coalition formation models. Furthermore, we show that, in the absence of comprehensiveness, f ‐core allocations may fail to exhibit equal treatment in payoffs for the same type of players. We also note that Scarf's nonemptiness result for the core of NTU games follows as a corollary of our main theorem.