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We consider a model where the same group of players is involved in more than one cooperative (transferable utility) game. A rule determines the payoffs per game, and for each player a utility function evaluates the resulting vector of payoffs. We assume that each player, independently, can make transfers of worth between different games, thereby affecting its payoff vector and, thus, utility. Two transfer systems are considered, resulting in two distinct noncooperative games, and the focus of the paper is on establishing existence and a characterization of Nash equilibria in these games.