We investigate the relationship between certain subsets of the core for TU market games and competitive payoff vectors of certain markets linked to that game. This

can be considered as the case in between the two extreme cases of Shapley and Shubik (1975). They remark already that their result can be extended to any closed convex subset of the core, but they omit the details of the proof which we present here. This more general case is in particular interesting, as the two theorems of

Shapley and Shubik (1975) are included as special cases.