Fixed-point approaches to the proof of the Bondareva–Shapley Theorem

We provide two new proofs of the Bondareva–Shapley theorem, which states that the core of a transferable utility cooperative game has a nonempty core if and only if the game is balanced. Both proofs exploit the fixed points of self-maps of the set of imputations, applying elementary existence arguments typically associated with noncooperative games to cooperative games.