Bondareva–Shapley theorem

The Bondareva–Shapley theorem states that a finite transferable-utility cooperative game has a nonempty core if and only if it is balanced.

Balanced game: A cooperative game in which every balanced collection of coalitions has total worth no greater than the grand coalition. The theorem’s condition is precisely that the game be balanced.
Transferable-utility game: A cooperative game in which coalition payoffs can be redistributed freely among members. The theorem applies to this class of games.
Assignment game: A cooperative game modeling gains from matching buyers and sellers in pairs. Its core captures stable prices and allocations in two-sided matching markets.
Shapley value: A cooperative-game payoff rule assigning each player their average marginal contribution across coalition orders. It selects a payoff, whereas the theorem only guarantees that some core payoff exists.
Balanced collection: A weighted collection of coalitions in which each player’s total weight across coalitions containing them equals one. These weights test whether coalition claims can jointly exceed the grand coalition’s worth.
Characteristic function: A function assigning each coalition the worth it can guarantee through cooperation. Its coalition values are the data tested by balancedness and the core.
Market game: A cooperative game whose coalition values arise from trading among coalition members. Balancedness helps establish nonempty cores for economies with transferable utility.
Nucleolus: A cooperative-game solution minimizing the largest coalition dissatisfaction in lexicographic order. It selects an allocation even when the core is empty, unlike the theorem’s existence criterion.
Core (game theory): The set of feasible payoff allocations that give every coalition at least its worth. The theorem characterizes exactly when this set is nonempty.
Coalition (game theory): A group of players considered as a unit capable of cooperating in a game. Balanced collections combine coalitions from the game’s player set.

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Cooperative game theoryRelated: It gives a precise condition for stable allocations to exist.