Chapter summary

This chapter presents the core, which is the most important set solution concept for coalitional games. The core consists of all coalitionally rational imputations: for every imputation in the core and every coalition, the total amount that the members of the coalition receive according to that imputation is at least the worth of the coalition.

The core of a coalitional game may be empty. A condition that characterizes coalitional games with a nonempty core is provided in Section 17.3. A game satisfying this condition is called a balanced game and the Bondareva–Shapley Theorem states that the core of a coalitional game is nonempty if and only if the game is balanced. This characterization is used in Section 17.4 to prove that every market game has a nonempty core. A game is called totally balanced if the cores of all its subgames are nonempty. It is proved that a game is totally balanced if and only if it is a market game. Similarly, a game is totally balanced if and only if it is the minimum of finitely many additive games.

In Section 17.6 it is proved that the core is a consistent solution concept with respect to the Davis–Maschler definition of a reduced game; that is, for every imputation in the core and every coalition, the restriction of the imputation to that coalition is in the core of the Davis–Maschler reduced game to that coalition.