The Core

Abstract

This chapter contains some basic results on the core of coalitional TU games. First the Bondareva-Shapley theorem which gives necessary and sufficient conditions for the nonemptiness of the core is proved. As an application of the foregoing theorem, we show that the core of a market game is nonempty. If the core of a game is nonempty, then the game is called balanced. A game is totally balanced if all of its subgames are balanced. The player set and the coalition function of a subgame are a subcoalition and the corresponding restriction of the coalition function of the game. In Section 3.3 we show that a coalitional game is a market game if and only if it is totally balanced. We prove in Section 3.4 that minimum cost spanning tree games and permutation games are totally balanced.