Associated Games to Optimize the Core of a Transferable Utility Game

Abstract

In view of the core optimization, this paper establishes a new associated game starting from one with a nonempty core and proposes a sequence of associated games recursively. We prove that the cores of the associated games are increasingly stable in two aspects. Firstly, the core of each game is contained in the one it precedes. Secondly, any allocation outside the core of the corresponding associated game is indirectly dominated by a certain allocation in it. Therefore, the last one of the nonempty cores in this sequence is the final optimized set. More interestingly, if this sequence does not encounter a game with an empty core, we show that it converges and that the limit game is a constant-sum one by the matrix approach. In this case, we can ideally select a unique point from the core of the original game, which is the core of such a limit game.