A Note on the Pre-Kernel and Pre-Nucleolus for Bankruptcy Games

A commonly held belief of game theorists is that the pre-nucleolus of the modest bankruptcy game is identical to the pre-nucleolus of its dual game, the greedy bankruptcy game. In this paper we provide some results that indicate that this belief is false. We establish some non-coincidence results of the pre-kernel and pre-nucleolus of the modest bankruptcy game with respect to the pre-kernel and pre-nucleolus of its corresponding greedy bankruptcy game.By clarifying the game theoretical solution concepts applied in the literature we are able to show that the pre-kernel and the pre-nucleolus coincide with the anti-pre-kernel and the anti-pre-nucleolus of its associated greedy bankruptcy game,respectively.Probably this is the cause of the belief that both nucleoli are identical.At the end of the paper,we highlight the results with some illustrative examples.