Characterization of the Pre-Kernel by Solution Sets

Abstract

We can derive several inclusion and interference results between the minimum sets of quadratic functions and the pre-kernel. These results enable us to give a full characterization of the pre-kernel in terms of constrained minimum sets, or restricted sub-differential of the corresponding conjugations of quadratic functions, that is, we implicitly base the representation of the pre-kernel on the Fenchel-Moreau conjugation of the characteristic function. In a further step additional results related to the vector spaces of balanced excesses are attained which allow us to give a replication result. Having worked out these auxiliary results we then turn our attention to the issue whether it is possible to replicate any arbitrary payoff vector on the domain as a pre-kernel element of a game constructed from a payoff equivalence class that contains this payoff vector. There, we provide an impossibility theorem. Moreover, we also address the reverse issue if any pre-kernel solution of a default game can be supported as a pre-kernel element of a related game from the same game space. This issue can be partly affirmed. It is shown that any pre-kernel belonging to a payoff set which satisfies the non-empty interior property is replicable as a pre-kernel element of a related game. From the replication result further results on the structure of the pre-kernel are established, for instance, on its disconnectedness.