In this chapter we look at a number of core concepts generalised for games in partition function form. When we talk about the core we mostly mean the coalition structure core but this should not lead to confusion. The reason for the multiplicity of solutions lies in the difficulty to generalise such a simple notion as the dominance relation. The core is driven by the assumption that each coalition has a well-defined value and will deviate or block the current proposal if this demand is not met. In the case of partition function form games, the payoff of a coalition depends on the partition it is embedded in; there is no reason to exclude the possibility that the same coalition C will have a very low payoff in some embedding partition 𝒫 but a very high payoff in partition 𝒬. Given a particular payoff-configuration, a deviation resulting in a payoff V(C,𝒬) may be desirable, while at the same time V(C,𝒫) is not. In order to understand the decision of coalition C, and thereby the stability of the status quo, it is crucial to understand the forces that shape the emerging partition. In the following we look at a number of models, each with different assumptions on what the partition will be.
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