This paper presents a generalization of the existing notion of satisfaction equilibrium (SE) for games in satisfaction form. The new equilibrium, which is referred to as the generalized SE (GSE), is particularly adapted for modeling problems such as service-level provisioning in decentralized self-configuring networks. Existence theorems for GSEs are provided for particular classes of games in satisfaction form and the problem of finding a pure strategy GSEs with a given number of satisfied players is shown to be NP-hard. Interestingly, for certain games there exist a dynamic, analogous to the best response of games in normal form, that is shown to efficiently converge to a pure strategy GSE under the given sufficient conditions. These contributions form a more flexible framework for studying self-configuring networks than the existing SE framework. This paper is concluded by a set of examples in wireless communications in which classical equilibrium concepts are shown to be not sufficiently adapted to model service-level provisioning. This reveals the relevance of the new solution concept of GSE.