In this paper, a generalization of the satisfaction equilibrium (SE) for games in satisfaction form (SF) is presented. This new solution concept is referred to as the generalized satisfaction equilibrium (GSE). In games in SF, players choose their actions to satisfy an individual constraint that depends on the actions of all the others. At a GSE, players that are unsatisfied are unable to unilaterally deviate to be satisfied. The concept of GSE generalizes the SE in the sense that it allows mixed-strategy equilibria in which there exist players who are unable to satisfy their individual constraints. The pure-strategy GSE problem is closely related to the constraint satisfaction problem and finding a pure-strategy GSE is proven to be NP-hard. The existence of at least one GSE in mixed strategies is proven for the class of games in which the constraints are defined by a lower limit on the expected utility. A dynamics referred to as the satisfaction response is shown to converge to a GSE in certain classes of games. Finally, Bayesian games in SF and the corresponding Bayesian GSE are introduced. These results provide a theoretical framework for studying service-level provisioning problems in communications networks as shown by several examples.