This paper proposes two generalization of the core and evaluates them on experimental data of assignment games (workers and firms negotiate wages and matching). The generalizations proposed allow for social utility components (e.g. altruism) and random utility components (e.g. logistic perturbations). These generalizations are well-established in analyses of non-cooperative games, and they prove to be both descriptive and predictive in the assignment games analyzed here. The "logit core" allows us to define a "stochastically more stable" relation on the outcome set, which has intuitive implications, and it fits better than alternative approaches such as random behavior cores and regression modeling.