Purpose: The usage of robot waiters in the hospitality industry is growing, thus increasing the number of human–robot interactions in frontline services. Focusing on robot waiters in restaurants, this study aims to propose the social cognition (SC)–psychological ownership (PO)–customer responses (CR) model, while examining the association between SC, PO, robot anthropomorphism and CR. Design/methodology/approach: The hypotheses of this study are tested using a three-step mixed-method approach that includes partial least squares structural equation modeling (PLS-SEM), necessary condition analysis (NCA) and fuzzy-set qualitative comparative analysis (fsQCA). Findings: PLS-SEM demonstrates the mediating role of psychological ownership on the relationship between SC, customer attitudes regarding being attended by a robot and revisiting intentions. Robot anthropomorphism enhances the relationship between SC and psychological ownership. NCA indicates that SC and psychological ownership are necessary conditions for the presence of favorable attitudes and revisiting intentions. FsQCA suggests that different configurations of the antecedent conditions lead to better attitudes and revisiting intentions. Practical implications: Frontline hospitality robots need to be perceived as warm, competent, responsive and adaptable to customer requests to elicit positive responses. Managers should employ attractive robots displaying anthropomorphic features. Managers need to ensure that customers have some knowledge about robots before interacting with them. Managers should also consider customer heterogeneity and the context in which the robots will be deployed. Originality/value: Based on the psychological ownership theory, this paper analyzes the relationship between SC, psychological ownership and CR. Anthropomorphism moderates the relationship between SC and psychological ownership.