Abstract Recently, much attention has been devoted to discrete preference games to model the formation of opinions in social networks. More specifically, these games model the agents’ strategic decision of expressing publicly an opinion, which is a result of an interplay between the agent’s private belief and the social pressure. However, these games have very limited expressive power; they can model only very simple social relations and they assume that all the agents respond to social pressure in the same way. In this paper, we define and study the novel class of generalized discrete preference games. These games have additional characteristics that can model social relations to allies or competitors and complex relations among more than two agents. Moreover, they introduce different levels of strength for each relation, and they personalize the dependence of each agent to her neighborhood. We show that these novel games admit generalized ordinal potential functions and, more importantly, that every two-strategy game that admits a generalized ordinal potential function is structurally equivalent to a generalized discrete preference game. This implies that the games in this novel class capture the full generality of two-strategy games in which the existence of (pure) equilibria is guaranteed by topological arguments.