In previous work, we have studied a very general formalism of two-player games relevant for applications such as model checking. We assume games in which strategies by the players lead to outcomes taken from a finite set, and each player strives for an outcome that is optimal according to his/her preferences. We have shown using the proof assistants Isabelle and Coq that if the game has a certain structure, then a Nash equilibrium exists; more precisely, any game can be abstracted by disregarding the preferences and simply saying that some outcomes are mapped to “win for player 1”, all the others to “win for player 2”. The particular structure we consider are those games for which every such abstraction leads to a game which has a determined winner. Here, we contribute several continuations of the work and their Isabelle formalisations.