This work is about formalising game-theoretic results using proof assistants. In previous work, we have formalised a result both in Coq and Isabelle. The result is as follows: starting from a two-player game with finitely many outcomes, one may derive a game by rewriting each of these outcomes with either the basic outcome “Player 1 wins” or “Player 2 wins”. If all ways of deriving such a win/lose (w/l) game lead to a game where one player has a winning strategy, then the original game has a Nash equilibrium (NE). Here, we present an application of this work to parity games and priority games using Isabelle. Dittmann has proven in Isabelle that parity games are positionally determined. First, we generalise this result to priority games, where parity is replaced by an arbitrary winning set W . Secondly, we consider preference priority games, i.e., sequential games where players have preferences over outcomes. We show that such games have an NE.