<p>We study the generalized ordinal Nash games defined over Banach spaces by employing variational techniques. To reformulate these games in terms of quasi-variational inequality problems, we will first form a suitable principal operator and study some significant properties of this operator. Then, we deduce the sufficient conditions to obtain an equilibrium for the proposed game by solving an auxiliary quasi-variational inequality. Based on this quasi-variational reformulation, we derive the existence of equilibrium for generalized ordinal Nash games with mid-point continuous preference maps. We apply the derived results to ensure the presence of Pareto equilibrium for multi-objective games and dynamic electricity markets.</p>