  The preferences of players in non-cooperative games represent their choice in
the set of available options, which meet the completeness property if players
are able to compare any pair of available options. In the existing literature,
the variational reformulation of generalized Nash games relies on the numerical
representation of players' preferences into objective functions, which is only
possible if preferences are transitive and complete. In this work, we first
characterize the jointly convex generalized Nash equilibrium problems in terms
of a variational inequality without requiring any numerical representation of
preferences. Furthermore, we provide the suitable conditions under which any
solution of a quasi-variational inequality becomes an equilibrium for the
generalized Nash game with non-ordered (incomplete and non-transitive)
non-convex inter-dependent preferences. We check the solvability of these games
when the strategy maps of players are (possibly) unbounded. As an application,
we derive the occurrence of competitive equilibrium for Arrow-Debreu economy
under uncertainty.
