  Any individual's preference represents his choice in the set of available
options. It is said to be complete if the person can compare any pair of
available options. We aim to initiate the notion of projected solutions for the
generalized Nash equilibrium problem with non-ordered (not necessarily complete
and transitive) preferences and non-self constraint map. We provide the
necessary and sufficient conditions under which projected solutions of a
quasi-variational inequality and the considered GNEP coincide. Based on this
variational reformulation, we derive the occurrence of projected solutions for
the considered GNEP. Alternatively, by using a fixed point result, we ensure
the existence of projected solutions for the considered GNEP without requiring
the compactness of choice sets.
