  We study generalized games defined over Banach spaces using variational
analysis. To reformulate generalized games as 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 under which an equilibrium for the generalized game can be obtained
by solving a quasi-variational inequality. Based on this variational
reformulation, we derive the existence of equilibrium for generalized games
with non-ordered (that is, associated weak preference relations need not be
complete and transitive) and mid-point continuous preference maps.
