The generalized Nash equilibrium problem (GNEP)
is an extension of the standard Nash equilibrium problem,
in which each player's strategy set is dependent on the rival
players' strategies. Departing from the traditional
consideration of the GNEP as a quasi-variational inequality,
we adopt a fresh view of the GNEP as a partitioned variational
inequality defined on a Cartesian product of lower-dimensional sets. This perspective allows us to fruitfully investigate (a) the local convergence of the Josephy-Newton method for solving the GNEP, and (b) the applicability of Lemke's classic method for solving the resulting linearized subproblems. Some recent applications of the GNEP to electricity modeling and multi-leader-follower games are also discussed.