Summary: We present a scheme which associates to a generalized quasi-variational inequality a dual problem and generalized Kuhn-Tucker conditions. This scheme allows to solve the primal and the dual problems in the spirit of the classical Lagrangian duality for constrained optimization problems and extends, in not necessarily finite dimensional spaces, the duality approach obtained by A. Auslender for generalized variational inequalities. An application to social Nash equilibria is presented together with some illustrative examples.