Definition of variational inequality (VI) and complementarity problem. KKT conditions. Relations to optimization problems. examples: Nash equilibria, generalized Nash equilibria, the electricity market, transportation models. Reformulation of a VI as a fixed point problem and existence results. Applications to traffic equilibrium problems. Monotonicity and its consequences. Reformualtion of a complementarity problem as a system of equations. Fischer-Burmeister function. Properties of the correspondent merit functions. Applications to the solution of the KKT conditions. Banach fixed point theorem. Projection algorithms for the solutions of VIs. Generalized Nash Equilibrium problems and taxation. Noncooperative game theory: existence of saddle points, bimatrix games and strictly competitive games. Existence of solutions to games of general form. Mixed strategies. Linear programs and VIs for the calculation of Nash equilibria. Elements of auction theory. Introduction to cooperative game theory: nucleus (equilibrated games and Bondareva-Shapley Theorem, market games), nucleulous and Shapley theory. Power indices.

Lecture Notes available at http://www.dis.uniroma1.it/~facchinei/didattica/GE