Abstract
In this chapter, we lay the foundation for turning our focus from dynamic optimization, which has been the subject of preceding chapters, to the notion of a dynamic game. To fully appreciate the material presented in subsequent chapters, we must in the present chapter review some of the essential features of the theory of finite-dimensional variational inequalities and static noncooperative mathematical games. Today many economists and engineers are exposed to the notion of a game-theoretic equilibrium that we study in this chapter, namely Nash equilibrium. Yet, the relationship of such equilibria to certain nonextremal problems known as fixed-point problems, variational inequalities and nonlinear complementarity problems is not widely understood. It is the fact that, as we shall see, Nash and Nash-like equilibria are related to and frequently equivalent to nonextremal problems that makes the computation and qualitative investigation of such equilibria so tractable. Although the static games discussed in this chapter are really steady states of dynamic games, we are, for the most part, indifferent in this chapter to any underlying dynamics. We also comment that readers familiar with finite-dimensional variational inequalities and static Nash games may wish to skip this chapter.
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List of References Cited and Additional Reading
Auchmuty, G. (1989). Variational principles for variational inequalities. Numerical Functional Analysis and Optimization 10, 863–874.
Auslender, A. (1976). Optimisation: Méthodes Numériques. Paris: Masson.
Bakusinskii, A. B. and B. T. Poljak (1974). On the solution of variational inequalities. Soviet Mathematics Doklady 15, 1705–1710.
Bertsekas, D. P. and E. M. Gafni (1982). Projection methods for variational inequalities with application to the traffic assignment problem. Mathematical Programming Study 17, 139–159.
Browder, F. E. (1966). Existence and approximation of solutions of nonlinear variational inequalities. Proceedings of the National Academy of Sciences 56, 1080–1086.
Cottle, R. W., J. S. Pang, and R. E. Stone (1992). The Linear Complementarity Problem. Boston: Academic Press.
Dafermos, S. C. (1980). Traffic equilibrium and variational inequalities. Transportation Science 14, 42–54.
Dafermos, S. C. (1983). An iterative scheme for variational inequalities. Mathematical Programming 26, 40–47.
Facchinei, F. and J.-S. Pang (2003a). Finite-Dimensional Variational Inequalities and Complementarity Problems, Volume I. New York: Springer-Verlag.
Facchinei, F. and J.-S. Pang (2003b). Finite-Dimensional Variational Inequalities and Complementarity Problems, Volume II. New York: Springer-Verlag.
Fiacco, A. V. and G. P. McCormick (1990). Nonlinear Programming: Sequential Unconstrained Minimization Techniques. Reprint. Philadelphia: Society for Industrial and Applied Mathematics.
Friesz, T., D., N. Bernstein, R. T. Mehta, and S. Ganjalizadeh (1994). Day-to-day dynamic network disequilibrium and idealized driver information systems. Operations Research 42, 1120–1136.
Friesz, T. L., P. A. Viton, and R. L. Tobin (1985). Economic and computational aspects of freight network equilibrium: a synthesis. Journal of Regional Science 25, 29–49.
Fukushima, M. (1992). Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems. Mathematical Programming 53, 99–110.
Goldsman, L. and P. T. Harker (1990). A note on solving general equilibrium problems with variational inequality techniques. Operations Research Letters 9, 335–339.
Hammond, J. H. (1984). Solving asymmetric variational inequality problems and systems of equations with generalized nonlinear programming algorithms. Ph. D. thesis, Department of Mathematics, MIT.
Harker, P. T. (1983). Prediction of intercity freight flows: theory and application of a generalized spatial price equilibrium model. Ph. D. thesis, University of Pennsylvania.
Harker, P. T. (1988). Accelerating the convergence of the diagonalization and projection algorithms for finite-dimensional variational ineqaulities. Mathematical Programming 41, 29–59.
Harker, P. T. and J.-S. Pang (1990). Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms, and applications. Mathematical Programming 48, 161–220.
Nagurney, A. (1987). Competitive equilibrium problems, variational inequalities, and regional science. Journal of Regional Science 27, 55–76.
Ortega, J. M. and W. C. Rheinboldt (2000). Iterative Solution of Nonlinear Equations in Several Variables (Reprint ed.). Society for Industrial and Applied Mathematics.
Pang, J.-S. and D. Chan (1982). Iterative methods for variational and complementary problems. Mathematical Programming 24, 284–313.
Peng, J.-M. (1997). Equivalence of variational inequality problems to unconstrained minimization. Mathematical Programming 78, 347–355.
Scarf, H. E. (1967). The approximation of fixed points of a continuous mapping. SIAM Journal of Applied Mathematics 15, 1328–1343.
Smith, T. E., T. L. Friesz, D. Bernstein, and Z. Suo (1997). A comparison of two minimum norm projective dynamic systems and their relationship to variational inequalities. In M. Ferris and J. S. Pang (Eds.), Complementarity and Variational Problems, pp. 405–424. SIAM.
Tobin, R. L. (1986). Sensitivity analysis for variational inequalities. Journal of Optimization Theory and Applications 48, 191–204.
Todd, M. J. (1976). The Computation of Fixed Points and Applications. New York: Springer-Verlag.
Wu, J. H., M. Florian, and P. Marcotte (1993). A general descent framework for the monotone variational inequality problem. Mathematical Programming 61, 281–300.
Yamashita, N., K. Taji, and M. Fukushima (1997). Unconstrained optimization reformulations of variational inequality problems. Journal of Optimization Theory and Applications 92(3), 439–456.
Zangwill, W. I. and C. B. Garcia (1981). Pathways to Solutions, Fixed Points, and Equilibria. Englewood Cliffs, NJ: Prentice-Hall.
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Friesz, T.L. (2010). Finite Dimensional Variational Inequalities and Nash Equilibria. In: Dynamic Optimization and Differential Games. International Series in Operations Research & Management Science, vol 135. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-72778-3_5
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