Dynamic Optimization and Differential Games pp 219-265 | Cite as

# Finite Dimensional Variational Inequalities and Nash Equilibria

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## 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.

## Keywords

Nash Equilibrium Variational Inequality Variational Inequality Problem Nonlinear Complementarity Problem User Equilibrium## Preview

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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.CrossRefGoogle Scholar - Auslender, A. (1976).
*Optimisation: Méthodes Numériques*. Paris: Masson.Google Scholar - Bakusinskii, A. B. and B. T. Poljak (1974). On the solution of variational inequalities.
*Soviet Mathematics Doklady**15*, 1705–1710.Google Scholar - 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.Google Scholar - Browder, F. E. (1966). Existence and approximation of solutions of nonlinear variational inequalities.
*Proceedings of the National Academy of Sciences**56*, 1080–1086.CrossRefGoogle Scholar - Cottle, R. W., J. S. Pang, and R. E. Stone (1992).
*The Linear Complementarity Problem*. Boston: Academic Press.Google Scholar - Dafermos, S. C. (1980). Traffic equilibrium and variational inequalities.
*Transportation Science**14*, 42–54.CrossRefGoogle Scholar - Dafermos, S. C. (1983). An iterative scheme for variational inequalities.
*Mathematical Programming**26*, 40–47.CrossRefGoogle Scholar - Facchinei, F. and J.-S. Pang (2003a).
*Finite-Dimensional Variational Inequalities and Complementarity Problems*, Volume I. New York: Springer-Verlag.Google Scholar - Facchinei, F. and J.-S. Pang (2003b).
*Finite-Dimensional Variational Inequalities and Complementarity Problems*, Volume II. New York: Springer-Verlag.Google Scholar - Fiacco, A. V. and G. P. McCormick (1990).
*Nonlinear Programming: Sequential Unconstrained Minimization Techniques*. Reprint. Philadelphia: Society for Industrial and Applied Mathematics.Google Scholar - 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.CrossRefGoogle Scholar - 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.CrossRefGoogle Scholar - Fukushima, M. (1992). Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems.
*Mathematical Programming**53*, 99–110.CrossRefGoogle Scholar - Goldsman, L. and P. T. Harker (1990). A note on solving general equilibrium problems with variational inequality techniques.
*Operations Research Letters**9*, 335–339.CrossRefGoogle Scholar - 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.Google Scholar - 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.Google Scholar - Harker, P. T. (1988). Accelerating the convergence of the diagonalization and projection algorithms for finite-dimensional variational ineqaulities.
*Mathematical Programming**41*, 29–59.CrossRefGoogle Scholar - 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.CrossRefGoogle Scholar - Nagurney, A. (1987). Competitive equilibrium problems, variational inequalities, and regional science.
*Journal of Regional Science**27*, 55–76.CrossRefGoogle Scholar - Ortega, J. M. and W. C. Rheinboldt (2000).
*Iterative Solution of Nonlinear Equations in Several Variables*(Reprint ed.). Society for Industrial and Applied Mathematics.Google Scholar - Pang, J.-S. and D. Chan (1982). Iterative methods for variational and complementary problems.
*Mathematical Programming**24*, 284–313.CrossRefGoogle Scholar - Peng, J.-M. (1997). Equivalence of variational inequality problems to unconstrained minimization.
*Mathematical Programming**78*, 347–355.Google Scholar - Scarf, H. E. (1967). The approximation of fixed points of a continuous mapping.
*SIAM Journal of Applied Mathematics**15*, 1328–1343.CrossRefGoogle Scholar - 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.Google Scholar - Tobin, R. L. (1986). Sensitivity analysis for variational inequalities.
*Journal of Optimization Theory and Applications**48*, 191–204.Google Scholar - Todd, M. J. (1976).
*The Computation of Fixed Points and Applications*. New York: Springer-Verlag.Google Scholar - Wu, J. H., M. Florian, and P. Marcotte (1993). A general descent framework for the monotone variational inequality problem.
*Mathematical Programming**61*, 281–300.CrossRefGoogle Scholar - 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.CrossRefGoogle Scholar - Zangwill, W. I. and C. B. Garcia (1981).
*Pathways to Solutions, Fixed Points, and Equilibria*. Englewood Cliffs, NJ: Prentice-Hall.Google Scholar