Abstract
The variational inequality problem has been utilized to formulate and study a plethora of competitive equilibrium problems in different disciplines, ranging from oligopolistic market equilibrium problems to traffic network equilibrium problems. In this paper we consider for a given variational inequality a naturally related ordinary differential equation. The ordinary differential equations that arise are nonstandard because of discontinuities that appear in the dynamics. These discontinuities are due to the constraints associated with the feasible region of the variational inequality problem. The goals of the paper are two-fold. The first goal is to demonstrate that although non-standard, many of the important quantitative and qualitative properties of ordinary differential equations that hold under the standard conditions, such as Lipschitz continuity type conditions, apply here as well. This is important from the point of view of modeling, since it suggests (at least under some appropriate conditions) that these ordinary differential equations may serve as dynamical models. The second goal is to prove convergence for a class of numerical schemes designed to approximate solutions to a given variational inequality. This is done by exploiting the equivalence between the stationary points of the associated ordinary differential equation and the solutions of the variational inequality problem. It can be expected that the techniques described in this paper will be useful for more elaborate dynamical models, such as stochastic models, and that the connection between such dynamical models and the solutions to the variational inequalities will provide a deeper understanding of equilibrium problems.
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References
J.P. Aubin and A. Cellina,Differential Inclusions (Springer, Berlin, 1984).
M. Beckmann, C.B. McGuire and C.B. Winsten,Studies in the Economics of Transportation (Yale University Press, New Haven, CT, 1956).
D. Bernstein, Programmability of continuous and discrete network equilibria, PhD Thesis, University of Pennsylvania, Philadelphia, PA (1990).
P. Billingsley,Convergence of Probability Measures (Wiley, New York, 1968).
K.C. Border,Fixed Point Theorems with Applications to Economics and Game Theory (Cambridge University Press, Cambridge, England, 1985).
E.A. Coddington and N. Levinson,Theory of Ordinary Differential Equations (McGraw-Hill, New York, 1955).
A.A. Cournot,Researches into the Mathematical Principles of the Theory of Wealth (1838; English transl.: MacMillan, London, England, 1897).
S. Dafermos, Traffic equilibrium and variational inequalities, Transp. Sci. 14 (1980) 42–54.
S. Dafermos, The general multimodal traffic equilibrium problem with elastic demand, Networks 12 (1982) 57–72.
S. Dafermos, An iterative scheme for variational inequalities, Math. Progr. 26 (1983) 40–47.
S. Dafermos, Exchange price equilibria and variational inequalities, Math. Progr. 46 (1990) 391–402.
S. Dafermos and A. Nagurney, Oligopolistic and competitive behavior of spatially separated markets, Reg. Sci. Urban Econ. 17 (1987) 245–254.
P. Dupuis, Large deviations analysis of reflected diffusions and constrained stochastic approximation algorithms in convex sets, Stochastics 21 (1987) 63–96.
P. Dupuis and H. Ishii, On Lipschitz continuity of the solution mapping to the Skorokhod Problem, with applications, Stochastics and Stochastic Reports 35 (1991) 31–62.
P. Dupuis and H. Ishii, SDEs with oblique reflection on nonsmooth domains, Ann. Prob. 21 (1993) 554–580.
P. Dupuis and H.J. Kushner, Stochastic approximation and large deviations: Upper bounds and w.p. 1 convergence, SIAM J. Control Optim. 27 (1989) 1108–1135.
P. Dupuis and R.J. Williams, Lyapunov functions for semimartingale reflected Brownian motions, Lefschetz Center for Dynamical Systems, Report # 92-5, Brown University, Providence, RI (1992), to appear in: Ann. Prob.
S.N. Ethier and T.G. Kurtz,Markov Processes: Characterization and Convergence (Wiley, New York, 1986).
A.N. Fillipov, Differential equations with discontinuous right-hand side, Mat. Sbornik (N.S.) 51 (1960) 99–128.
S.D. Flam, Solving convex programs by means of ordinary differential equations, Math. Oper. Res. 17 (1992) 290–302.
S.D. Flam and A. Ben-Israel, A continuous approach to oligopolistic market equilibrium, Oper. Res. 38 (1990) 1045–1051.
M. Florian and M. Los, A new look at static spatial price equilibrium problems, Reg. Sci. Urban Econ. 12 (1982), 579–597.
M.I. Freidlin and A.D. Wentzell,Random Perturbations of Dynamical Systems (Springer, New York, 1984).
D. Gabay and H. Moulin, On the uniqueness and stability of Nash equilibria in noncooperative games, in:Applied Stochastic Control of Econometrics and Management Science, eds. A. Bensoussan, P. Kleindorfer and C.S. Tapiero (North-Holland, Amsterdam, The Netherlands, 1980).
C.B. Garcia and W.I. Zangwill,Pathways to Solutions, Fixed Points, and Equilibria (Prentice-Hall, Englewood Cliffs, NJ, 1981).
P. Hartman,Ordinary Differential Equations (Wiley, New York, 1964).
P. Hartman and G. Stampacchia, On some nonlinear elliptical differential functional equations, Acta Math. 115 (1966) 271–310.
A. Haurie and P. Marcotte, On the relationship between Nash-Cournot and Wardrop equilibria, Networks 15 (1985) 295–308.
D. Kinderlehrer and G. Stampacchia,An Introduction to Variational Inequalities and Their Applications (Academic Press, New York, 1980).
H.J. Kushner and D.S. Clark,Stochastic Approximation Methods for Constrained and Unconstrained Systems (Springer, New York, 1978).
S. Lefschetz,Differential Equations. Geometric Theory (Interscience, New York, 1957).
A. Mas-Colell,The Theory of General Economic Equilibrium. A Differential Approach (Cambridge University Press, Cambridge, England, 1985).
A. Nagurney, Migration equilibrium and variational inequalities, Econ. Lett. 31 (1989) 109–112.
A. Nagurney,Network Economics: A Variational Inequality Approach (Kluwer Academic, Boston, MA, 1993).
A. Nagurney, J. Dong and M. Hughes, The formulation and computation of general financial equilibrium, Optimization 26 (1992) 339–354.
J.F. Nash, Equilibrium points inn-person games, Proc. National Acad. Sci. 36 (1950) 48–49.
J.F. Nash, Noncooperative games, Ann. Math. 54 (1951) 286–298.
K. Okuguchi,Expectations and Stability in Oligopoly Models, Lecture Notes in Economics and Mathematical Systems 138 (Springer, Berlin, 1976).
K. Okuguchi, The Cournot oligopoly and competitive equilibrium as solutions to non-linear complementarity problems, Econ. Lett. 12 (1983) 127–133.
K. Okuguchi and F. Szikdarovsky,The Theory of Oligopoly with Multi-Product Firms, Lecture Notes in Economics and Mathematical Systems 342 (Springer, Berlin, 1990).
P.A. Samuelson, The stability of equilibrium: Comparative statics and dynamics, Econometrica 9 (1941) 97–120.
A.V. Skorokhod, Stochastic equations for diffusions in a bounded region, Theory Prob. Appl. 6 (1961) 264–274.
S. Smale, A convergence process of price adjustment and global Newton methods, J. Math. Econ. 3 (1976) 107–120.
M.J. Smith, The existence and calculation of traffic equilibria, Transp. Res. 17B (1983) 291–303.
H.R. Varian, Dynamical systems with applications to economics, in:Handbook of Mathematical Economics 1, eds. K.J. Arrow and M.D. Intrilligator (North-Holland, Amsterdam, The Netherlands, 1981) pp. 93–110.
L.T. Watson, Solving the nonlinear complementarity problem by a homotopy method, SIAM J. Control Optim. 17 (1979) 36–46.
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Dupuis, P., Nagurney, A. Dynamical systems and variational inequalities. Ann Oper Res 44, 7–42 (1993). https://doi.org/10.1007/BF02073589
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DOI: https://doi.org/10.1007/BF02073589