This chapter deals with the problem of finding out an equilibrium in an oligopolistic market model where several subjects supply a single homogeneous product in a non-cooperative manner. The problem is reduced to a nonlinear equation some terms of which are determined by solving nonlinear complementarity problems. An algorithm is presented that combines the Newton method steps with the dichotomy techniques. Under certain assumptions, the algorithm is shown to be convergent at the quadratic rate. Finally, the algorithm is extended to the case of nonlinear production costs, and its linear convergence is demonstrated.
KeywordsNewton Method Complementarity Problem Linear Complementarity Problem Nonlinear Complementarity Problem Inverse Demand Function
Unable to display preview. Download preview PDF.
- Bakhvalov, N.S. Numerical Methods (in Russian), Part 1, Moscow: Nauka, 1973.Google Scholar
- Berezin, I.S., and Zhidkov, N.G. Computational Methods (in Russian), Part 2, Moscow: Fizmatgiz, 1962.Google Scholar
- Bulavsky VA and Kalashnikov VV. 1. Parameter driving method to study equilibrium. Ekonomika i Matematicheskie Metody (in Russian), 1994; 30:129–138.Google Scholar
- Bulavsky VA and Kalashnikov VV. 2. Equilibria in generalized Cournot and Stackelberg models. Ekonomika i Matematicheskie Metody (in Russian). 1995; 31:164–176.Google Scholar
- Bulavsky VA and Kalashnikov VV. 4. ‘A Newton-like approach to solving an equilibrium problem’. -In: High Performance Algorithms and Software in Nonlinear Optimization, F. Giannessi e.a., eds. Dordrecht: Kluwer Academic Publishers, 1998.Google Scholar
- Cohen G. Nash equilibria: Gradient and decomposition algorithms. Large Scale Systems. 1984; 12:173–184.Google Scholar
- Harker PT and Pang JS 3. ‘A damped-Newton method for the linear complementarity problem.’- In: Computational Solution of Nonlinear Systems of Equations, E.L. Allgower and K. Georg, eds. AMS Lectures on Applied Mathematics. 1990; 26: 265–284.Google Scholar
- Kalashnikov VV and Kalashnikova NI. 1. Global convergence of inexact Newton method solving complementarity problems.- In: “Optimizatsia”, Institute of Mathematics, Novosibirsk. 1988; 42(59): 66–85 (in Russian). Google Scholar
- Kalashnikov VV and Kalashnikova NI. 2. Step accuracy control for Newton method solving nonlinear complementarity problem.- In: “Optimizatsia”, Institute of Mathematics, Novosibirsk. 1988; 43(60): 27–40 (in Russian). Google Scholar
- Kalashnikov VV and Kalashnikova NI. 3. Kalashnikov VV and Kalashnikova NI. Optimal control of inner accuracy in bilevel iteration process. -In: “Optimizatsia”, Institute of Mathematics, Novosibirsk, 1988; 44(61): 27–55 (in Russian). Google Scholar
- Polyak BT. Gradient methods to minimize functionals. Zhurnal Vychislit. Matemat. i Matem. Fiziki (Comput. Maths and Math. Phys.), 1963; 3: 643–653.Google Scholar
- Stackelberg, H. Marktform und Gleichgewickt. Vienna: Julius Springer, 1934.Google Scholar