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Relaxation algorithms to find Nash equilibria with economic applications

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Abstract

Recent theoretical studies have shown that a relaxation algorithm can be used to find noncooperative equilibria of synchronous infinite games with nonlinear payoff functions and coupled constraints. In this study, we introduce an improvement to the algorithm, such as the steepest-descent step-size control, for which the convergence of the algorithm is proved. The algorithm is then tested on several economic applications. In particular, a River Basin Pollution problem is considered where coupled environmental constraints are crucial for the relevant model definition. Numerical runs demonstrate fast convergence of the algorithm for a wide range of parameters.

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References

  1. J.P. Aubin, Mathematical Methods of Game and Economic Theory (Elsevier, Amsterdam, 1980).

    Google Scholar 

  2. V.Z. Belenkii, V.A. Volkonskii, S.A. Ivankov, A.B. Pomaranski and A.D. Shapiro, Iterative Methods in Game Theory and Programming (Nauka, Moscow, 1974) (in Russian).

    Google Scholar 

  3. T. Başar, Relaxation techniques and asynchronous algorithms for online computation of non-cooperative equilibria, Journal of Economic Dynamics and Control 11 (1987) 531-549.

    Article  Google Scholar 

  4. S. Berridge and J.B. Krawczyk, Relaxation algorithms in finding Nash equilibria, Economic Working papers Archive (1997). URL: http://econwpa.wustl.edu/eprints/comp/papers/9707/9707002. abs

  5. M.C. Ferris and J.S. Pang, eds., Complementarity and Variational Problems: State of the Art (SIAM, Philadelphia, PA, 1997).

    Google Scholar 

  6. A. Haurie and J.B. Krawczyk, Optimal charges on river effluent from lumped and distributed sources, Environmental Modeling and Assessment 2 (1997) 93-106.

    Article  Google Scholar 

  7. W.D. Montgomery, Markets in licenses and efficient pollution control programs, Journal of Economic Theory 5 (1972) 395-418.

    Article  Google Scholar 

  8. H. Nikaido and K. Isoda, Note on noncooperative convex games, Pacific Journal of Mathematics 5, Supp. 1 (1955) 807-815.

    Google Scholar 

  9. J.B. Krawczyk, ECON 407-Economic Dynamics B, Introduction to Dynamic Games with Application-Course Notes, Works and Services, Printing Unit, Victoria University of Wellington (May 1995) p. 1774.

  10. J.B. Krawczyk, Modelling and economics of effluent management in municipalities, in: Modelling Change in Integrated Economic and Environmental Systems, eds. S. Mahendrarajah, A.J. Jakeman and M.J. McAleer (Wiley, New York, 1998).

    Google Scholar 

  11. J.B. Krawczyk, O. Pourtallier and M. Tidball, A steady-state satisfactory solution to an environmental game with piece-wise defined payoffs, Natural Resource Modeling 11(4) (winter 1998) 301-329.

    Google Scholar 

  12. S. Li and T. Başar, Distributed Algorithms for the Computation of Noncooperative Equilibria, Automatica 23 (1987) 523-533.

    Article  Google Scholar 

  13. A. Nagurney and K. Dhanda, Variational inequalities for marketable pollution permits with technological investment opportunities: the case of oligopolistic markets, Mathematics and Computer Modeling 26(2) (1997) 1-25.

    Article  Google Scholar 

  14. A. Nagurney, S. Thore and J. Pan, Spatial market policy modeling with goal targets, Operations Research 44 (1996) 393-406.

    Article  Google Scholar 

  15. A. Nagurney, Network Economics: A Variational Inequality Approach (Kluwer Academic, Boston, MA, 1993).

    Google Scholar 

  16. E.A. Nurminski, Subgradient method for minimizing weakly convex functions and ɛ-subgradient methods of convex optimization, in: Progress in Nondifferentiable Optimization: Collaborative Proceedings CP-82-S8 (International Institute for Applied Systems Analysis, Laxenburg, Austria, 1982) pp. 97-123.

    Google Scholar 

  17. G.P. Papavassilopoulos, Iterative techniques for the Nash solution in quadratic games with unknown parameters, SIAM J. Control and Optimization 34(4) (1986) 821-834.

    Article  Google Scholar 

  18. A. Randall, Resource Economics (Wiley, New York, 1987).

    Google Scholar 

  19. J.B. Rosen, Existence and uniqueness of equilibrium points for concave n-person games, Econometrica 33(3) (1965) 520-534.

    Article  Google Scholar 

  20. N.Z. Shor, Minimization Methods for Non-Differentiable Functions (Springer, Berlin, 1985).

    Google Scholar 

  21. S. Uryasev, On the anti-monotonicity of differential mappings connected with general equilibrium problem. Optimization, A Journal of Mathematical Programming and Operations Research (Berlin) 19(5) (1988) 693-709.

    Google Scholar 

  22. S. Uryasev, Adaptive Algorithms for Stochastic Optimisation and Game Theory (Nauka, Moscow, 1990) (in Russian).

    Google Scholar 

  23. S. Uryasev and R.Y. Rubinstein, On relaxation algorithms in computation of noncooperative equilibria, IEEE Transactions on Automatic Control 39 (6) 1263-1267.

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Krawczyk, J.B., Uryasev, S. Relaxation algorithms to find Nash equilibria with economic applications. Environmental Modeling & Assessment 5, 63–73 (2000). https://doi.org/10.1023/A:1019097208499

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