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Complete stability of noncooperative games

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Abstract

This paper is concerned with a class of noncooperative games ofn players that are defined byn reward functions which depend continuously on the action variables of the players. This framework provides a realistic model of many interactive situations, including many common models in economics, sociology, engineering, and political science. The concept of Nash equilibrium is a suitable companion to such models.

A variety of different sufficient conditions for existence, uniqueness, and stability of a Nash equilibrium point have been previously proposed. By sharpening the noncooperative aspect of the framework (which is really only implicit in the original framework), this paper attempts to isolate one set of “natural” conditions that are sufficient for existence, uniqueness, and stability. It is argued thatl ∞ quasicontraction is such a natural condition. The concept of complete stability is introduced to reflect the full character of noncooperation. It is then shown that, in the linear case, the condition ofl ∞ quasicontraction is both necessary and sufficient for complete stability.

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References

  1. Frank, C. R., Jr., andQuant, R. E.,On the Existence of Cournot Equilibrium, International Economic Review, Vol. 4, pp. 92–96, 1963.

    Google Scholar 

  2. Marschak, J., andRadner, R.,The Economic Theory of Teams, Yale University Press, New Haven, Connecticut, 1972.

    Google Scholar 

  3. Simonovitis, A.,Some Properties of a Decentralized Adjustment Process, Management Science, Vol. 22, pp. 883–891, 1976.

    Google Scholar 

  4. Luenberger, D. G., andPhilippi, B.,Neighborhood Effects and Maintenance of the Urban Housing Stock, Journal of Mathematical Sociology, Vol. 5, pp. 151–172, 1977.

    Google Scholar 

  5. Bush, W. C., andMayer, L. S.,Some Implications of Anarchy for the Distribution of Property, Journal of Economic Theory, Vol. 8, pp. 401–412, 1974.

    Google Scholar 

  6. Okuguchi, K.,Natural Equilibrium in the Bush-Mayer Model of Anarchy: A Dynamic Analysis, Public Choice, Vol. 26, pp. 129–134, 1976.

    Google Scholar 

  7. Spense, M.,Competitive and Optimal Response to Signals: An Analysis of Efficiency and Distribution, Journal of Economic Theory, Vol. 7, pp. 296–332, 1974.

    Google Scholar 

  8. Ho, Y. C., andKastner, M. P.,Market Signalling: An Example of a Two-Person Decision Problem with a Dynamic Information Structure, Proceedings of the 1976 IEEE Conference on Decision and Control, Clearwater, Florida, 1976.

  9. Quirk, J., andSaposnik, R.,Introduction to General Equilibrium Theory and Welfare Economics, McGraw-Hill Book Company, New York, New York, 1968.

    Google Scholar 

  10. Malishevskii, A. V.,Models of Joint Operation of Many Goal-Oriented Elements, I, Automation and Remote Control, Vol. 33, pp. 1828–1845, 1972.

    Google Scholar 

  11. Malishevskii, A. V.,Models of Joint Operation of Many Goal-Oriented Elements, II, Automation and Remote Control, Vol. 34, pp. 1749–1762, 1973.

    Google Scholar 

  12. Rosen, J. B.,Existence and Uniqueness of Equilibrium Points for Concave N-Person Games, Econometrica, Vol. 33, pp. 520–534, 1965.

    Google Scholar 

  13. Williams, J. B.,Stability of Noncooperative Games, Operations Research, Vol. 19, pp. 774–783, 1971.

    Google Scholar 

  14. Belenky, V. Z., Ivankov, S. A., andVolkonsky, V. A.,On a Certain Approach to the Investigation of the Convergence of Iterative Processes, Computing Equilibria: How and Why, Edited by J. Los and M. W. Los, North-Holland Publishing Company, Amsterdam, Holland, 1976.

    Google Scholar 

  15. Maschler, M., andPeleg, B.,Stable Sets and Stable Points of Set-Valued Dynamic Systems with Applications to Game Theory, SIAM Journal of Control and Optimization, Vol. 14, pp. 985–995, 1976.

    Google Scholar 

  16. Goffman, C., andPedrick, G.,First Course in Functional Analysis, Prentice-Hall, Englewood Cliffs, New Jersey, 1965.

    Google Scholar 

  17. Berge, C.,Topological Spaces, The Macmillan Company, New York, New York, 1963.

    Google Scholar 

  18. Edelstein, M.,On Fixed and Periodic Points Under Contractive Mappings, Journal of The London Mathematical Society, Vol. 37, pp. 74–79, 1962.

    Google Scholar 

  19. Varga, R. S.,Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, New Jersey, 1962.

    Google Scholar 

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Authors and Affiliations

  1. Department of Engineering-Economic Systems, Stanford University, Stanford, California

    D. G. Luenberger (Professor)

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  1. D. G. Luenberger
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Additional information

This research was supported by the Air Force Office of Scientific Research under Grant No. AFOSR 77-3141 and by the National Science Foundation under Grant No. GK18748.

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Luenberger, D.G. Complete stability of noncooperative games. J Optim Theory Appl 25, 485–505 (1978). https://doi.org/10.1007/BF00933516

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