The New Palgrave Dictionary of Economics

2018 Edition
| Editors: Macmillan Publishers Ltd

Non-cooperative Games (Equilibrium Existence)

  • Philip J. Reny
Reference work entry
DOI: https://doi.org/10.1057/978-1-349-95189-5_2572

Abstract

This article provides a brief overview of equilibrium existence results for continuous and discontinuous non-cooperative games.

Keywords

Auctions Bertrand price-competition models Convexity Cournot oligopoly models Discontinuous games Endogenous sharing rules Equilibrium Equilibrium existence Finite-action games Fixed point theorems Infinite-action games Mixed strategy Nash equilibria Nash equilibrium Non-cooperative games Pure strategy Nash equilibria Quasi-concavity Quasi-convexity Spatial economics Strategic form games 

JEL Classifications

C7 
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Bibliography

  1. Baye, M., G. Tian, and J. Zhou. 1993. Characterizations of the existence of equilibria in games with discontinuous and non-quasiconcave payoffs. Review of Economic Studies 60: 935–948.CrossRefGoogle Scholar
  2. Bertrand, J. 1883. Théorie mathématique de la richesse sociale. Journal des Savants 67: 499–508.Google Scholar
  3. Billingsley, P. 1968. Convergence of probability measures. New York: John Wiley and Sons.Google Scholar
  4. Cournot, A. 1838. In Researches into the mathematical principles of the theory of wealth, ed. N. Bacon, 1897. New York: Macmillan.Google Scholar
  5. Dasgupta, P., and E. Maskin. 1986. The existence of equilibrium in discontinuous economic games, I: Theory. Review of Economic Studies 53: 1–26.CrossRefGoogle Scholar
  6. Debreu, G. 1952. A social equilibrium existence theorem. Proceedings of the National Academy of Sciences 38: 386–393.CrossRefGoogle Scholar
  7. Fudenberg, D., R. Gilbert, J. Stiglitz, and J. Tirole. 1983. Preemption, leapfrogging, and competition in patent races. European Economic Review 22: 3–31.CrossRefGoogle Scholar
  8. Glicksberg, I. 1952. A further generalization of the Kakutani fixed point theorem. Proceedings of the American Mathematical Society 3: 170–174.Google Scholar
  9. Hotelling, H. 1929. The stability of competition. Economic Journal 39: 41–57.CrossRefGoogle Scholar
  10. Jackson, M., and J. Swinkels. 2005. Existence of equilibrium in single and double private value auctions. Econometrica 73: 93–139.CrossRefGoogle Scholar
  11. Jackson, M., L. Simon, J. Swinkels, and W. Zame. 2002. Communication and equilibrium in discontinuous games of incomplete information. Econometrica 70: 1711–1740.CrossRefGoogle Scholar
  12. Milgrom, P., and J. Roberts. 1990. Rationalizability, learning, and equilibrium in games with strategic complementarities. Econometrica 58: 1255–1277.CrossRefGoogle Scholar
  13. Milgrom, P., and R. Weber. 1982. A theory of auctions and competitive bidding. Econometrica 50: 1089–1122.CrossRefGoogle Scholar
  14. Milgrom, P., and R. Weber. 1985. Distributional strategies for games with incomplete information. Mathematics of Operations Research 10: 619–632.CrossRefGoogle Scholar
  15. Nash, J. 1950. Equilibrium points in n-person games. Proceedings of the National Academy of Sciences 36: 48–49.CrossRefGoogle Scholar
  16. Nash, J. 1951. Non-cooperative games. Annals of Mathematics 54: 286–295.CrossRefGoogle Scholar
  17. Osborne, M., and A. Rubinstein. 1994. A course in game theory. Cambridge, MA: MIT Press.Google Scholar
  18. Reny, P. 1999. On the existence of pure and mixed strategy Nash equilibria in discontinuous games. Econometrica 67: 1029–1056.CrossRefGoogle Scholar
  19. Robson, A. 1994. An ‘informationally robust’ equilibrium in two-person nonzero-sum games. Games and Economic Behavior 2: 233–245.CrossRefGoogle Scholar
  20. Schafer, W., and H. Sonnenschein. 1975. Equilibrium in Abstract economies without ordered preferences. Journal of Mathematical Economics 2: 345–348.CrossRefGoogle Scholar
  21. Simon, L. 1987. Games with discontinuous payoffs. Review of Economic Studies 54: 569–597.CrossRefGoogle Scholar
  22. Simon, L., and W. Zame. 1990. Discontinuous games and endogenous sharing rules. Econometrica 58: 861–872.CrossRefGoogle Scholar
  23. Sion, M. 1958. On general minimax theorems. Pacific Journal of Mathematics 8: 171–176.CrossRefGoogle Scholar
  24. Vives, X. 1990. Nash equilibrium with strategic complementarities. Journal of Mathematical Economics 19: 305–321.CrossRefGoogle Scholar
  25. von Neumann, J. 1928. Zur Theorie der Gesellshaftspiele. Mathematische Annalen 100, 295–320. Trans. S. Bargmann [On the theory of games of strategy]. In Contributions to the theory of games, vol. 4, ed. R. Luce and A. Tucker, Princeton: Princeton University Press, 1959.Google Scholar

Copyright information

© Macmillan Publishers Ltd. 2018

Authors and Affiliations

  • Philip J. Reny
    • 1
  1. 1.