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On Existence of Nash Equilibria of Games with Constraints on Multistrategies

Author

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  • C. Pensevalle

    (Universita' di Sassari)

  • G. Pieri

    (Universita' di Genova)

Abstract

In this paper, sufficient conditions are given, which are less restrictivethan those required by the Arrow–Debreu–Nash theorem, on theexistence of a Nash equilibrium of an n-player game {1, . . . , Yn,f1, . . . , fn} in normal form with a nonempty closedconvex constraint C on the set Y=Πi Yi of multistrategies. Theith player has to minimize the function fi with respect to the ithvariable. We consider two cases. In the first case, Y is a real Hilbert space and the loss function class isquadratic. In this case, the existence of a Nash equilibrium is guaranteedas a simple consequence of the projection theorem for Hilbert spaces. In thesecond case, Y is a Euclidean space, the loss functions are continuous, andfi is convex with respect to the ith variable. In this case, the techniqueis quite particular, because the constrained game is approximated with asequence of free games, each with a Nash equilibrium in an appropriatecompact space X. Since X is compact, there exists a subsequence of theseNash equilibrium points which is convergent in the norm. If thelimit point is in C and if the order of convergence is greater than one,then this is a Nash equilibrium of the constrained game.

Suggested Citation

  • C. Pensevalle & G. Pieri, 2000. "On Existence of Nash Equilibria of Games with Constraints on Multistrategies," Journal of Optimization Theory and Applications, Springer, vol. 107(3), pages 601-613, December.
  • Handle: RePEc:spr:joptap:v:107:y:2000:i:3:d:10.1023_a:1026403417009
    DOI: 10.1023/A:1026403417009
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    References listed on IDEAS

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    1. Partha Dasgupta & Eric Maskin, 1986. "The Existence of Equilibrium in Discontinuous Economic Games, I: Theory," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 53(1), pages 1-26.
    2. M. B. Lignola, 1997. "Ky Fan Inequalities and Nash Equilibrium Points without Semicontinuity and Compactness," Journal of Optimization Theory and Applications, Springer, vol. 94(1), pages 137-145, July.
    3. Partha Dasgupta & Eric Maskin, 1986. "The Existence of Equilibrium in Discontinuous Economic Games, II: Applications," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 53(1), pages 27-41.
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