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On Multicriteria Games with Uncountable Sets of Equilibria

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Abstract

The famous Harsanyi's (1973) Theorem states that generically a finite game has an odd number of Nash equilibria in mixed strategies. In this paper, we show that for finite multicriteria games (games with vector-valued payoffs) this kind of result does not hold. In particular, we show, by examples, that it is possible to find balls in the space of games such that every game in this set has uncountably many equilibria so that uncountable sets of equilibria are not nongeneric in multicriteria games. Moreover, we point out that, surprisingly, all the equilibria of the games cor- responding to the center of these balls are essential, that is, they are stable with respect to every possible perturbation on the data of the game. However, if we consider the scalarization stable equilibrium concept (introduced in De Marco and Morgan (2007) and which is based on the scalarization technique for multicriteria games), then we show that it provides an effective selection device for the equilibria of the games corresponding to the centers of the balls. This means that the scalarization stable equilibrium concept can provide a sharper selection device with respect to the other classical refinement concepts in multicriteria games.

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  • Giuseppe De Marco & Jacqueline Morgan, 2009. "On Multicriteria Games with Uncountable Sets of Equilibria," CSEF Working Papers 242, Centre for Studies in Economics and Finance (CSEF), University of Naples, Italy.
  • Handle: RePEc:sef:csefwp:242
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    1. Kohlberg, Elon & Mertens, Jean-Francois, 1986. "On the Strategic Stability of Equilibria," Econometrica, Econometric Society, vol. 54(5), pages 1003-1037, September.
    2. Peter Borm & Freek van Megen & Stef Tijs, 1999. "A perfectness concept for multicriteria games," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 49(3), pages 401-412, July.
    3. Jacqueline Morgan, 2005. "Approximations and Well-Posedness in Multicriteria Games," Annals of Operations Research, Springer, vol. 137(1), pages 257-268, July.
    4. Giuseppe De Marco & Jacqueline Morgan, 2007. "A Refinement Concept For Equilibria In Multicriteria Games Via Stable Scalarizations," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 9(02), pages 169-181.
    5. Ritzberger, Klaus, 1994. "The Theory of Normal Form Games form the Differentiable Viewpoint," International Journal of Game Theory, Springer;Game Theory Society, vol. 23(3), pages 207-236.
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