On Multicriteria Games with Uncountable Sets of Equilibria
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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- 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.
- van Megen, F. & Borm, P.E.M. & Tijs, S.H., 1995. "A perfectness concept for multicriteria games," Discussion Paper 1995-28, Tilburg University, Center for Economic Research.
- Borm, P.E.M. & van Megen, F.J.C. & Tijs, S.H., 1999. "A perfectness concept for multicriteria games," Other publications TiSEM 322bd1a7-90e0-400d-b167-6, Tilburg University, School of Economics and Management.
- E. Kohlberg & J.-F. Mertens, 1998.
"On the Strategic Stability of Equilibria,"
Levine's Working Paper Archive
445, David K. Levine.
- 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.
- 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 207-36.
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