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On the Set of Proper Equilibria of a Bimatrix Game

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  • Jansen, Mathijs

Abstract

In this paper it is proved that the set of proper equilibria of a bimatrix game is the finite union of polytopes. To that purpose we split up the strategy space of each player into a finite number of equivalence classes and consider for a given [epsilon] [greater than] 0 the set of all [epsilon]-proper pairs within the cartesian product of two equivalence classes. If this set is non-empty, its closure is a polytope. By considering this polytope as [epsilon] goes to zero, we obtain a (Myerson) set of proper equilibria. A Myerson set appears to be a polytope.

Suggested Citation

  • Jansen, Mathijs, 1993. "On the Set of Proper Equilibria of a Bimatrix Game," International Journal of Game Theory, Springer;Game Theory Society, vol. 22(2), pages 97-106.
    • Handle: RePEc:spr:jogath:v:22:y:1993:i:2:p:97-106
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    Cited by:

    1. Kleppe, John & Borm, Peter & Hendrickx, Ruud, 2012. "Fall back equilibrium," European Journal of Operational Research, Elsevier, vol. 223(2), pages 372-379.
    2. Fiestras-Janeiro, G. & Borm, P.E.M. & van Megen, F.J.C., 1996. "Protective Behavior in Games," Other publications TiSEM 0f0d5aed-021d-45d8-9776-0, Tilburg University, School of Economics and Management.
    3. Dieter Balkenborg & Josef Hofbauer & Christoph Kuzmics, 2015. "The refined best-response correspondence in normal form games," International Journal of Game Theory, Springer;Game Theory Society, vol. 44(1), pages 165-193, February.
    4. John Kleppe & Peter Borm & Ruud Hendrickx, 2013. "Fall back equilibrium for $$2 \times n$$ bimatrix games," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 78(2), pages 171-186, October.
    5. van Beek, Andries & Borm, Peter, 2024. "Entangled Equilibria for Bimatrix Games," Other publications TiSEM 834facb8-8e33-4046-a452-7, Tilburg University, School of Economics and Management.
    6. A. J. Vermeulen & M. J. M. Jansen, 1994. "On the set of (perfect) equilibria of a bimatrix game," Naval Research Logistics (NRL), John Wiley & Sons, vol. 41(2), pages 295-302, March.
    7. van Beek, Andries & Borm, Peter, 2024. "Entangled Equilibria for Bimatrix Games," Discussion Paper 2024-016, Tilburg University, Center for Economic Research.

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