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On noncooperative games, minimax theorems and equilibrium problems

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  • Frenk, J.B.G.
  • Kassay, G.

Abstract

In this chapter we give an overview on the theory of noncooperative games. In the first part we consider in detail for zero-sum (and constant-sum) noncooperative games under which necessary and sufficient conditions on the payoff function and different (extended) strategy sets for both players an equilibrium saddlepoint exists. This is done by using the most elementary proofs. One proof uses the separation result for disjoint convex sets, while the other proof uses linear programming duality and some elementary properties of compact sets. Also, for the most famous saddlepoint result given by Sion's minmax theorem an elementary proof using only the definition of connectedness is given. In the final part we consider n-person nonzero-sum noncooperative games and show by a simple application of the KKM lemma that a so-called Nash equilibrium point exists for compact strategy sets and concavity conditions on the payoff functions.

Suggested Citation

  • Frenk, J.B.G. & Kassay, G., 2006. "On noncooperative games, minimax theorems and equilibrium problems," Econometric Institute Research Papers EI 2006-21, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    • Handle: RePEc:ems:eureir:7756
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    1. Yang, Z.F., 1996. "Simplicial fixed point algorithms and applications," Other publications TiSEM 60fbb5f7-785c-4c91-8b84-5, Tilburg University, School of Economics and Management.
    2. Frenk, J. B. G. & Kassay, G. & Kolumban, J., 2004. "On equivalent results in minimax theory," European Journal of Operational Research, Elsevier, vol. 157(1), pages 46-58, August.
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