On Noncooperative Games, Minimax Theorems and Equilibrium Problems

Hans Frenk and G. Kassay

ERIM Report Series Research in Management from Erasmus Research Institute of Management (ERIM), ERIM is the joint research institute of the Rotterdam School of Management, Erasmus University and the Erasmus School of Economics (ESE) at Erasmus University Rotterdam

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.

Keywords: Equilibrium Problems; KKM Lemma; Minimax Theorems; Nash Equilibrium Point; Non-Cooperative Game Theory (search for similar items in EconPapers)
JEL-codes: C72 M M11 R4 (search for similar items in EconPapers)
Date: 2006-06-10
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