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.

Equilibrium Problems, KKM Lemma, Minimax Theorems, Nash Equilibrium Point, Non-Cooperative Game Theory
Noncooperative Games (jel C72), Business Administration and Business Economics; Marketing; Accounting (jel M), Production Management (jel M11), Transportation Systems (jel R4)
Erasmus Research Institute of Management
hdl.handle.net/1765/7809
ERIM Report Series Research in Management
ERIM report series research in management Erasmus Research Institute of Management
Erasmus Research Institute of Management

Frenk, J.B.G, & Kassay, G. (2006). On Noncooperative Games, Minimax Theorems and Equilibrium Problems (No. ERS-2006-022-LIS). ERIM report series research in management Erasmus Research Institute of Management. Erasmus Research Institute of Management. Retrieved from http://hdl.handle.net/1765/7809