Strongly Stable Equilibrium Points of N -Person Noncooperative Games

Abstract

We introduce a concept of a strongly stable (Nash) equilibrium point of an n-person noncooperative game in normal form. Roughly speaking, an equilibrium point is strongly stable if it changes continuously and uniquely against any small perturbation to payoffs of players. We establish a necessary and sufficient condition for an equilibrium point to be strongly stable. By the Karush-Kuhn-Tuckcr optimality condition for nonlinear programs, an equilibrium point is shown to correspond to a zero-point of a certain PC 1 -mapping. Then our condition is stated in terms of the local nonsingularity of the Jacobian matrix of this mapping. Our proof is based on the degree theory of mappings. Finally, we give an example of a 3-person game with two nonquasi-strong equilibrium points, only one of which is strongly stable.

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