Econometrica: Nov, 1990, Volume 58, Issue 6

On the Definition of the Strategic Stability of Equilibria

https://doi.org/0012-9682(199011)58:6<1365:OTDOTS>2.0.CO;2-Y
p. 1365-1390

John Hillas

The concept of strategically stable equilibria introduced by Kohlberg and Mertens (1986) is motivated by a number of requirements which Kohlberg and Mertens argue a satisfactory solution concept for noncooperative games should satisfy. They defined a number of different solution concepts of the following general form: a stable set of equilibria is a minimal closed set of equilibria such that all small perturbations of the game have equilibria close to the stable set. Their definitions varied with the definition of a small perturbation. Unfortunately none of their definitions satisfied all of their requirements. We show that by changing the definition of a perturbation it is possible to define a solution concept that does satisfy all of the requirements. In the main definition of this paper we look not at perturbations of the payoffs or the strategy space but directly at perturbations of the best reply correspondence. With the appropriate topology on this space of perturbations the resulting definition does satisfy all of the requirements. We also show that one does not have much freedom in the topology one uses.