Existence theorems for Nash equilibrium and equilibrium in secure strategies

Abstract

The paper proposes a method for constructing existence theorems for equilibrium in secure strategies (EinSS) on the basis of the well-known existence theorems for Nash equilibrium. The system of defi nitions of the EinSS is interpreted as the development of the logic of rational behavior in the defi nition of Nash equilibrium. It is proved that under the condition of «strong threats» for the existence of EinSS, it is suffi cient to fulfi ll the conditions of the known existence theorem for the Nash equilibrium only on the sets of secure strategies. This statement is formulated both globally and locally, and is an effective tool for application to practical problems. It opens up the possibility of constructing various particular existence theorems for EinSS. As a demonstration of the proposed approach, from the Debreu’s theorem of the existence of social equilibrium, the corresponding theorem of the existence of the EinSS is obtained.