Set Stability of Probabilistic Logical Networks and Its Application to State-Based Games

Set stability of probabilistic multivalued logical networks (PMLNs) is investigated in this article. Different from the existing concepts of set stability, the set stability considered in this article is starting from a given initial set to a given destination set. First, an algebraic expression of PMLNs is established by using the semitensor product (STP) method. Then, a matrix method is proposed to obtain the largest invariant subset contained in the given destination set. Based on the invariant subset, a necessary and sufficient condition for set stability from a given initial set to a destination set is presented. Finally, application to state-based games is considered, and the criterion proposed for PMLNs is applied to verify the stability of recurrent state equilibrium.