Equilibium contained by counter-threats and complex equilibrium in secure strategies

We present two generalizations of the concept of equilibrium in secure strategies. In equilibrium contained by counter-threats (ECCT) no player can increase his or her payoff by a unilateral deviation without creating a threat to lose more than he or she wins. We assume that this condition must be satisfied for any pseudo-equilibrium in the generalized sense, and, therefore, any such equilibrium must belong to the set of ECCT. The second generalization is the complex equilibrium in secure strategies. The proposed concept allows us identifying a hierarchical structure of mutual threats between players and will be useful for the analysis of problems with asymmetric behavior of the players. Search algorithms for proposed equilibria and their examples in the matrix games are provided.