Equilibrium 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 its payoff by a unilateral deviation without creating a threat to lose more than it wins. 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 identifying a hierarchical structure of mutual threats between players and will be useful for the analysis of problems with asymmetric behavior of players. Search algorithms for the proposed equilibria and their examples in matrix games are provided.