We formalize the adversarial process between defender and attackers as a game and study the non-cooperative evolutionary game mechanism under bounded rationality. We analyze the long-term dynamic process between the attacking and defending parties using the evolutionary stable strategies derived from the evolutionary game model. First, we construct a multi-player evolutionary game model consisting of a defender and multiple attackers, formally describe the strategies, and construct a three-player game payoff matrix. Then, we propose two punishment schemes, i.e., static and dynamic ones. Moreover, through the combination of mathematical derivation with simulation, we obtain the evolutionary stable strategies of each player. Different from previous work, in this paper, we consider the influence of strategies among different attackers. The simulation shows that (1) in the static punishment scheme, increasing the penalty can quickly control the occurrence of network attacks in the short term; (2) in the dynamic punishment scheme, the game can be stabilized effectively, and the stable state and equilibrium values are not affected by the change of the initial values.

This paper focuses on the investigation of filters of pseudo BCK-algebra and BL-algebra, important and popular generic commutative and non-commutative logical algebras. By characterizing Boolean filter and implicative filter in pseudo BCK-algebra, the essential equivalent relation between these two filters is revealed. An open problem that "In pseudo BCK-algebra or bounded pseudo BCK-algebra, is the notion of implicative pseudo-filter equivalent to the notion of Boolean filter?" is solved. Based on this, this paper explores the essential relations between the implicative (Boolean) filter and implicative pseudo BCK-algebra. A complete solution to an open problem that "Prove or negate that pseudo BCK-algebras is implicative BCK-algebras if and only if every filter of them is implicative filters (or Boolean filter)" is derived. This paper further characterizes the fantastic filter and normal filter in BL-algebra, then gets the equivalent relation between the two filters, and completely solves two open problems regarding the relationship between these two filters: 1. Under what suitable condition a normal filter becomes a fantastic filter? and 2. (Extension property for a normal filter) Under what suitable condition extension property for normal filter holds?