## 2 Background of Immune Game Theory
### Game Theory  
Game theory studies the strategy and decision-making problems of two or more participants, emphasizing the direct interaction between the behaviors of decision subjects.  
A complete game structure should contain the elements of a player, strategy set, and payoff function or utility function [18].  
1. Player  
The multiple players in the game model refer to the parties involved in the game. It is the decision-making body of the game model and the maker of the game strategy. The participants have independent decision-making ability and are responsible for the corresponding consequences. \(P\) represents a set. Participants may be individuals or groups or groups with common interests.
2. Strategy set  
The strategy collection includes strategies that decision-making individuals may adopt. \(S_{i}\) is the policy set of participant \(i\). It should contain at least two policies, and must be a practical action plan.
3. Payoff function  
Under the game set of specific rules, the expected income obtained by decision-making individual \(i\) is represented by payment function \(U_{i}\). Utility refers to the gains and losses of participants. Decision-making individuals use different strategies to obtain different results. The result is influenced by strategies of all decision-making individuals. Utility functions represent the level of income of different decision-making individuals, and are functions of the strategies of all such individuals.  
The game model is established by the attack and defense of both sides, strategy set, and utility function as  
\[\mathrm{G}=\{N,\{S_{i}\},\{U_{i}\}\}. \tag{3}\]  
* Nash Equilibrium [19]  
A strategy by which the Nash equilibrium can be achieved is called optimal. This occurs when no decision-making individual can increase its benefits by changing strategy.  
[PERSON] defined equilibrium [20] as follows. For a game model \(G=\big{\{}N,\{T_{i}\}_{i\in N},\{U_{i}\}_{i\in N}\big{\}}\), \(N\) is a set of players, and \(S_{i}\) is a strategy selected by the \(i\)th player with utility \(i\)th \(U_{i}\). The decision space is \(S=\times S_{i},\ i\in N\), and the utility function set is \(U\). In \(G\), the utility of the \(i\)th player can be represented as a function of \(S_{i}\): \(U(S_{i},S_{-i})\), or \(U(S)\), where, \(S_{-i}\) is the strategy set of players other than \(i\) in \(N\). The strategies selected by all players form a set of strategies \(S\). \(S=[s_{1},s_{2},\cdots,s_{N}]\). When \(S\) meets the formula (4), a _Nash_ equilibrium is obtained [21].  
\[U_{i}(S)\geq U_{i}(s_{i},s_{-i}),\ \forall i\in N,\ s_{i}\in S_{i} \tag{4}\]  
In a multimedia mobile network, the operation of each node is random and mobile. The network resources occupied by the security detection and recovery of the system are uncertain. The most important point of node security is to ensure the safe operation of the node. Obviously, it is not enough to be able to detect anomalies and issue safety warnings. In order to effectively allocate network resources, a novel idea in node security is to establish the order of node detection and operation by a game based on immune detection.  
Self-organization, dynamic topology, and limited resources are the main characteristics of mobile multimedia network. These characteristics determine that each node has its own decision when communicating. The network nodes are correlated. Compared with dominant-strategy equilibrium and cooperative equilibrium, it is suitable to adopt Nash equilibrium because of the non-cooperative and complete information relationship between nodes in a mobile multimedia network. Therefore, Nash equilibrium is chosen as the game strategy in this paper. In the Nash equilibrium, each node can be regarded as a player making game decisions by utility calculation.