## 4 Slp-macgt Model Design
### Game Process between Sensor Nodes and Attackers
#### 4.5.1 Basic Assumptions of the Game Model  
The relationship between the source node and the attacker can be regarded as a zero-sum game. In general, the source node knows its own position, while the attacker can only estimate or infer the position information through certain methods [33]. In this paper, we consider a scenario where there are multiple attackers in the network, involving interactions and decisions among multiple participants. Source location privacy in underwater acoustic sensor networks usually involves multiple rounds of the game, which includes repeated strategy selection, confrontation, and adjustment between the source node and attackers. The strategies of source nodes and attackers are often dynamic and can evolve over time, and evolutionary game theory can capture these changes [34] and allow analysis strategies to adapt to changing environments over time. Unlike traditional static games, evolutionary games focus on how players choose strategies in multiple rounds of the game, and these strategies can spread and mutate through a series of evolutionary mechanisms. At the same time, evolutionary game theory no longer treats both players of the game as superrational players but believes that players usually reach game equilibrium through trial and error. The main components of game theory include several elements: players, game strategies, and game benefits. The participant set is \(PLAYER=(player_{X},player_{Y},player_{Z},\ldots)\). The source node \((player_{X})\) is a player that actively protects its location privacy in the network, and its goal is to select a strategy that maximizes location privacy while maintaining communication performance. This paper takes the existence of two attackers as an example: A \((player_{Y})\) and B \((player_{Z})\) are two players who actively try to obtain the location information of the source node, and the goal is to choose a strategy to infer the location of the source node to the greatest extent. The game strategy set is \(STR=(S_{X},S_{Y},S_{Z})\). \(S_{X}\) is the strategy space of the source node, and the source node can choose different protection strategies, such as a virtual coordinate system, relay node selection, and network coding, to form the source node strategy space \(S_{X}=\{X_{1},X_{2}\}\) (weak defense, strong defense). \(S_{Y}\) is the strategy space of attacker A, who can adopt methods, such as eavesdropping attack, traceback attack, traffic analysis, or cooperation with other attackers, to form the attacker's strategy space \(S_{Y}=\{Y_{1},Y_{2}\}\) (single attack, cooperative attack). The strategy space of attacker B, \(S_{Z}=\{Z_{1},Z_{2}\}\) (single attack, cooperative attack), is the same as that of attacker A. Only when both attackers choose a cooperative strategy can cooperation between the attackers be achieved. The probability \(P=(P_{X},P_{Y},P_{Z})\) of game strategy selection is the probability set of the source node and attacker strategies, where \(P_{X}=\{x,1-x\}\) represents the probability of weak defense and strong defense of the source node, \(P_{Y}=\{y,1-y\}\) represents the probability of attacker A's single attack and cooperative attack, and \(P_{Z}=\{z,1-z\}\) represents the probability of attacker B's single attack and cooperative attack. The game return can be set as \(RETURN=(R_{X},R_{Y},R_{Z})\). \(R_{X}\) represents the return set of the source node, \(R_{Y}\) represents the return set of attacker A, and \(R_{Z}\) represents the return set of attacker B. The return of the source node is measured by the information entropy, and the return of the attacker is measured by the information gain, indicating that the attacker's uncertainty about the location of the source node is reduced after obtaining a certain amount of information. The game cost can be set as \(COST=(C_{X},C_{Y},C_{Z})\). \(C_{X}\) represents the payoff set of the source node, \(C_{Y}\) represents the payoff set of attacker A, and \(C_{Z}\) represents the payoff set of attacker B.