This paper focuses on a group pursuit game problem involving a pursuer P and multiple evaders Ei(i = 1,..., 4). In a model highly disadvanta¬geous to evaders, an additional velocity a + is introduced for pursuer. The pursuer possesses a velocity a- for capturing evaders moving in prescribed directions and a velocity a+ for capturing evaders deviating from the pre¬scribed direction. However, a+ can only be utilized once throughout the entire game.
The game model is described in the form of differential equations, with strategies and payoff functions defined for both the pursuer and evaders. We assume the pursuer employs a discriminatory strategy, while the evaders’ movement directions are highly disadvantageous to them¬selves. In the nonzero-sum game, Nash equilibrium of the game is found, and conditions for the effectiveness of the pursuers’ punishment strategy are proved, as presented in the paper. In the matrix game, strategies of the evaders under rational and irrational behavior are separately studied. Through simulation, payoff matrices for evaders are obtained under scenar¬ios of both equal and unequal velocity. It is revealed that for the evader group, even under irrational behavior, better payoffs can be achieved than under rational behavior. This situation may exist but might not necessarily materialize, as it also depends on the strategy of the pursuer.
In the master’s thesis, we delve into the pursuit game problem between one pursuer and four evaders in game theory, exploring variations in strategies within nonzero-sum games and matrix games. Introducing the pursuer a+, with a velocity that exacerbates the survival conditions for evaders in a game model significantly unfavorable to them.
In nonzero-sum game, we establish the Nash equilibrium, in which the pursuer chooses a target to pursue, prompting the other evaders to move in prescribed directions. If any evader deviates from the prescribed direction, the pursuer adjusts the pursuit order to capture the deviant evader first as a form of punishment. We derive geometric formulas for calculating the coordinates of capture points and evaders at different time instant, and identify effective punishment strategy conditions for the pursuer.
In matrix game, we analyze the strategy variations under rational and irrational behavior of evaders. Keeping a+and a unchanged respec¬tively while increasing the other velocity for experiments, interestingly, we observe that despite the higher velocity of the newly introduced velocity a+, the overall survival time of the evaders remains nearly unchanged, un¬like the scenario when studying the survival time of individual evaders. Additionally, Under irrational behavior, there exists at least one strategy whereby the evaders’ survival time surpasses that achieved by adhering to the prescribed direction of movement.
Through the lens of game theory, we aim to gain deeper insights into strategic behaviors and the dynamics of complex systems. Through theo¬retical modeling and empirical analysis, i hope to contribute to a broader understanding of strategic interactions and their impacts on reality.
