This paper investigates the containment control problem of signed networks with competitive leaders for discrete-time multi-agent systems under the leader-follower framework. Firstly, a non-cooperative game model is formulated, where two groups of competitive leaders are regarded as two players in the game. Each player aims to adjust its strategy to attract followers, thereby minimizing the distance between the followers' final states and its own. Secondly, the properties of the game are analyzed, and a strategic approach is proposed for players to optimize their interests where leaders can strategically select followers and add positive or negative edges to steer followers' final states closer to their own. Furthermore, for structurally balanced networks, the relationship between players' payoffs and the number of followers and connecting edges within two cells is revealed. Specifically, for the directed tree graph, a graph-theoretic method is proposed to determine whether a player's strategy constitutes a Nash equilibrium solution. Finally, numerical simulation examples are provided to illustrate the effectiveness of the theoretical results.