Proportional–integral–derivative (PID) controllers remain among the most widely used control strategies in industrial applications due to their simple structure and reliable performance. However, optimal tuning of PID parameters, particularly for time-delay systems with conflicting performance requirements, remains a challenging problem. This study proposes an integrated game-theoretic framework for multi-objective optimal PID controller design. In the proposed approach, two conflicting performance criteria, namely settling time and maximum overshoot, are modeled as independent players in a non-cooperative game, and the Nash equilibrium solution is determined as a competitive outcome. Subsequently, this equilibrium point is employed as the disagreement point in a Nash bargaining process to obtain a compromise solution from the Pareto-optimal set. The PID controller parameters are considered as design variables, while frequency-domain stability constraints are incorporated into the optimization procedure. The effectiveness of the proposed framework is evaluated through comparative studies with conventional PID tuning methods. In the case study, the Nash bargaining solution achieves a settling time of 5.19 s and a maximum overshoot of 0.41%, representing improvements of approximately 34% and 78%, respectively, compared with the Nash equilibrium. Compared with the Ziegler–Nichols tuning, the settling time is reduced from 16.81 to 5.19 s, while the maximum overshoot decreases from 0.51% to 0.41%.