This paper develops a multi-event-triggered (MET) control for multiplayer mixed zero-sum (MZS) differential games of linear systems. The game consists of two groups of controllers: a group of non-competitive players with distinct performance objectives, and a competitive group whose players jointly compete with a maximizing adversarial disturbance over a shared value function. Together, these two groups form the nonzero-sum (NZS) component of the game, while the competitive group and the disturbance constitute the zero-sum (ZS) interaction. The proposed approach guarantees exponential stability while enabling players to employ distinct triggering rules, leading to a decentralized MET control architecture with asynchronous control updates and reduced communication and computational burden. Leveraging Lyapunov stability analysis, a global triggering condition is first derived, where the functionals of group measurement errors are constrained by a global triggering budget. Subsequently, sufficient conditions to derive local triggering rules are established. A hierarchical budget allocation is further proposed to systematically distribute the global triggering threshold between groups and then among the players. Zeno behavior exclusion is established, and simulation results on a two-area interconnected power system load frequency control demonstrate the effectiveness of the proposed approach. Comparative studies with two centralized event-triggered methods show that the proposed approach achieves similar regulation performance while requiring fewer policy updates and enabling asynchronous triggering.