In　this　article,　by　exploiting　the　event-triggered　neural　critic　control　scheme,　the　optimal　control　question　of　nonlinear　continuous-time　multiplayer　nonzero-sum　games　with　asymmetric　control　constraints　is　investigated.　First,　to　satisfy　the　asymmetric　constraints,　we　propose　an　appropriate　nonquadratic　function.　Also,　the　time-based　coupled　Hamilton-Jacobi　(HJ)　equations　and　the　optimal　control　policies　are　deduced.　To　save　communication　resources　and　improve　control　efficiency,　the　event-driven　mechanism　is　innovated.　Further,　the　event-triggered　optimal　control　strategies　and　the　event-triggered　coupled　HJ　equations　are　given.　Afterwards,　a　novel　triggering　condition　is　established,　and　the　critic　neural　networks　are　built　to　evaluate　the　optimal　cost　functions,　thereby　obtaining　the　event-triggered　near-optimal　controls.　During　neural　critic　learning,　a　new　weight　adjustment　rule　is　proposed.　Moreover,　by　exploiting　the　Lyapunov　approach,　the　stability　of　the　system　state　and　the　weight　estimation　errors　is　guaranteed.　Finally,　two　examples　are　given　to　verify　the　efficacy　of　the　established　event-triggered　neural　critic　control　scheme.