WAM-OPD: Joint Video-Action Supervision for World Action Model Post-Training with On-Policy Distillation

World Action Models (WAMs) generate both future video and robot actions, offering two connected outputs for post-training supervision. How can a pretrained WAM learn from a stronger Teacher on the histories it encounters during execution? We present WAM-OPD, which collects Student rollout histories and queries a Teacher for paired video and action targets. The Student learns from both targets while retaining its one-step video and action generation at deployment. Across 12 RoboTwin 2.0 tasks, WAM-OPD improves average success from 33.8% to 65.7%; across four real-robot tasks, it improves average success from 51.4% to 64.6%. With the collected Student histories held fixed, joint video-action supervision achieves the highest observed success on all three ablation tasks, while either modality alone also improves performance. A separate comparison with Teacher-generated histories finds task-dependent differences between the two history sources. These results demonstrate the value of paired video-action supervision for improving WAM policies without increasing their deployed sampling budget.