What job duties does a video game programmer have

a video game programmer would have the same duties as any programmer. Unless they are a lead or project manager, they would typically code from specs(specifications), which identify what functions a particular program should do. The testing of these functions is also an important part of their role.
the design and following specs to tell exactly what should be changed is in the hands of the lead analyst and business consultants or experts (sometimes called SMEs - Subject Matter Experts

Sometimes to get a VGP position you need industry experience. Several Test Video Games during their summers.