Obama is willing to put his picks in because everybody knows that almost no one will get every game right. Plus, if a team he picked loses, he can just say that they didn't play to their level or the team that upset them played over their head. Either way, he is absolved from all blame and is not wrong.
Obama is a narcissist. He is pathologically incapable of being wrong. That's why he doesn't want to produce a budget. When people point out, as they invariably will, that his premise of governance is wrong, his assumptions are wrong, and his conclusions are at odds with reality, there really isn't anyone else to blame. That's what he can't deal with.