[mannika]

No one in their right mind would open just Box B. The reliable predictor has already made his choice about what is in each box. As long as the boxes cannot contain negative money, why would you only be opening one box? It is completely ridiculous. However, if this predictor is that great, he knows that you are going to do this, because any rational person would, and therefore would only put nothing in Box B.

So, bottom line, predictor will place $1000 in box A, and $0 in box B, and you should choose both in order to get anything at all.

EDIT: If he/she is indeed a reliable predictor, I think this is a Nash equilibrium.