Prediction rankings

I am trying to predict rankings for these players for any season ahead.
For example, how could I predict the ranking for player 1 for season #34?
ur=unranked for that season
You can estimate a random utility probit model, assuming you have observed

1) players' performance,
2) various characteristics of the players (age, experience, success rate, publicity, compensation, etc)

over many seasons. 17 seasons is too little for any intelligent inference. Basing one's inference on 17 observations only is essentially guessing, glorified with formulas.