I am trying to build a model that predicts if a user is going to purchase certain product. I also want to predict which users are new users (those who have not made purchase in x period of times). I think this can be seen as a multi-class classification problem but my concern is that one of the class is entailed by the other. Does multi-class classification algorithms work in this situation? And is there any research in similar problems?
What makes this problem even more interesting to me is that if I treat it as a binary classification, then I can potentially have the same user with similar feature in different days and force the model to predict positive in the first day and negative in the others.