Help me choose a regression model?

#1
Hi everyone, I have the following problem in mind:

Suppose m items are displayed to a user and the user can buy any one of them. Each of the items has a feature set associated with it. I want to know which are the features that most influence the purchase of the item.

If m were equal to 1, I would have used logistic regression, associating a purchase with 1, and non-purchase with 0. But I have m > 1, so the output variable belongs to the set {0,1,...,m}, where 0 indicates non-purchase, and 1 through m indicate ID of the item purchased.

Is there something like logistic regression with multiple outputs? Will something like that even work in this case? Any other models that I might want to apply here?