A group may appreciate recommendations on items that fit their joint preferences. When the members' actual preferences are unknown, a recommendation can be made with the aid of collaborative filtering methods. We offer to narrow down the recommended list of items by eliciting the users' actual preferences. Our final goal is to output top-N preferred items to the group out of the top-N recommendations provided by the recommender system (K < N), where one of the items is a necessary winner. We propose an iterative preference elicitation method, where users are required to provide item ratings per request. We suggest a heuristic that attempts to minimize the preference elicitation effort under two aggregation strategies. We evaluate our methods on real-world Netflix data as well as on simulated data which allows us to study different cases. We show that preference elicitation effort can be cut in up to 90% while preserving the most preferred items in the narrowed list.