In this paper, we propose an interactive version of the Borda method for collective decision-making (social choice) when the alternatives are described with respect to multiple attributes and the individual preferences are unknown. More precisely, assuming that individual preferences are representable by linear multi-attribute utility functions, we propose an incremental elicitation method aiming to determine the Borda winner while minimizing the communication effort with the agents. This approach follows the recent work of Lu and Boutilier [8] relying on the minimax regret as a criterion for dealing with uncertainty in the preferences. We show that, when preferences are expressed on a multi-attribute domain and are additively separable over attributes, regret-based incremental elic-itation methods can be made more efficient to determine or approximate the Borda winner. Our approach relies on the representation of incomplete preferences using convex polyhedra of possible utilities and is based on linear programming both for minimizing regrets and selecting informative preference queries. It enables to incrementally collect preference judgements from the agents until the Borda winner can be identified. Moreover, we provide an incremental technique for eliciting a collective ranking instead of a single winner.