The broken Borda rule and other refinements of approval ranking

We study the social aggregation problem in the preference-approval model of Brams and Sanver (The mathematics of preference, choice and order: essays in honor of Peter C. Fishburn. Springer, Berlin, 2009). Each voter reports a linear ordering of the alternatives and an acceptability threshold. A rule transforms every profile of such “opinions” into a social ordering. The approval rule ranks the alternatives according to the number of voters who find them acceptable. The broken Borda rule ranks them according to the total score they receive; the scores assigned by a voter follow the standard Borda scale except that a large break is introduced between the score of her worst acceptable alternative and the score of her best unacceptable alternative. We offer an axiomatization of this rule and other lexicographic combinations of the approval rule and a fixed social welfare function.