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The broken Borda rule and other refinements of approval ranking

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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.

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  1. List works in Sen’s (1970) social welfare functionals framework. He proposes the so-called ONC+0 axiom, which requires that the social ordering be invariant under increasing transformations of the individual utility functions that preserve the sets of alternatives with positive, negative, and zero utility respectively. This means that a social welfare functional is allowed to use precisely the information encoded in Brams and Sanver’s preference-approval model.

  2. Axiomatizations of the Borda count (or the Borda choice rule) were proposed by Young (1974), Nitzan and Rubinstein (1981), Mihara (2017), Sato (2017), Maskin (2020), Heckelman and Ragan (2021).

  3. This axiom is now incompatible with the Pareto criterion \([aP_{i}b\) for all \(i\in N]\Rightarrow \left[ a{\mathbf {P}}(P,C)b\right] .\) In the current framework, however, the Pareto criterion is not compelling because the voters’ views about the acceptability of the alternatives may conflict with their preferences.


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Correspondence to Yves Sprumont.

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We thank S. Horan, J.-F. Laslier, R. Sanver, and two anonymous referees for very helpful feedback.

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Barokas, G., Sprumont, Y. The broken Borda rule and other refinements of approval ranking. Soc Choice Welf 58, 187–199 (2022).

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