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Öğe Monotonicity properties and their adaptation to irresolute social choice rules(Springer, 2012) Sanver, M. Remzi; Zwicker, William S.What is a monotonicity property? How should such a property be recast, so as to apply to voting rules that allow ties in the outcome? Our original interest was in the second question, as applied to six related properties for voting rules: monotonicity, participation, one-way monotonicity, half-way monotonicity, Maskin monotonicity, and strategy-proofness. This question has been considered for some of these properties: by Peleg and BarberA for monotonicity, by Moulin and P,rez et al, for participation, and by many authors for strategy-proofness. Our approach, however, is comparative; we examine the behavior of all six properties, under three general methods for handling ties: applying a set extension principle (in particular, Gardenfors' sure-thing principle), using a tie-breaking agenda to break ties, and rephrasing properties via the t-a-t approach, so that only two alternatives are considered at a time. In attempting to explain the patterns of similarities and differences we discovered, we found ourselves obliged to confront the issue of what it is, exactly, that identifies these properties as a class. We propose a distinction between two such classes: the tame monotonicity properties (which include participation, half-way monotonicity, and strategy proofness) and the strictly broader class of normal monotonicity properties (which include monotonicity and one-way monotonicity, but not Maskin monotonicity). We explain why the tie-breaking agenda, t-a-t, and Gardenfors methods are equivalent for tame monotonicities, and how, for properties that are normal but not tame, set-extension methods can fail to be equivalent to the other two (and may fail to make sense at all).Öğe One-way monotonicity as a form of strategy-proofness(Springer Heidelberg, 2009) Sanver, M. Remzi; Zwicker, William S.Suppose that a vote consists of a linear ranking of alternatives, and that in a certain profile some single pivotal voter v is able to change the outcome of an election from s alone to t alone, by changing her vote from P-v to P'(v). A voting rule F is two-way monotonic if such an effect is only possible when v moves to from below s (according to P-v) to above s (according to P'(v)). One-way monotonicity is the strictly weaker requirement forbidding this effect when v makes the opposite switch, by moving s from below t to above t. Two-way monotonicity is very strong-equivalent over any domain to strategy proofness. One-way monotonicity holds for all sensible voting rules, a broad class including the scoring rules, but no Condorcet extension for four or more alternatives is one-way monotonic. These monotonicities have interpretations in terms of strategy-proofness. For a one-way monotonic rule F, each manipulation is paired with a positive response, in which F offers the pivotal voter a strictly better result when she votes sincerely.