One-way monotonicity as a form of strategy-proofness

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{P^prime_{v}} . A voting rule F{mathcal{F}} is two-way monotonic if such an effect is only possible when v moves t from below s (according to P _v to above s (according to P￠_v{P^prime_{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{mathcal{F}} , each manipulation is paired with a positive response, in which F{mathcal{F}} offers the pivotal voter a strictly better result when she votes sincerely.