One-way monotonicity as a form of strategy-proofness
International Journal of Game Theory
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 σ to τ. A voting rule F is two-way monotonic if such an effect is only possible when v moves t from below s (according to σ) to above s (according to τ). One-way monotonicity is the strictly weaker requirement that such an effect never occur when v makes the opposite switch, by
... oving s from below t to above t. Two-way monotonicity is a very strong property, equivalent, over any domain, to strategy proofness. It thus cannot be satisfied by any "reasonable" resolute voting rule over the full domain. One-way monotonicity fails for every Condorcet extension; in this respect, and in others, it resembles Moulin's participation property, although the two properties are independent. One-way monotonicity holds for all sensible voting rules -those for which the election outcome is determined by the numerical value of a function called a sensible virtue. Total score, as determined by any scoring rule, is a sensible virtue, but the class of sensible rules is larger than that of scoring rules. The names for these monotonicities arise from their interpretations in terms of manipulability. We may think of either σ or τ as representing v's sincere preferences. For a two-way monotonic function, neither of these interpretations ever yields a successful manipulation. For a one-way monotonic rule F, whenever one of the interpretations yields a successful manipulation, the other yields a positive response, in which F offers v a strictly better result when she votes sincerely. For such a rule F, each manipulation can thus be seen as part of the cost to be paid for appropriate responsiveness to the sincere will of the electorate.