A social choice rule (SCR) is monotonic if raising a single alternative in voters’ preferences while leaving the rankings otherwise unchanged is never detrimental to the prospects for winning of the raised alternative. Monotonicity is rather weak but well-known to discriminate against scoring elimination rules, such as plurality with a run off and single transferable vote. We define the minimal monotonic extension of an SCR as its unique monotonic supercorrespondence that is minimal with respect to set inclusion. After showing the existence of the concept, we characterize, for every non-monotonic SCR, the alternatives that its minimal monotonic extension must contain. As minimal monotonic extensions can entail coarse SCRs, we address the possibility of refining them without violating monotonicity provided that this refinement does not diverge from the original SCR more than the divergence prescribed by the minimal monotonic extension itself. We call these refinements monotonic adjustments and identify conditions over SCRs that ensure unique monotonic adjustments that are minimal with respect to set inclusion. As an application of our general findings, we consider plurality with a runoff, characterize its minimal monotonic extension as well as its (unique) minimal monotonic adjustment. Interestingly, this adjustment is not coarser than plurality with a runoff itself, hence we suggest it as a monotonic substitute to plurality with a runoff.

如果在选民的偏好中提出一个替代方案，而在其他情况下保持排名不变，则社会选择规则（SCR）是单调的，这决不会损害所提出替代方案的获胜前景。单调性相当弱，但是众所周知，它会歧视计分消除规则，例如，有票数众多的票数和可转让单票的票数。我们将SCR的最小单调扩展定义为其唯一的单调超对应，这对于集合包含而言是最小的。在显示了该概念的存在之后，我们针对每个非单调SCR表征其最小单调扩展必须包含的替代方案。由于最小的单调扩展可能需要粗略的SCR，我们提出了在不违反单调性的情况下优化它们的可能性，只要这种改进与原始SCR的差异不超过最小单调扩展本身所规定的差异。我们称这些细化为单调调整，并确定SCR上的条件，以确保唯一的单调调整相对于集合包含而言是最小的。作为我们一般发现的一种应用，我们考虑带有径流的复数，表征其最小单调扩展及其（唯一）最小单调调整。有趣的是，这种调整并不比带有径流本身的复数粗，因此我们建议将其作为带径流的复数的单调替代。我们称这些细化为单调调整，并确定SCR上的条件，以确保唯一的单调调整相对于集合包含而言是最小的。作为我们一般发现的一种应用，我们考虑带有径流的复数，表征其最小单调扩展及其（唯一）最小单调调整。有趣的是，这种调整并不比带有径流本身的复数粗，因此我们建议将其作为带径流的复数的单调替代。我们称这些细化为单调调整，并确定SCR上的条件，以确保唯一的单调调整相对于集合包含而言是最小的。作为我们一般发现的一种应用，我们考虑带有径流的复数，表征其最小单调扩展及其（唯一）最小单调调整。有趣的是，这种调整并不比带有径流本身的复数粗，因此我们建议将其作为带径流的复数的单调替代。