Non-monotonic Voting Methods: An Overview

Voting is a necessary ingredient of democratic governance. Yet, there are numerous voting rules that differ from each other in processing the ballots into voting results. This is illustrated by examining a setting where five well-known rules each lead to a different election winner when operating on a fixed distribution of voter opinions. This observation has led to the study of voting procedures in terms of various desiderata. Our focus is on a specific set of desiderata that stem from the notion of monotonicity which characterizes those systems where any winning alternative remains a winner when its support is enlarged, ceteris paribus. This definition pertains to fixed electorates where the only possible changes involve improving the standing of the winning alternative with respect to some others barring all other types of changes. This concept should be kept distinct from Maskin monotonicity which maintains that whenever a winner improves its position with respect to some other alternatives and maintains its position vis-á-vis the remaining ones, it should remain the winner. These monotonicity concepts are defined for fixed electorates. There are other monotonicity-related properties definable in variable electorates. The best-known is undoubtedly the invulnerability to the no-show paradox. The paradox occurs whenever a group of voters obtains a better outcome by not voting at all than by voting according to its preferences, ceteris paribus. Of particular interest from the view-point of democratic governance are two extreme cases of the no-show paradox: the P-TOP and P-BOT ones. These will be focused upon in this book.