Five Voting Rules Susceptible to Types of Monotonicity Failure Under Both Fixed and Variable Electorates

In voting theory, monotonicity is the axiom that an improvement in the ranking of a candidate by voters cannot cause a candidate who would otherwise win to lose. The participation axiom states that the sincere report of a voter’s preferences cannot cause an outcome that the voter regards as less attractive than the one that would result from the voter’s non-participation. This chapter identifies three binary distinctions in the type of circumstances in which failures of monotonicity or participation can occur under five voting procedures (Plurality with Runoff, Alternative Vote, Dodgson’s, Nanson’s, and Coombs’ methods) either when the electorate is of fixed or of variable size. The distinction that is unique to monotonicity is whether the voters whose changed rankings demonstrate non-monotonicity are better or worse off. The distinction that is unique to participation is whether the marginally participating voter causes his first choice to lose or his last choice to win. The overlapping distinction is whether the profile of voters’ ranking has a Condorcet winner or a cycle at the top. The chapter traces the occurrence of all the resulting combinations of characteristics in the voting methods that can exhibit failures of monotonicity.