Abstract
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.
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Notes
- 1.
Miller (2012) proved that double monotonicity failure in three-candidate elections under the Alternative Vote (and the Plurality with Runoff) procedure can occur only when a top cycle exists. Thus under the Plurality with Runoff (and the Alternative Vote) procedure a necessary condition for double monotonicity failure to occur when a Condorcet winner exists initially is that there be at least four candidates.
- 2.
At least 16 voters of group 2 must change their ranking from a \( \succ \) d \( \succ \) b… to b \( \succ \) a \( \succ \) d…, i.e., a total of 32 rank inversions.
- 3.
At least 19 voters of group 5 must change their ranking from b \( \succ \) e \( \succ \) c… to c \( \succ \) b \( \succ \) e…, i.e., a total of 38 rank inversions.
- 4.
To minimize the total number of rank inversions at least the four voters of group #5 must change their ranking from … c \( \succ \) h \( \succ \) a … to … a \( \succ \) c \( \succ \) h…, (8 rank inversions); the 10 voters of group #7 must change their ranking from c \( \succ \) b \( \succ \) a… to a \( \succ \) c \( \succ \) b… (20 rank inversions); and one of the 38 voters in group #6 must change his or her ranking from …c \( \succ \) h \( \succ \) j \( \succ \) a …to …a \( \succ \) c \( \succ \) h \( \succ \) j… (3 inversions), i.e., a total of 31 rank inversions (8 + 20 + 3).
- 5.
To minimize the total number of rank inversions at least the eight voters of group #1 must change their ranking from a \( \succ \) b… to b \( \succ \) a (8 rank inversions), and 16 of the 43 voters in group #3 must change their ranking from a \( \succ \) d \( \succ \) b… to b \( \succ \) a \( \succ \) d… (32 rank inversions), i.e., a total of 40 rank inversions (8 + 32).
- 6.
To minimize the total number of rank inversions the eight voters of group #1 must change their ranking from a \( \succ \) b \( \succ \) c… to a \( \succ \) c \( \succ \) b… (8 rank inversions), and 11 of the 38 voters in group #6 must change their ranking from b \( \succ \) e \( \succ \) c… to c \( \succ \) b \( \succ \) e (22 rank inversions), i.e., a total of 30 rank inversions (8 + 22).
- 7.
To minimize the total number of rank inversions the two voters of group #4 must change their ranking from …c \( \succ \) h \( \succ \) a …to … a \( \succ \) c \( \succ \) h (four inversions), the 10 voters of group #6 must change their ranking from c \( \succ \) b \( \succ \) a… to a \( \succ \) c \( \succ \) b… (20 inversions), and the three of the 38 voters of group #5 must change their ranking from … c \( \succ \) h \( \succ \) j \( \succ \) a… to … a \( \succ \) c \( \succ \) h \( \succ \) j…(nine inversions), i.e., a total 33 rank inversions (4 + 20 + 9).
- 8.
To minimize the total number of rank inversions at least 16 of the 43 voters in group #2 must change their ranking from a \( \succ \) d \( \succ \) b… to b \( \succ \) a \( \succ \) d… , i.e., a total of 32 rank inversions.
- 9.
To minimize the number of rank inversions at least 17 of the 43 voters in group #2 must change their ranking from … b \( \succ \) g \( \succ \) c … to … c \( \succ \) b \( \succ \) g…, i.e., a total of 34 rank inversions.
- 10.
Note that the sum of all the voters’ Borda scores can also be obtained by multiplying the number of voters (16 in this example) by the number of paired comparisons among all the candidates (6 · 5/2 = 15 paired comparisons in this example).
- 11.
For example, in order for b to become a Condorcet winner a total of 5 preference inversions are needed: 3 inversions from a ≻ b to b ≻ a and another 2 inversions from d ≻ b to b ≻ d.
- 12.
References
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Felsenthal, D.S., Nurmi, H. (2017). Five Voting Rules Susceptible to Types of Monotonicity Failure Under Both Fixed and Variable Electorates. In: Monotonicity Failures Afflicting Procedures for Electing a Single Candidate. SpringerBriefs in Economics. Springer, Cham. https://doi.org/10.1007/978-3-319-51061-3_4
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