Interacting double monotonicity failure with direction of impact under five voting methods

• “Monotonicity failure” is regarded as a very serious failing in a voting rule.
• The P-R, AV, Coombs’, Dodgson’s and Nanson’s rules exhibit monotonicity failure.
• N. Miller showed that the P-R and AV rules exhibit “Double Monotonicity Failure”.
• We expand Miller’s finding to all five non-monotonic voting rules.
• We exemplify several sub-types of double monotonicity failure for each voting rule.

Monotonicity failure is widely considered a severe pathology in a voting method, and some authors regard a voting method that suffers from this pathology to be totally unacceptable. Of the various voting methods discussed prominently in the literature, five methods are subject to monotonicity failures: Alternative Vote (AV), Plurality Runoff (P-R), and Dodgson’s, Nanson’s, and Coombs’ methods. Two of these methods (AV and P-R) are used in practice. In the 2nd World Congress of the Public Choice Societies conducted in March 2012 in Miami, Fl., Nicholas Miller presented a paper in which he defined, inter alia, a novel type of monotonicity failure which he called ‘double monotonicity failure’ and investigated some of its properties in three-candidate elections under the AV and P-R methods. The present paper extends Miller’s investigation by showing that all the aforementioned voting methods may display double monotonicity failure, as well as by interacting double monotonicity failure with the question of whether the monotonicity failures improve or worsen the apparent welfare of the voters whose votes change. We determine which of the four resulting cases can arise in each of the aforementioned five voting methods.