The likelihood of monotonicity paradoxes in run-off elections

A monotonicity paradox occurs when a voting system reacts in a perverse way to a change in individual opinions. The vulnerability of a voting system to monotonicity paradoxes is defined as the proportion of voting situations that can give rise to such paradoxes. In this paper we provide analytical representations of this vulnerability in the three-alternative case for two voting systems, i.e. plurality with run-off (f(1)) and anti-plurality with run-off (f(2)). Our results suggest that the vulnerability to monotonicity paradoxes is lower with f(1) than with f(2).