Beyond neutrality: Extended difference of votes rules

In a voting situation where there are two alternatives, simple majority rule outputs the alternative with the most votes or outputs a tie if both alternatives receive the same number of votes. For any nonnegative integer k, the difference of votes rule Mk outputs the alternative that beats the competing alternative by more than k votes. If the two alternatives are not necessarily treated equally, then we get the class of Mk,l rules where the integers k and l are the thresholds for when one alternative beats the other. Llamazares (2006) characterized the class of Mk rules with the conditions of anonymity, neutrality, monotonicity, weak Pareto and cancellation. We extend Llamazares' Theorem by proving that the Mk,l rules are the only voting rules satisfying anonymity, monotonicity, and cancellation. In addition, we describe the class of voting rules that satisfy only monotonicity and cancellation.