Consider what happens to the results of an election using a ranked method if we add ballots marked W>X>Y>Z. If the original winner is Z, then the following table shows some possible progressions of winners as the number of added ballots increases, and whether those progressions are allowed under the Participation, Monotonicity (traditional definition for ranked ballots), and Consistency criteria.
Progression Participation Monotonicity Consistency ------------------------------------------------------------- Z, Y, X, W Y Y Y Z, X, Y, W N ? Y Z, Y, X, W, X N ? N The question mark under the Monotonicity column indicates that, for these cases, the answer depends on whether ballot addition can be modeled as ballot substitution (e.g., adding 2*N ballots of type B is the same as removing N ballots of type A and adding N ballots of type B, if A and B exactly cancel each other out in the election method in question; in the case of ranked ballots the obvious choice for A is Z>Y>X>W). If ballot addition can't, be modeled as substitution, then monotonicity says nothing about the effect of added ballots, so the ? becomes a Y; if it can, then adding N2 ballots cannot reverse an outcome caused by adding N1 ballots (for N2 > N1), and so the ? becomes an N. In the Consistency column, note that, once the winner has progressed to W, adding more W>X>Y>Z ballots cannot reverse this result. However, any reversals before W becomes the winner are not Consistency violations. So Consistency isn't really as strong as I had assumed, and can be weaker than Monotonicity for some classes of methods (those that allow addition to be modeled as substitution). Therefore, it seems like Mike's suggestion, using Participation rather than Consistency as the reference method criterion in defining Monotonicity, is probably a good one. BTW, does anybody know of any methods that meet one or two of these criteria, but not all three? -- Richard
