# [EM] monotonicity, participation, and consistency

Richard Moore rmoore4 at home.com
Fri Jan 11 18:20:45 PST 2002

```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

```