[EM] Basic Monotony

[Richard Moore]

Joe Weinstein wrote:

> SIDE QUIBBLE  (before we get down to business!).  In our EM-list 
> discussions we don't really mean 'monotone'!  That's because there are 
> two ways a process (or, mathematically, a function) can be monotone: in 
> the 'same' direction ('increasing': more input yields more output) and 
> in the 'opposite' direction ('decreasing': more input yields less 
> output).   The respective terms are 'isotone' and 'antitone' (or, if you 
> like, 'isotonic' and 'antitonic').   EM-list 'monotone' really means 
> 'isotone'!  For  familiarity, I too will use 'monotone' rather than 
> 'isotone'.


I have a quibble with your quibble. While it's true, mathematically,
that "isotone" and "antitone" are distinct cases of "monotone", I don't
think the distinction is needed in election method theory. The reason
is, we are talking about whether the truth value of "X is a winner"
never decreases when support for X on the ballots increases. If we had
a method that was antitone, i.e., "X is a winner" never increases
when support for X increases, then we simply reverse the sense of the
ballots and get an equivalent isotone method. Just as when a poll asks
you to rate various candidates on a scale of 1 to 10. Some polls may
define "1" as the best rating and "10" as the worst, and others may do
just the opposite. But that doesn't make one poll isotone and the other
antitone, with respect to your support for that candidate -- at least,
not as long as you know which type of poll you're participating in.
(True, one method is antitone with respect to the numerical rating you
assign, but that numerical rating is not what I refer to when I say
"your support for that candidate").

So, in the context of election methods, it is possible to define
monotonicity as if we were defining isotonicity, without any loss of
generality.

> STRONGER-THAN-BASIC MONOTONY?   Basic-monotone methods include Lone-mark 
> plurality and other Cumulative methods, Approval and other unconstrained 
> Cardinal Ratings methods; and Borda.  It turns out, however, that many 
> such methods with MAXGRADE>1 - and almost all usual ones with MAXGRADE>3 
> - FAIL to be monotone in the simplest stronger-than-basic sense!
> 
> Suppose we have a simplest imaginable NON-basic swap.  The swap changes 
> just one marked ballot - ballot B1 is changed to ballot B2 - and 
> moreover there are two candidates (rather than just at most one) X and Y 
> whose grades are both made higher, or are both made lower.  Almost every 
> usual method M with MAXGRADE > 3 admits such an example which violates 
> monotony.


My example of this is CR where one candidate's grade is increased by 5
points, another's is increased by 10 points, and everyone else's stays
the same. This swap can change the first candidate from a loser to a
winner in some circumstances, and it can change that candidate from
a winner to a loser in other circumstances. So CR is not monotone with
respect to this swap. Yet, I don't think that that is a strike against
CR, so this particulary monotonicity test is not of any interest with
respect to CR.

On the other hand, I question whether the basic swaps are sufficient
for truly meaningful tests of monotonicity for CR-based methods. For
instance, in straight (additive or linear) CR, if we change one ballot by
decreasing the ratings of everyone but A, we know this swap will never
cause A to lose. But if, in some non-linear variation of CR, this swap
could cause A to lose, then I would call that a serious monotonicity
violation. Maybe the basic swaps are sufficient to catch all such
variations of CR (by combining a series of swaps, perhaps), but I'm not
sure we can fully test for all serious monotonicity failures of all
methods that use CR ballots in this way. Then again, I could be wrong.


 -- Richard


PS -- You could model the swap I described in the last paragraph as
a combination of basic swaps, each one decreasing the rating of some
candidate other than A and leaving A alone. However, what about the
case where A's rating is decreased by 5 points, and every other
candidate's rating is decreased by 10 points? There is no basic
swap that decreases A's rating *and* favors A, so I don't see how
you can model this change with basic swaps. So I'm still pretty
convinced that the basic swaps are insufficient for CR-based methods.