Monotonicity comments

[Forest Simmons]

See Martin Harper's treatment of monotonicity in dyadic ballots in his
EM posting dated 4 Apr 2001 at the URL:

http://groups.yahoo.com/group/election-methods-list/message/7165


Forest

On Tue, 8 Jan 2002, Richard Moore wrote:

> MIKE OSSIPOFF wrote:
> 
> > It seems to me that that gets my definition out of that problem, doesn't it?
> > After all,
> > just as in a ranking the status of being ranked #5 has nothing to do with
> > who the other
> > candidates are, or how they're ranked compared to eachother, so A's (4,10)
> > or 10/14 says
> > something about A's status in the same way.
> 
> Your solution to the dyadic ballots problem is on the right track,
> I think. However, it probably needs to be a little less simplistic.
> Not only the number of ">" marks above and below the candidate, but
> the way those marks are grouped, is important. Recall that Forest
> mentioned the dyadic ballot maps to a binary tree. The leaves of
> the tree can be assigned unique numbers, providing an order to the
> possible ways each candidate can be marked. There are more leaves
> than candidates, so moving a candidate to a different leaf will
> increase or decrease the candidate's rating but not necessarily
> change the relative order of the candidates.
> 
> > Yes, I see the problem now. It's something that hadn't occurred to me, and
> > that
> > I'll have to study. Monotonicity is more difficult that it seemed. It seemed
> > that it was
> > only necessary to make its obvious meaning explicit, but it turns out that
> > its meaning
> > is far from clear. Might Monotonicity have to be dropped, if no satisfactory
> > definition
> > is found?
> 
> I brought up the self-reference problem, not to discourage work
> on the definition of MC, but to stiumlate some "out of the box"
> thinking. This criterion is one of the most important ones (IMHO)
> and should not be dropped because it's hard to define.
> 
> The limited-scope definitions (that apply only to ranked ballots,
> or only to CR ballots, etc.) work well within their respective
> scopes. The difficulty is in writing one definition that has
> universal scope.
> 
> The mathematical (non-EM) definition of monotonicity is that
> a function is monotonic if it is order-preserving. That is,
> if, when x2 > x1, it is always true that f(x2) >= f(x1), then
> f is monotonic (specifically, it is monotonically increasing;
> it could be monotonically decreasing if f(x2) <= f(x1) for
> x2 > x1). This definition is easily understood if there is
> just one independent variable. But what if there are multiple
> independent variables? How do you decide if (x2, y2) > (x1, y1)?
> 
> One approach would be to define x and y (and any other
> independent variables) in terms of a single parameter (let's call it t).
> This constrains the evaluation of monotonicity to a path
> through the domain. Then, since x = x(t) and y = y(t), you could
> replace f(x, y) with g(t). We could then make a meaningful
> determination of monotonicity subject to those constraints.
> 
> The limited-scope definitions already set constraints. For CR ballots,
> for example, it is assumed that, when we vote X higher, we are
> giving X a higher rating while leaving the other candidates'
> ratings untouched or at least not increasing them as much as
> we increase X's rating. Having such constraints makes a useful
> definition possible. The trick is not to have too much constraint,
> or the resulting criterion could fail to detect some types of
> monotonicity failure that we care about.
> 
> Solving the dyadic ballot problem by mapping the series of
> ">" symbol groups above and below a candidate to a position on a
> binary tree, which position can be assigned a number, is another
> way of ordering the domain.
> 
> If the path taken by (x(t), y(t)) does not intersect itself anywhere,
> then we can also define t in terms of x and y, so the monotonicity
> definition for a two-dimensional mathematical function f would
> become: "f(x, y) is monotone (w/r/t the function t(x, y)) iff either
> 
> (1) t(x2, y2) > t(x1, y1) implies that f(x2, y2) >= f(x1, y1),
> or
> (2) t(x2, y2) > t(x1, y1) implies that f(x2, y2) <= f(x1, y1)."
> 
> We can think of t(x, y) as an ordering function for the domain.
> This can be generalized to more than two variables, naturally.
> 
> I suggested that the reference method we rely on as the ordering
> function should be one that passes the Consistency Criterion.
> Consistency and monotonicity are closely related, but consistency
> is much easier to define.
> 
> Here is the last definition I suggested to Forest (the one that
> unfortunately makes proofs difficult):
> 
> Method M is monotone iff there exists a method C such that:
> 
> (1) C is a consistent method;
> and
> (2) For every pair (S,S') of multisets of ballots, if M(S') != M(S), then
> there exists a multiset T for which at least one of the following is true:
> 
>  (a) M(S') != C(S+T) and M(S') = C(S'+T)
>  or
>  (b) M(S) = C(S+T) and M(S) != C(S'+T)
> 
> Putting condition 2a in English, S is the original set of ballots and S'
> is the new set of ballots, so M(S) is the original winner (Jones) and
> M(S') is the new winner (Smith). If S and S' are such that they elect
> two different people (Jones and Smith), then we need to find some set of
> ballots (T) such that if the old ballots (S) are combined with T, the
> winner under method C will not be Smith, but if the new ballots (S') are
> combined with with T then the winner under method C will be Smith. In
> other words, according to method C (our reference method), changing
> from S to S' favors Smith.
> 
> The alternate condition 2b is similar, with the meaning that, according to
> method C, changing from S to S' "disfavors" Jones (i.e., could prevent
> Jones from being elected when combined with the T ballots).
> 
> So under this definition, saying M is monotonic means that there exists
> a method C that passes the consistency criterion and such that, if
> changing the ballots (from S to S') causes the winner in method M to
> change (from Jones to Smith), then according to method C, the change
> from S to S' either favors Smith or disfavors Jones.  This is the same
> as saying that a change that neither favors Smith nor disfavors Jones
> (under method C) should never change the winner from Jones to Smith
> (under method M). It might, however, change the winner from Jones to
> Johnson (by favoring Johnson), or it might change the winner from
> Thompson to Smith (by disfavoring Thompson).
> 
> Note that C is not in any way specified; the definition just says that
> C must exist. Thus, even if methods M and N use the same type of ballots,
> the reference method that is used to determine if M is monotone might not
> be identical to the reference method that is used to determine if N is
> monotone. But note that, if C is used to evaluate whether changing S to
> S' favors Smith for method M, then the same C must be used to evaluate
> whether changing R to R' favors Smith for method M.
> 
> I'm not satisfied that this definition is correct (it could be too loose, or it
> could be too stringent), and as I noted it's not easy to apply, so any new
> insights will be welcome.
> 
>  -- Richard
> 
> 
>