Order dependence vs. independence

You may recall the commutative property of math-
ematics, which essentially speaks of changing the order in which we do the math. This property holds for addition and multiplication; for example:

You'll may also remember that it doesn't hold for all mathematical operations; for example:

Technically, there ARE times when these rules hold (e.g., when A and B are the same value), but these are exceptions.

I'm reviewing a client's performance attribution process, which includes their reporting of contribution (a.k.a. "absolute attribution"), and discovered that their method to link contribution effects over time is "order dependent." What I mean by this is that if we link January's contribution to February's, we get different answers than if we link February's to January's. 

This caused me to recall a rather lengthy series of discussions that occurred about 10 years ago, when we saw several attribution linking methods introduced. I have confirmed with a few folks that the concept of order independence is an important one; I agree.

What is particularly troubling to me is that this contribution linking method is WIDELY used within our industry. But just because everyone else does something doesn't make it right. 

I am going to pursue this matter further, and will offer further comments both here as well as in our newsletters. Any thoughts you have will be welcome.