Most, if not all scientific research in the past fifteen years has debunked the idea of alpha/beta/omega ranking in dogs. There's a great book that covers recent canine research:
Amazon.com: Dog Sense: How the New Science of Dog...
One thing that "alpha-ness" doesn't address is that different resources may be more or less important to a given dog at any time. A non-confrontational dog may change his stripes when he's hungry. Or maybe one dog always gets human affection first, but doesn't care about toys particularly.
I have seen two contradicting arguments based on alpha theory on possessiveness. The first says that you should allow dogs to work it out among themselves, because dogs are happier when they know their place. The second says that you should not allow this kind of behavior, or even force the possessive dog into "last" place, because not intervening displaces you as the ultimate alpha.
Since there's no evidence for either side of the argument as far as I know, I vote for doing whatever you want. But of course, the wife is always right.
Up with four-kneed elephants!