An oft-recommended approach to handling a multi-dog household is to reinforce the ranking hierarchy -- to feed the "alpha" dog first, the middle-ranking dog second, the low-ranking dog last; to greet or pet the "top" dog first, and so on.
But behaviorists are moving away from that technique, and most now advise reinforcing appropriate behavior without taking hierarchy into account. As scientists explore more about dog cognition and social behavior, we're discovering that dog hierarchies are a lot more fluid and complex than the straight alpha - beta - omega designations once used for wolf packs might suggest.
In my experience, the average pet owner rarely understands dog behavior well enough to really know who the "top dog" is. Most think that the dog who fights and snarks at the others the most is the "top dog" -- they've never even heard of "alpha wannabe's" (dogs who are status seekers but are not confident enough to actually fulfill the role of "top dog"). So the wrong dog gets reinforced.
Dog hierarchies are also subject to change -- especially when other dogs enter the mix (whether that's adding a new dog to a household or simply having other dogs visiting).
I advise my clients that it doesn't matter who's "top dog" within the dog hierarchy, as long as they all defer to the humans in the house, and those humans have the ability to direct and control those dogs (whether that's through management or good training).
Need some guidelines for good management and training for multi-dog household situations? I recommend "Feeling Outnumbered?" by Karen London and Patricia McConnell.
In my own household, I have two dogs: Nico, a large neutered male mutt who flips between being completely calm and a couch potato to being a high-stress high-energy dog in a heartbeat (the switch back isn't quite so fast), and Brandy, a medium-sized spayed female mutt who thinks everything should be hers. No resource guarding against humans, but she does resource guard against other dogs -- which is fine with me as long as it's not overdone and the other dog respects the warnings to "stay away from my stuff." However, in the past Nico has deferred to Brandy, walking away from toys if she said she wanted them. That's emboldened her to escalate her demands from toys to other things like pigs' ears, but Nico's having none of that. So there have been some ferocious-sounding tussles (the latest over a pig's ear that had slid under a piece of furniture).
It's hard to tell which dog is "top dog" as each defers to the other in different circumstances. Brandy used to be the only one who cared about toys (and food, for that matter), so Nico would just walk away, letting her have what she wanted. But Nico's decided he likes toys and food, too, so their relationship is changing so far as "sharing" (i.e. Brandy getting everything) is concerned. Nico still leads when they're exploring (in the yard or on long walks in the woods), Brandy takes the lead in play a lot of the time. Nico takes the lead in "guarding" the house when people are involved; Brandy's the one who'll raise the alarm when there are other dogs on the property.
I don't think that trying to raise one over the other would work in my situation. So I use management (no more pigs' ears under furniture) and training (they both defer to me and know "off," "leave it," "go
lie down," and the usual sit, stay, settle, etc.) to keep peace in the house.