Actually, the dog on your sofa or floor is not a wild dog or a wolf and has not been for somewhere between 10,000 and 30,000 years. That isn't to say that today's dogs are not pack animals- they are in fact, quite well suited to live in a family group.
I prefer the term family instead of pack, because when we start tossing around the word 'pack', somebody inevitably feels the need to talk about a 'pack leader'. Then they tend to jump to the need to "establish dominance" or "assert dominance". The problem with those terms is that the come from observations on captive wolves from the 1940's. There are a few fallacies here, so I'll briefly put it in human terms to highlight the important ones.
Think about how human behavior was understood in the 1940's, how well understood were human psychological issues and human development compared to today? (Take a look here) They were starting to turn away from old traditions that we would pretty much consider barbaric or torturous today and look into science- starting.
Now, if we didn't even understand our own brains, how could we assume to understand those of dogs, especially wolves who were in a situation very different from a family/pack in the wild? How could we assume to think that those wolves in captivity, living in what was quite different from a wild pack of wolves and were thousands of years different genetically from our own dogs, could be accurately compared to our pets at home? We were wrong. It's ok though, everyone makes mistakes, right?
The problem is that these beliefs are still haunting us and our dogs today. It's right under our noses at the vet clinic or the groomer who roughly handles dogs in an effort to be efficient, though they do have the dog's best interest at heart. We even see it in the hands of some dog trainers, who believe that coercion is the best way to train a dog.
It is true that wolves and dogs are social animals, just like people are social animals. It's true that there is a need for some balance within any social group of animals, it's just not as serious and intense as the 'alpha' folks would have you believe. The fact is that even in packs of wolves or in groups of animals put together in captivity (as long as they are grouped well), there shouldn't be frequent violent displays or pain inflicted regularly. In groups of wolves, just as many familial groups of animals, there is a general acceptance of those in the family and a general (at least initial) apprehension towards outsiders. Simply from an evolutionary standpoint, it makes sense to be suspect of someone new wandering into your pack- if they are a good fit personality-wise, it's still another mouth to feed. This could explain some of the more severe aggressive displays we observe at times, but there's more.
Within all social groups, there needs to be communication for everyone to get along. Good communication includes signals on what is ok and not ok, what the animal intends to do, and even random movements at times (after all, animals are not machines!) Good communication is clear, precise and effectively gets the message across, right?
Right, it gets the message across with minimal effort on the part of the animal(s) communicating. Now, this is where those who believe in alpha rolls and other threat displays are put in a funny place. As a pack animal, living in the wild, why would a wolf use any more energy than necessary in a social situation? They don't; it wouldn't make sense. Energy is conserved and used when necessary, not just to prove they are bog and strong. They use exactly what the situation dictates, so in the wild and even in domesticated dogs, you will see a lot more subtle forms of communication in the majority of situations. Using something dramatic is usually not necessary, except in an extreme circumstance; so using an alpha roll or a shock on a dog who simply did not know that he wasn't supposed to pull on leash is certainly an overreaction. Redirecting with a u-turn for pulling is a lot easier and less dramatic. Using an alpha roll on a dog who is just confused will only confuse it more and can lead to fear-based aggression.
Dogs and wolves do utilize a hierarchy in social settings, but it is more fluid than many humans tend to think; many aspects of behavior are situational and are not the only way differences are sorted out.
So many trainers out there use more forceful (or intentionally painful and life-threatening) techniques in a misguided effort to dissuade dogs from trying to establish 'dominance' over us. This is another one that baffles me. Again in human terms, how many of us out there really want to be a president or world leader, or even a mayor? I can't tell you exact numbers, but I guarantee that every person you see walking down the street wants to be in charge of every other person walking down the street. We are not all built that way. Dogs are the same, wolves are the same, all animals are the same- we don't all want to be in charge. Most of the time when I see a dog that another trainer describes as 'dominant', it's either an adolescent who simply does not have manners or a grasp on basic cues or a dog who is so terrified that it is reacting defensively to everything as if it's a threat. With both, the problem can be solved with simply teaching basic cues, impulse control and by the human(s) being consistent in their expectations and training. This can all be done without inflicting pain, by the way.
Since we are not dogs and we can't possibly perceive everything as a dog does, we should not go around tossing in phrases like 'dominance', 'pack leader', or 'alpha' because we only know them in human terms, not canine. This and our ignorance to much of the subtle body language of dogs and wolves put us in a dangerous place when we try treating our dogs like wolves. Worse, it puts our dogs in dangerous situations when they are treated like something they are not- they are not wolves and they are not humans; simply dogs. Let's learn more about them and treat them accordingly, doesn't the pup keeping your feet warm deserve that?
Aggression in Dogs; Practical Management, Prevention & Behavior Modification by Brenda Aloff Pages 30-33
https://vimeo.com/19472436 (NOVA documentary on canine understanding of people, compared to other animals; differences between dogs and wolves
http://www.anchorholdfilms.com/ (short Documentary on the fallacies of punishment training and "alpha" fallacies)
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3143860/ (NIH study on Clinical Psychology in the United States between 1940 and 2010)