There is no doubt about it. Hamm is a tyrant. He always needs to be in control, and he is very cruel to the people around him. Though he is blind and in a wheelchair, he reigns over the small room as if it were his kingdom. There is a suggestion that Hamm was, at one time, an actual monarch, but we are not sure whether or not we can take Hamm's word for it.
There's that moment when Hamm says to Clov, "And your rounds? When you inspected my paupers. Always on foot?" (1.72) A guy who has paupers is probably a king, right? From our perspective, though, we think Hamm could just as easily have been a landlord who thought of himself as a king and called his tenants "paupers." The point is that he thinks of himself as a king, not that he actually is one.
That said, Hamm does, somehow, maintain power over the entire stage. He keeps his parents in trash bins, and when they misbehave, he shouts to Clov, "Bottle them!" Indeed, it is his power over Clov that is given the most attention in the play. Hamm is completely dependent on his servant, and yet somehow, he gets Clov to obey him completely.
There is a hint as to how this authority over Clov came about in one of Hamm's stories: he speaks of a young man who came to him and begged for bread for his child. Hamm agreed to take the man in, but "in the end [the man] asked [Hamm] would [Hamm] consent to take in his child as well – if he were still alive. It was the moment [Hamm] was waiting for" (1.537). Though he never says it explicitly, there is a strong suggestion that Clov is this child, a child who Hamm has now provided with a life of indentured servitude.
How does Hamm maintain his power? Late in the play, Clov wonders why he always obeys Hamm. Hamm suggests, "Perhaps it's compassion. A kind of great compassion" (1.746). This is, in a way, a kind of political skill on Hamm's part. He does not tell Clov, "You obey me because you are weak and a coward." He tells him that he is a compassionate human being, and that is why he always listens to Hamm. Hamm allows Clov to feel good about himself while still remaining Hamm's servant.
That said, imagine Hamm as the last piece – the king – belonging to a player who is losing a game of chess. As he moves trivially from one square to another, he maintains great vanity about being king. Yet the end is inevitable. What does it mean to be a tyrant in a world where everyone else has given up? Not much.
Hamm has toys. One of his most valued possessions is his stuffed dog. When Clov brings it to him and sets it beside him, he asks if the dog is looking up at him, "As if he were asking me to take him for a walk?" (1.433) A moment later, he tells Clov to "leave [the dog] like that, standing there imploring me" (1.435). Hamm's vanity is ridiculous. The dog is stuffed! The way that he imagines his toys admiring him suggests that he is very much like a child.
Hamm is tremendously needy. He often calls for Clov when Clov is standing right next to him. At another point, Hamm asks Clov to kiss him, and when Clov won't kiss him on the lips he asks him to kiss him on the forehead. When Clov won't do that, he asks him to hold his hand. Hamm wants to be held and comforted. Like a playground bully, his vulnerability is obvious. He wants to reach out to those around him, but he doesn't understand how.
During one of his monologues, Hamm actually draws a direct comparison between himself and a child. He says, "Then babble, babble, words, like a solitary child who turns himself into children, two, three, so as to be together, and whisper together, in the dark" (1.688). Hamm always wants to play, but in general, he plays with words instead of with actual toys (the dog being the exception that proves the rule). Yet, the line also draws attention to just how alone Hamm is, which brings us to our last point…
Have you ever heard the word solipsism? The first time we heard it, we thought it was just a fancy word for something like selfishness. But what solipsism technically means is that you believe that nothing exists outside of your own mind. The world that passes before your eyes is nothing but a fantasy. You are the only thing that actually exists. Now, when we take a close look at Hamm's situation, we realize just how easy it is for Hamm's vanity to teeter over into this conviction.
Hamm is blind. He is also in a wheelchair. Hamm can do practically nothing for himself. He has only Nagg and Clov to converse with, and the two often only respond at their leisure. As for the earth, it is silent. The sea does not even have a tide for Hamm to listen to. Imagine what it is like to be inside Hamm's head. Having trouble? Consider what he tells Clov when Clov is acting unappreciative.
"One day you'll be blind, like me. You'll be sitting there, a speck in the void, in the dark, for ever, like me… Infinite emptiness will be all around you, all the resurrected dead of all the ages wouldn't fill it and there you'll be like a little bit of grit in the middle of the steppe." (1.379)
This is what it is like to be Hamm, to be "a speck in the void." It is difficult for him to maintain touch with the outside world under such conditions, and this may, in part, be why he maintains some iota of hope that the destruction of the world is not complete. His own sanity may be shaky enough that he's not certain that the world is as bad as it seems.
One more thing: Hamm and Clov go for a tour around the room, and when they get back, Hamm insists that he be right in the center of the room. He says, "I feel a little too far to the left.'' Then, "Now I feel a little too far to the right." Then, "I feel a little too far forward." And then, "Now I feel a little too far back" (1.281). You might be thinking, "Hamm has a big fat ego and he thinks that he is the center of the world." But here is another way to think of it: imagine how hard it is for Hamm to situate himself in relation to the real world. Tyrant that he is, this just might give us reason to feel compassion toward him.