Tyson has a need to submit. He always puts himself in positions that put control in another people’s hands: the army, Sebastian, the Jims. He isn’t a leader, he needed to have someone else in charge to have lasting order in his life. He needs that structure to feel safe. Everything in Tyson’s life if built around routines and patterns. He has made himself a world where he is protected by routines all the time but without someone, something, else in charge Tyson feels helpless. He’s needs the structure that having someone else in charge provides.
His routines, patterns, twitches, etc. only go so far. They work well enough to keep him sane and safe, but not happy. Without someone else in charge, without his owners, he becomes overwhelmed. They give him something else to focus on, they give him a purpose. Having the two Moriarty’s in charge takes away that stress, they make him happy and he will be loyal to them because of it. He doesn’t understand the reasons behind it but he knows he needs it. This is why he’s free to stop being their pet at any time, they can offer him that because he’ll never take that option. He is and will always be their Pup.