Two-bit fights because it is a good way to blow off steam, and when someone deserves it.
Two-bit is “the oldest of the gang and the wisecracker of the bunch,” and his nickname comes from the fact that he always has to tell his perspective. He liked to fight, and shoplift. He does not need or want most of what he steals, but he likes the thrill.
"A fair fight isn't rough," Two-Bit said. "Blades are rough. So are chains and heaters and pool sticks and rumbles. Skin fighting isn't rough. It blows off steam better than anything. There's nothing wrong with throwing a few punches. Socs are rough….” (Ch. 2, p. 30)
Two-bit does not like the Socs because he does not think they play fair. They gang up on greasers, and they bring weapons to fights. As far as Two-bit is concerned, there is nothing wrong with fair fighting. No one can get seriously hurt and it is good entertainment. Greasers sometimes get into conflicts with each other, but they still stick up for one another when needed.
Two-bit has a strong sense of justice. He says that it’s all right if Dally gets what’s coming to him for slashing a car’s tires, because it’s fair. He also slaps Pony for making a snide remark to Johnny.
When Pony gets upset with Johnny and tells him he isn’t wanted at home, Johnny is clearly hurt. This upsets Two-bit’s sense of justice, and he lets Pony know.
Two-Bit slapped me a good one across the side of the head, and hard. "Shut your mouth, kid. If you wasn't Soda's kid brother I'd beat the tar out of you. You know better than to talk to Johnny like that." He put his hand on Johnny's shoulder. (Ch. 3, p. 44)
Pony apologizes. Although this is not a real fight, Two-bit implies that it could easily have been one if Pony wasn’t special to him as Soda’s brother. Two-bit is not afraid to fight with other greasers if he feels that they deserve it.
In many ways, Two-bit is the conscience of the gang. He has a clear moral center, and although his sense of right and wrong is different than many of us are used to, he espouses and reinforces real values.
To calm down his mind, to relieve stress.