Fights don't start immediately; they evolve over a period from angry words into physical action. This evolution will usually take a few minutes, so each person has the opportunity stop the escalation by just leaving.
Fights are about dominance
Fights are about one person wanting to show that they are tougher, more powerful, or more dominant than the other person. Each person feels the need to prove his or her superiority over the other. If one person does not need to prove his or her superiority and walks away, then the fight does not occur.
Fighters do not care about the fighting ability of their opponents. The purpose in a fight is prove superiority, so even when fighters lose, they rationalize and see themselves as superior since they did not back down from a fight with a better fighter.
In a fight, the participants usually are not trying to seriously injure or kill the other person, they merely want to win the fight to show their superiority. However, fighting is dangerous and serious injury, permanent injury, or death is possible.